(A) the annual rate of change between 1992 and 2006 was 0.0804
(B) r = 0.0804 * 100% = 8.04%
(C) value in 2009 = $11,650
What is the rate of change?
The rate of change is a mathematical concept that measures how much one quantity changes with respect to a change in another quantity. It is the ratio of the change in the output value of a function to the change in the input value of the function. It describes how fast or slow a variable is changing over time or distance.
A) The initial value is $44,000 and the final value is $15,000. The time elapsed is 2006 - 1992 = 14 years.
Using the formula for an annual rate of change (r):
final value = initial value * [tex](1 - r)^t[/tex]
where t is the number of years and r is the annual rate of change expressed as a decimal.
Substituting the given values, we get:
$15,000 = $44,000 * (1 - r)¹⁴
Solving for r, we get:
r = 0.0804
So, the annual rate of change between 1992 and 2006 was 0.0804 or approximately 0.0804.
B) To express the rate of change in percentage form, we need to multiply by 100 and add a percent sign:
r = 0.0804 * 100% = 8.04%
C) Assuming the car value continues to drop by the same percentage, we can use the same formula as before to find the value in the year 2009. The time elapsed from 2006 to 2009 is 3 years.
Substituting the known values, we get:
value in 2009 = $15,000 * (1 - 0.0804)³
value in 2009 = $11,628.40
Rounding to the nearest $50, we get:
value in 2009 = $11,650
Hence, (A) the annual rate of change between 1992 and 2006 was 0.0804
(B) r = 0.0804 * 100% = 8.04%
(C) value in 2009 = $11,650
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Find the value of x.
Answer:
x = 4,5
Step-by-step explanation:
These two triangles are similar (according to 3 sides):
The three sides of one triangle are proportional to the three sides of the other triangle:
[tex] \frac{7.5}{5} = \frac{6}{4} = \frac{x}{3} [/tex]
[tex] \frac{6}{4} = \frac{x}{3} [/tex]
Use the property of proportion to find x (cross-multiply):
[tex]4x = 6 \times 3[/tex]
[tex]4x = 18[/tex]
Divide both sides by 4 to make x the subject:
[tex]x = 4.5[/tex]
Choose the best answer to the following question. Explain your reasoning with one or more complete sentences.
Which of the following is an example of exponential decay?
A. The population of a rural community decreasing by 100 people per year is an example of exponential decay because population growth and decay are represented exponentially, and in this case it is decreasing.
B. Government support for education decreasing by 1% per year is an example of exponential decay because the support decreases by the same percentage each year.
C. The price of gasoline decreasing by $0.02 per week is an example of exponential decay because the price decreases very slowly at first.
D. The price of gasoline decreasing by $0.02 per week is an example of exponential decay because the amount at which the price is decreasing increases by $0.02 each week.
E. The population of a rural community decreasing by 100 people per year is an example of exponential decay because the population decreases at a steady rate.
F. Government support for education decreasing by 1% per year is an example of exponential decay because the amount of support is decreasing over time.
B. Government support for education decreasing by 1% per year is an example of exponential decay because the support decreases by the same percentage each year.
What is exponential decay?Exponential decay is a mathematical concept that describes the decrease of a quantity over time at a rate proportional to its current value. In other words, the amount by which the quantity decreases over a certain period of time is proportional to the current value of the quantity. This is often represented by an exponential function with a negative exponent, where the exponent is a function of time. The concept of exponential decay is used in many fields, including physics, chemistry, biology, and finance, among others.
Reasoning: It is because the support decreases by the same percentage each year. Exponential decay refers to a process in which a quantity decreases over time at a rate proportional to its current value. In this case, the government support is decreasing by a constant percentage each year, which is a characteristic of exponential decay.
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suppose that f( n ) = f(n/5) + 3n^2 when n is a positive integer divisible by 5, and f(1) = 4. Find f(125)
According to the given information, f(125) is equal to 48479.
What is function?
A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.
To find f(125), we need to apply the recursive formula for f(n) when n is a positive integer divisible by 5. Since 125 is divisible by 5, we can use the formula:
f(125) = f(125/5) + 3(125)²
f(125/5) = f(25) = f(25/5) + 3(25)²
f(25/5) = f(5) = f(5/5) + 3(5)²
f(5/5) = f(1) = 4
Now we can substitute the values back into the formula and simplify:
f(5) = f(5/5) + 3(5)² = 4 + 75 = 79
f(25) = f(25/5) + 3(25)² = 79 + 3(625) = 2054
f(125) = f(125/5) + 3(125)² = 2054 + 3(15625) = 48479
Therefore, f(125) is equal to 48479.
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WILL GIVE TRUE 100 POINTS AND BRAINLYEST FOR THE CORRECT ANSWER
Answer:
y intercept = +4
slope = +2
domain = (-2,0)
range = (4,0)
Graph is increasing
x intercept = -2
Step-by-step explanation:
Line hits y axis at point 4
Slope is calculated by rise over run (2/1)
Domain is x,y listed
Range is y,x listed
Line is going up
Line hits x axis at -2
Soit ABCD un carré de côté a, I le milieu de [BC] et J celui de [DC].
On se propose d'évaluer l'angle IAJ de mesure 8.
1) Exprimer AI AJ en fonction de cos (8) et de a.
2) a). Exprimer Al et AJ à l'aide des vecteurs AB et AD.
b) Donner une autre expression de AI-AJ.
3) Déduire des questions précédentes la valeur exacte de cos() et une valeur valeur
approchée à 102 près par défaut, de (en degrés).
D
Answer:
Step-by-step explanation:
A retiree invests $9,000 in a savings plan that pays 4% per year. What will the account balance be at the end of the first year?
[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$9000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ t=years\dotfill &1 \end{cases} \\\\\\ A = 9000[1+(0.04)(1)] \implies A = 9360[/tex]
California has three times as many airports as Georgia.
a. Underline the key math word in the problem.
. Circle the operation you would use to find how many airports
Georgia has. Explain.
+
X
Hence, by multiplying the number of airports in California by a factor of three-quarters, we may get the number of airports in Georgia.
How to find how many airports Georgia has?To find how many airports Georgia has, we need to use division. We can divide the total number of airports in California by 3 to find the number of airports in Georgia.
The key math word in the problem is "three times."
Let's say the number of airports in California is "C". Then, we can represent the number of airports in Georgia as "G," and we know that:
C = 3G
To solve for G, we can divide both sides of the equation by 3:
C/3 = G
Therefore, the number of airports in Georgia is equal to one-third of the number of airports in California.
The key math word in the problem is "three times."
We need to divide in order to determine the number of airports in Georgia. The number of airports in Georgia can be calculated by dividing the total number of airports in California by 3.
Let's say the number of airports in California is "C". Then, we can represent the number of airports in Georgia as "G," and we know that:
C = 3G
To solve for G, we can divide both sides of the equation by 3:
C/3 = G
Therefore, the number of airports in Georgia is equal to one-third of the number of airports in California.
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the official world record for the men's marathon is 2 hours and 3 minutes 59 seconds set by Haile Gebrselassie of Ethiopia. This berlin, marathon, held in September 2008, was 42.195 kilometers long. Write Gebrselassie's average speed for this marathon in meters per second. round your answer to the nearest tenth
Step-by-step explanation:
2 hrs = 7200 seconds
3 min 59 = 239 s
total seconds = 7439 s
42.195 km = 42159 m
42159 m / 7439s = 5.7 m/s
which sequence of transformations was most likely applied to figure QRSTU to form Q’R’S’T’U’ as shown below?
D. Reflection across the y-axis, translation 4 units to the right. sequence of transformation is most likely applied to figure QRSTU to form Q'R'S'T'U'.
What is transformation?
To determine the sequence of transformations applied to figure QRSTU to form Q'R'S'T'U', we can compare the corresponding vertices of the two pentagons:
Q and Q': Q' is to the right of Q and has a positive x-coordinate.
R and R': R' is to the left of R and has a negative x-coordinate.
S and S': S' is above S and has a positive y-coordinate.
T and T': T' is above T and has a positive y-coordinate.
U and U': U' is to the right of U and has a positive x-coordinate.
Based on these comparisons, it appears that the figure has been reflected across the y-axis and then translated 4 units to the right. Therefore, the most likely sequence of transformations applied to QRSTU to form Q'R'S'T'U' is:
D. Reflection across the y-axis, translation 4 units to the right.
What are vertices?
A vertex (plural: vertices) is a point where two or more line segments or curves meet. Vertices are the corner points of a polygon or the endpoints of a line segment or ray. In three-dimensional geometry, vertices are the corner points of a polyhedron or the endpoints of an edge.
Vertices are important in defining the properties of a shape, such as its angles, sides, and symmetry. They are also used in graph theory to represent points in a graph or network.
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Complete question is: D. Reflection across the y-axis, translation 4 units to the right. sequence of transformation is most likely applied to figure QRSTU to form Q'R'S'T'U'.
Find the volume of the oblique rectangular prism below. (2 points)
15
10
8
Enter your answer and your work or explanation in the space provided. Round your answer to the nearest hundredth.
Using the formula we know that the volume of the oblique rectangular prism is 1,200units³ respectively.
What is an oblique rectangular prism?A prism with oblique bases is one in which the bases are not parallel to one another.
An oblique rectangular prism is a rectangular prism with bases that are not arranged one over the other.
The following qualities describe an oblique prism: It seems to be slanted or skewed. Parallelograms make up an oblique prism's faces.
The height is gauged from the outside.
The two bases are not positioned next to one another exactly.
So, the volume formula for the oblique rectangular prism:
V=l⋅w⋅h
Now, insert values:
V=l⋅w⋅h
V=15*10*8
V=1,200 units ³
Therefore, using the formula we know that the volume of the oblique rectangular prism is 1,200units³ respectively.
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Help need correct answer asap
Two skaters are practicing at the same time on the same rink. A coordinate grid is superimposed on the ice. One skater follows
the path y = - 2x + 14, while the other skater follows the curve y = - 2x^2+ 14x.
Find all the points where they might collide if they
are not careful.
The two skaters might collide at the points (1, 12) and (7, 0).
What are the collide points?
To find the points where the two skaters might collide, we need to find the values of x and y that satisfy both equations simultaneously.
Setting the two expressions for y equal to each other, we get:
-2x + 14 = -2x² + 14x
Simplifying this equation, we get:
2x² - 16x + 14 = 0
Dividing both sides by 2, we get:
x² - 8x + 7 = 0
This equation can be factored as:
(x - 1)(x - 7) = 0
Therefore, the possible values of x where the skaters might collide are x=1 and x=7.
To find the corresponding values of y, we can substitute each value of x into one of the original equations.
For x=1, we get:
y = -2(1) + 14 = 12
Therefore, one possible point of collision is (1, 12).
For x=7, we get:
y = -2(7)² + 14(7) = 0
Therefore, another possible point of collision is (7, 0).
Therefore, To find the points where the two skaters might collide, we need to find the values of x and y that satisfy both equations simultaneously.
Setting the two expressions for y equal to each other, we get:
-2x + 14 = -2x² + 14x
Simplifying this equation, we get:
2x² - 16x + 14 = 0
Dividing both sides by 2, we get:
x² - 8x + 7 = 0
This equation can be factored as:
(x - 1)(x - 7) = 0
Therefore, the possible values of x where the skaters might collide are x=1 and x=7.
To find the corresponding values of y, we can substitute each value of x into one of the original equations.
For x=1, we get:
y = -2(1) + 14 = 12
Therefore, one possible point of collision is (1, 12).
For x=7, we get:
y = -2(7)² + 14(7) = 0
Therefore, another possible point of collision is (7, 0).
Therefore, the two skaters might collide at the points (1, 12) and (7, 0).
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10b + 12 = 62
what is b?
Step-by-step explanation:
10b = 62 - 12
10b = 50
b = 50/10
b = 5
Answer:
5
Step-by-step explanation:
When trying to find the value of an unknown variable, it is important to take into factors on how to reverse an effect being done to the variable, which is how we can isolate b and figure out the value.
- Addition reverses subtraction
- Multiplication reverses division
We have a simple equation, [tex]10b+12=62[/tex] , and we need to work around the variable b in order to isolate it. By doing this, we are able to find the true value of b.
We can subtract 12 on each side to get the term away from the constant.
[tex]10b+12=62[/tex]
- 12 -12
[tex]10b=50[/tex]
Then, to isolate the variable from the coefficient, which is 10, we can divide it to negate the multiplication factor on the term.
[tex]\frac{10b}{10} = \frac{50}{10}[/tex]
b=5
Therefore, the variable b=5.
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5^a - 3^a = 2b a and b are less then 5 and positive integers work out all the possible values of a and b
Step-by-step explanation:
pls send a picture for better clarification of the questions
A television game show has 15 doors, of which the contestant must pick 3. Behind 3 of the doors are expensive cars,
and behind the other 12 doors are consolation prizes. The contestant gets to keep the items behind the 3 doors she
selects. Determine the probability that the contestant wins at least one car.
The probability that the contestant wins at least one car is 13/35.
How to calculate the probabilityA television game show has 15 doors, of which the contestant must pick 3. Behind 3 of the doors are expensive cars, and behind the other 12 doors are consolation prizes
probability of an event =total favorable outcome for event/total possible outcomes
P(contestant wins at least one car) will be:
= 1 - 12/15 × 11/14
=13/35
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Solve both triangles below, where A= 45 degrees, c= 4.2 degrees, C= 40 degrees, N= 45 degrees, and m= 2. Round to the nearest tenth.
Triangle 1:
A = 45 degrees, C = 40 degrees, c = 4.2
First, we need to find angle B.
B =[tex] 180 - (A + C) = 180 - (45 + 40) = 95[/tex] degrees
Now, we can use the Law of Sines to find side b.
[tex]b/sin(B) = c/sin(C)[/tex]
b =[tex] (c * sin(B)) / sin(C)[/tex]
b =[tex] (4.2 * sin(95)) / sin(40)[/tex]
b ≈ [tex]5.6[/tex]
Triangle 2:
N = 45 degrees, M = 180 - N = 135 degrees, m = 2
To find side n, we can use the Law of Sines again.
[tex]n/sin(N) = m/sin(M)[/tex]
n =[tex] (m * sin(N)) / sin(M)[/tex]
n =[tex] (2 * sin(45)) / sin(135)[/tex]
n ≈ [tex]1.4[/tex]
So, the solutions for the two triangles are:
Triangle 1: A = 45 degrees, B = 95 degrees, C = 40 degrees, b ≈ 5.6, c = 4.2
Triangle 2: N = 45 degrees, M = 135 degrees, n ≈ 1.4, m = 2
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- Background Layout Theme Transition
ck to add speaker notes
DAY THREE: WATER
While you are on the island you will have the ability to collect units of
Water. You can collect a total of 1 unit of water per day. To be fair, I'll let
you start off with 1 unit of water. Everyday, you will only need to
consume 1 unit of water per day in order to survive.
1. What is your total units of water collected for day 3?
How much water did you consume?
2.
3. How much water do you have leftover?
4.
How did you go about collecting your water?
5. What tools did you use in this process?
To find the total units of water collected for day 3, we can simply add up the units of water collected on each day, starting with the 1 unit of water we already have:
Day 1: 1 unit of water
Day 2: 1 unit of water
Day 3: 1 unit of water
1. Total units of water collected for day 3 = 1 + 1 + 1 = 3 units of water
2. Since we only need to consume 1 unit of water per day to survive, we would have consumed a total of 3 units of water in 3 days.
3. To find the amount of water leftover, we can subtract the units of water consumed from the total units of water collected:
Total units of water collected = 3 units
Units of water consumed = 3 units
Water leftover = Total units of water collected - Units of water consumed
= 3 - 3
= 0 units of water leftover
4. We could collect water in a number of ways, depending on what resources were available on the island. We could collect rainwater using containers, or we could find a freshwater stream or well and collect water from there. We could also try to collect dew or moisture using a cloth or other absorbent material.
5. The tools we would use to collect water would depend on the method we chose. For example, if we were collecting rainwater, we would need containers to catch the water, such as buckets or barrels. If we were collecting water from a stream or well, we would need a way to transport the water back to our shelter, such as a jug or canteen. If we were collecting moisture, we would need a cloth or other absorbent material to collect the water.
Last week, a candy store sold 6 1/3
pounds of white chocolate. It sold 4 times as much milk chocolate as white chocolate. DUE NOWW HELP
Question 1
How many pounds (in mixed number form) of milk chocolate did the candy store sell last week?
Responses
A 24 2/3
B. 25 2/3
C. 25 1/3
D. 24 1/3
Question 2
How many more pounds of milk chocolate were sold than white chocolate at the candy store last week?
Responses
A. 19 2/3
B. 18
C. 19
D. 18 2/3
1)The number of milk chocolate that sold last week is = [tex]25\frac{2}{3}[/tex] B is correct
2)white chocolate at the candy store last week is 19.
Last week, a candy store sold [tex]5\frac{1}{3}=\frac{16}{3}[/tex] pounds of white chocolate. It sold 4 times as much milk chocolate as white chocolate.
The number of milk chocolate did the candy store sell last week.
Since the candy store sold 4 times as much milk chocolate as white chocolate, it sold:
[tex]4 *\frac{ 19}{3 } =\frac{ 76}{3}\\ pounds\ of\ milk\ chocolate.= 25 \frac{2}{3}[/tex]
option B is correct.
2)
To find out how many more pounds of milk chocolate were sold than white chocolate at the candy store last week, we can subtract the amount of white chocolate sold from the amount of milk chocolate sold:
[tex]\frac{76}{3} - \frac{19}{3} \\= \frac{57}{3}\\ = 19\ pounds.[/tex]
option C is correct
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All the following will appear in a cash budget EXCEPT:
A. Depreciation
B. Corporation tax payment
C. Purchase of new vehicle
D. Receipt of dividend from investment
Answer:
The answer is A. Depreciation.
A cash budget is a financial statement that projects a company's future cash inflows and outflows. It is used to help the company manage its cash flow and avoid running out of money. The cash budget includes all of the company's expected cash receipts and payments, including sales, purchases, expenses, and taxes. Depreciation is not a cash flow, so it does not appear on a cash budget.
Depreciation is an accounting concept that allows companies to spread the cost of an asset over its useful life. It is not a cash expense, because the company does not actually pay out any money when it depreciates an asset. Instead, the company reduces the value of the asset on its balance sheet.
The cash budget is a valuable tool for businesses of all sizes. It can help businesses to plan for future cash needs, avoid running out of money, and make sound financial decisions.
Step-by-step explanation:
A plant measured x inches tall last week and 8 inches tall this week. Represent the number of inches the plant grew what you do do you miltiply , divide, ect
The number of inches the plant grew this week would be 8-x. As the length increased this week the length is substracted from this week to last week.
Subtraction (which is signified by the minus sign −) is one of the four arithmetic operations along with addition, multiplication and division. Subtraction is an operation that represents removal of objects from a collection. For example, in the adjacent picture, there are 5 − 2 peaches—meaning 5 peaches with 2 taken away, resulting in a total of 3 peaches.
Therefore, the difference of 5 and 2 is 3; that is, 5 − 2 = 3. While primarily associated with natural numbers in arithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.
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A magazine picture of a purse is shown below. How much area does the purse (not including the handle) take up in the magazine layout?
The area of the purse can be calculated using the formula for the area of a trapezoid. The purse (not including the handle) takes up 34 cm² in the magazine layout.
What is the formula of area of a trapezoid?The formula states that the area of a trapezoid is equal to one half of the product of the length of the two parallel sides, multiplied by the height. Thus, the area of the purse can be calculated as:
Area = (1/2) x (11cm + 6cm) x 4cm
Area = (1/2) x (17cm) x 4cm
Area = 34 cm²
Therefore, the purse (not including the handle) takes up 34 cm² in the magazine layout.
It is important to note that the formula for the area of a trapezoid can be used in this instance because the purse has a shape of a trapezoid.
This is due to the fact that the line parallel to the 11cm base has a length of 6cm. The line is not parallel to the 4cm height and thus, the shape of the purse is a trapezoid and not a rectangle.
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If 40% of x is 111, what is 80% of x?
Answer: 222
Step-by-step explanation:
So you know 40% of x is 111
and we need to find 80%
80% is twice of 40%
so we multiply 111 by 2 and get 222
x=222
Find the value of z such that 0.8904
of the area lies between −z
and z. Round your answer to two decimal places.
The value of z that lies in a given area through which the given complex number expression is satisfied is z = 1.6
What about complex number?
A complex number is a number that can be expressed in the form a + bi, where "a" and "b" are real numbers and "i" is the imaginary unit, which is defined as the square root of -1. The real part of a complex number is "a" and the imaginary part is "b".
Complex numbers are important in many areas of mathematics, science, and engineering. They are used to represent and manipulate quantities that involve both real and imaginary components, such as alternating currents in electrical engineering and wave functions in quantum mechanics. Complex numbers also have a geometric interpretation, where they can be represented as points in the complex plane, which is a two-dimensional coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
Define area:
Area is a measure of the size of a two-dimensional region or surface. It is usually measured in square units, such as square meters, square centimeters, or square feet.
The area of a shape can be calculated using different formulas depending on the shape's characteristics.
According to the given information:
⇒ Here, the z scores be z1 and z2
⇒ In which Z1 = 0.5 - 0.8904/2 = 0.0548
⇒ And, Z2 = 0.5 + 0.8904/2 = 0.9452
⇒ Using standard normal distribution table,
⇒ We, have that, Z1 = -1.6
⇒ Z2 = 1.6
⇒ So, the value of Z = 1.6
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1.3 1.4 Thabang save money by putting coins in a money box. The money box has 600 coins that consist of 20 cents and 50 cents. 1.3.1 1.3.2 Determine the total amount of cents in the money box. Convert the amount in 1.3.2 to rands (R).
1.3.1 There are 380 pieces of 50 cents coins.
1.3.2 The total amount of cents in the money box are 23,400.
1.3.3 The amount in rands is 234.
The solution has been obtained by using unit conversion.
What is unit conversion?
Unit conversion is a multi-step procedure that requires multiplying or dividing by a numerical factor, or more specifically, a conversion factor. Furthermore, rounding and choosing the right number of significant digits may be required. Various metrics are measured using various conversion units.
1.3.1 Let the number of 50 cents coin be x.
We are given that total number of coins are 600.
This gives us an equation as x + 220 = 600.
So,
x = 600 - 220
x = 380
1.3.2 The total cents is given by the following expression
⇒ (20 * 220) + 50x
⇒ 4400 + (50 * 380)
⇒ 4400 + 19000
⇒ 23,400 cents
1.3.3 We know that 100 cents = 1 rand.
So, using unit conversion, we get
⇒ 1 cent = [tex]\frac{1}{100}[/tex] rands
⇒ 23400 cents = [tex]\frac{1}{100}[/tex] * 23400 rands
⇒ 23400 cents = 234 rands
Hence, the required solutions have been obtained.
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The complete question has been attached below.
Simplify the expression 3x + 2y - x + y
Answer:
2x + 3y
Step-by-step explanation:
= 3x + 2y - x + y
= 2x + 2y + y
= 2x + 3y
hope this helps
the cost of two brands of a household is given. Find the unit price and tell which is better to buy. A) $6.15 for 24 fl oz of dishwashing liquid B: $9.60 for 42fl oz of dishwashing liquid A: $4.29 for 4 oz of shoe polish B: $6.33 for 6 oz of shoe polish
Answer:
it is better to buy brand B for both dish washing liquid and shoe polish.
Step-by-step explanation:
 please help!!!!!!!!!!!!!!!!
Answer:
I CANT SEE WHAT IT SAYS
Step-by-step explanation:
Find where f(x) = g(x), when f(x) = √40x and g(x)=√x + 3.
Therefore, f(x) = g(x) when function x is approximately 0.235 or 33.765.
Quadratic equations?A quadratic equation is a polynomial equation of the second degree, which means that the highest power of the variable in the equation is 2. A general quadratic equation can be written in the form:
[tex]ax^2 + bx + c = 0[/tex]
Where 'a', 'b', and 'c' are constants, and 'x' is the variable. The value of 'x' that satisfies the equation is called the root or solution of the equation.
To find where f(x) = g(x), we need to solve the equation:
[tex]\sqrt40x = \sqrt{x} + 3[/tex]
Squaring both sides, we get:
[tex]40x = (x + 3)^2[/tex]
Expanding the right-hand side, we get:
40x = x² + 6x + 9
Bringing all terms to one side, we get:
[tex]x^2- 34x - 9 = 0[/tex]
We can solve this quadratic equation using the quadratic formula:
[tex]x = (-b± \sqrt{(b^2 - 4ac))}/2a[/tex]
where a = 1, b = -34, and c = -9.
Plugging in these values, we get:
x = (34 ± √ (34² + 419)) / 2
x = (34 ± √1165) / 2
x ≈ 0.235 or x ≈ 33.765
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if point A(a,b) lies on the line 3x-4y=26 and B(b,a) lies on the line 5x-3y+31=0. Find the equation and length of AB.
The equation of the line passing through points [tex]A(a,b)[/tex] and [tex]B(b,a)[/tex] is [tex]x + y = a + b[/tex] , and the length of [tex]AB[/tex] is [tex]\sqrt(2(a - b)^2).[/tex]
What is the slope of the line?To find the equation of the line passing through points A(a,b) and B(b,a), we can first find the slope of the line. using the two given points, and then use the point-slope form of a line equation.
The slope of a line passing through two points ([tex]x_{1} , y_{1}[/tex]) and ([tex]x_{2}, y_{2}[/tex]) is given by:
[tex]m = (y_{2} - y_{1} ) / (x_{2} - x_{1} )[/tex]
Applying this formula, we can find the slope of the line passing through points A(a,b) and B(b,a):
Slope [tex](m) = (a - b) / (b - a)[/tex]
Since a and b can take any real values, we can see that the slope of the line passing through points A and B is -1.
Now, let's find the equation of the line passing through points A and B using the point-slope form of a line equation:
Point-slope form: [tex]y - y_{1} = m(x - x_{1} )[/tex]
Substituting the slope (-1) and the coordinates of point A(a,b) into the point-slope form, we get:
[tex]y - b = -1(x - a)[/tex]
Expanding the equation, we get:
[tex]y - b = -x + a[/tex]
Rearranging the terms, we get the equation of the line passing through points A(a,b) and B(b,a):
[tex]x + y = a + b[/tex]
Now, let's find the length of AB using the distance formula:
The distance between two points [tex](x_{1} , y_{1} )[/tex] and [tex](x_{2} , y_{2} )[/tex] is given by:
[tex]d = \sqrt((x_{2} - x_{1} )^2 + (y_{2} - y_{1} )^2)[/tex]
Applying this formula, we can find the length of AB using the coordinates of points A(a,b) and B(b,a):
Length of [tex]AB (d) = \sqrt((b - a)^2 + (a - b)^2)[/tex]
Simplifying the expression, we get:
Length of [tex]AB (d) = \sqrt(2(a - b)^2)[/tex]
Therefore, The equation of the line passing through points [tex]A(a,b)[/tex] and [tex]B(b,a)[/tex] is [tex]x + y = a + b[/tex] , and the length of [tex]AB[/tex] is [tex]\sqrt(2(a - b)^2).[/tex]
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Solve the following inequalities, giving your answer in interval notation and sketch the solution:
−14 < −7(3 + 2) < 1
The final answer is:
-1/7 < 5 < 2
Or, in interval notation:
(−1/7, 2)
What is interval?
In mathematics, an interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. An interval can be written using interval notation, which uses parentheses, square brackets, or a combination of both to indicate whether the endpoints of the interval are included or excluded. For example:
First, let's simplify the expression inside the parentheses:
3 + 2 = 5
Now we can substitute this value into the expression and simplify further:
-14 < -7(5) < 1
-14 < -35 < 1
To solve this compound inequality, we need to isolate the variable (in this case, there is no variable) to find the range of values that satisfy the inequality. We can do this by dividing all three parts of the inequality by -7. However, we need to reverse the direction of the inequality signs since we are dividing by a negative number:
-14/-7 > -35/-7 > 1/-7
2 > 5 > -1/7
Therefore, the solution to the inequality is:
-1/7 < 5 < 2
In interval notation, this can be written as:
(−1/7, 2)
To sketch the solution, we can plot the interval on a number line. The interval does not include the endpoints, so we use open circles to indicate this:
<-------------(○======○)------------->
-1/7 5 2
The open circle at -1/7 indicates that -1/7 is not included in the interval, and the open circle at 2 indicates that 2 is also not included in the interval. The line segment between the circles represents the range of values that satisfy the inequality.
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A study was done to estimate the mean commute time for a population of employed adults. A random sample of
25 employed adults in a particular city was taken, and the 95% confidence interval for mean commute time was
calculated to be 34.42 ± 8.14 minutes. A councilman is advocating for road repairs and states that, in this city,
people with jobs commute an average of 45 minutes to work each day. Does the confidence interval justify the
councilman's statement?
No, because 45 minutes is greater than the upper bound of the confidence interval.
Yes, because 45 minutes is greater than the upper bound of the confidence interval.
No, because the mean commute time from the 25 sampled adults was 34.42 minutes.
O Yes, because there were almost certainly some people in the sample of 25 adults whose commute was greater
than 45 minutes.
The confidence interval does not justify the councilman's statement because 45 minutes is greater than the upper bound of the confidence interval.
What is a confidence interval?
A confidence interval is a statistical range that is used to estimate an unknown population parameter based on a sample from that population. It is a measure of the degree of uncertainty or precision associated with the estimated value of the parameter.
The confidence interval of 34.42 ± 8.14 minutes means that we are 95% confident that the true population mean commute time falls between 26.28 minutes (34.42 - 8.14) and 42.56 minutes (34.42 + 8.14). Since the councilman's statement of 45 minutes falls outside of this range, we cannot say that it is justified by the confidence interval.
Additionally, the fact that the mean commute time from the sample of 25 adults was 34.42 minutes does not necessarily mean that the true population mean commute time is also 34.42 minutes. It only provides an estimate, and the confidence interval takes into account the variability of the sample mean.
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Answer:
A. No, because 45 minutes is greater than the upper bound of the confidence interval.
Step-by-step explanation: