The principle of mathematical induction is a method of proof used in mathematics to prove that a statement is true for all natural numbers.
It is based on the idea that if the statement is true for one number, then it can be used to prove that it is true for the next number. Mathematical induction can be expressed mathematically as follows:
Let P(n) be a statement involving an integer n
Base Case: P(m) is true for some m
Induction Hypothesis: Assume P(k) is true for some k>m.
Induction Step: Show that P(k+1) is true.
Therefore, P(n) is true for all n>m
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The proof shows that ABCD is a rhombus. Which of the following is the
missing reason?
A. Reflective property
B. Symmetric property
C. Transitive property
D. Addition property
The correct answer is B. Symmetric property.
The symmetric property states that if a = b, then b = a. In the context of geometry, this property can be used to show that if one side of a figure is congruent to another side, then the second side is also congruent to the first. In the case of the given proof, it is possible that the symmetry of the figure is used to show that opposite sides of the rhombus are congruent.
The reflective property (A) is not typically used to prove that a figure is a rhombus, as it relates to the reflection of a figure across a line. The transitive property (C) and the addition property (D) are also unlikely to be used in this context, as they relate to the properties of equality and addition, respectively, rather than geometric properties of figures.
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How will the product change if one number is decreased by a factor of 2 and the other is decreased by a factor of 8 ?
The product is decreased by a factor of 16.
What is a factor?
In mathematics, a factor is a number or quantity that, when multiplied with another number or quantity, produces a given result. For example, in the expression 3 x 4 = 12, 3 and 4 are factors of 12. Factors can also refer to algebraic expressions, where they are the expressions that are multiplied together to obtain a larger expression.
Let's say we have two numbers, A and B, and we want to find the product of A and B.
The product of A and B is AB.
If we decrease A by a factor of 2, the new value of A becomes A/2. If we decrease B by a factor of 8, the new value of B becomes B/8.
So the new product of A/2 and B/8 is:
(A/2)(B/8) = AB/16
Therefore, the product is decreased by a factor of 16.
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Find the simplified form of the expression. Give your answer in scientific notation.
(8 × 10^4)(9 × 10^–8)
Answer:
7.2×10^-3
Step-by-step explanation:
8 × 10^4×9 × 10^–8
=8×9×10^(4-8)
=72×10^-4
=7.2×10^-3
Bro leave me alone pls
Answer:
Step-by-step explanation:
Find the missing length indicated. * A) 12 C) 8 A OB D 16 24 36 B) 18 D) 15
Answer:
Is D
Step-by-step explanation:
The Gordon family plans to buy a TV. One TV has a purchase price of $330 and an estimated yearly operating cost of $14. The other has a purchase price of $369 and an estimated yearly operating cost of $9. Which TV should the Gordons buy if they plan to keep it for 8 years? Use rational functions to help justify your answer.
ANSWER:rational functions that model the average annual cost of each television if it is owned for x years, such as f(x)= (330+14x)/x and
g(x)= (369+9x)/x
calculation proving that over 8 years, the annual cost for the $330 television is $55.25; for the $369 television, it is $55.13
statement that the Gordons should buy the $369 television because it is less expensive overall
step by step: edge 2023
Answer:
The Gordon's should buy TV2
Step-by-step explanation:
To determine which TV the Gordons should buy, we need to compare the total cost of each TV over the 8-year period. Let's define some variables:
- Let x be the number of years the TV is used.
- Let y1 be the total cost of TV 1 over x years.
- Let y2 be the total cost of TV 2 over x years.
- Let P1 be the purchase price of TV 1.
- Let P2 be the purchase price of TV 2.
- Let O1 be the annual operating cost of TV 1.
- Let O2 be the annual operating cost of TV 2.
Using these variables, we can write the following equations:
y1 = P1 + x * O1
y2 = P2 + x * O2
To compare the total cost of each TV over 8 years, we need to find y1 and y2 when x = 8. Plugging in the given values, we get:
y1 = 330 + 8 * 14 = 462
y2 = 369 + 8 * 9 = 441
Therefore, the total cost of TV 1 over 8 years is $462, and the total cost of TV 2 over 8 years is $441. Since TV 2 has the lower total cost, the Gordons should buy TV 2.
From a mathematical standpoint, we can also use rational functions to analyze this problem. The total cost of each TV is a linear function of x, so we can write:
y1(x) = P1 + x * O1
y2(x) = P2 + x * O2
The ratio of these functions is:
y1(x) / y2(x) = (P1 + x * O1) / (P2 + x * O2)
To determine which TV is cheaper over 8 years, we need to compare the ratios when x = 8:
y1(8) / y2(8) = (330 + 8 * 14) / (369 + 8 * 9) ≈ 1.051
Since this ratio is greater than 1, TV 1 is more expensive than TV 2 over 8 years. Therefore, the Gordons should buy TV 2.
Taylor wants to take out a mortgage of $380,000 with interest that compounds monthly. Use the formula A = P(1 +)*t to find which of these loans will have the lowest total cost. A) 20 years at 10% interest B) 25 years at 6.5% interest C) 30 years at 5% interest D) 35 years at 4% interest
Using the compound interest formula it is obtained that the loans which will have the lowest total cost is option C) - 30 years at 5% interest.
What is Compound Interest?
The interest that is calculated using both the principal and the interest that has accrued during the previous period is called compound interest. It differs from simple interest in that the principal is not taken into account when determining the interest for the subsequent period with simple interest. Compound interest is commonly abbreviated C.I. in mathematics.
We can use the formula of compound interest [tex]A = P\big(1 + \frac{r}{n} \big)^{(nt)}[/tex] to calculate the total cost (A) of each loan, where P is the principal (the initial amount borrowed), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the total number of years.
For all loans, the principal P is $380,000.
For Loan A, the annual interest rate is 10%, and the interest is compounded 12 times per year (monthly), so r = 0.1/12 and n = 12.
The total time is 20 years, so t = 20.
Plugging in the values, we get -
[tex]A = $380,000(1 +0.1/12)^{(12 \times20)} = $1,245,956.70[/tex]
For Loan B, the annual interest rate is 6.5%, and the interest is compounded 12 times per year (monthly), so r = 0.065/12 and n = 12.
The total time is 25 years, so t = 25.
Plugging in the values, we get -
[tex]A = $380,000(1 +0.065/12)^{(12 \times25)} = $844,275.92[/tex]
For Loan C, the annual interest rate is 5%, and the interest is compounded 12 times per year (monthly), so r = 0.05/12 and n = 12.
The total time is 30 years, so t = 30.
Plugging in the values, we get -
[tex]A = $380,000(1 +0.05/12)^{(12 \times30)} = $766,726.73[/tex]
For Loan D, the annual interest rate is 4%, and the interest is compounded 12 times per year (monthly), so r = 0.04/12 and n = 12.
The total time is 35 years, so t = 35.
Plugging in the values, we get -
[tex]A = $380,000(1 +0.04/12)^{(12 \times35)} = $810,780.25[/tex]
Therefore, Loan C with 30 years at 5% interest has the lowest total cost, with a total cost of $766,726.73.
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Write 2/8 in lowest terms.
Answer: 1/4
Step-by-step explanation:
2 divided by 2 is 1 and 8 divided by 2 is 4 so 1/4
Answer: 1/4
Step-by-step explanation: We can simplify by divideing both the numerator and the denominator by 2. That will leave us with the answer of 1/4.
(PLEASE HELP MY GRADES WRE DUE TMRO)Scientists are preparing two satellites to be launched. The graph below represents the
number of miles, y, that the satellite, Space Explorer A, flies in a hours.
Miles
The rate of change is equal to 6,800 miles per hour.
What is the rate of change?In Mathematics and Geometry, the rate of change can be determined by using the following mathematical equation (formula);
Rate of change, m = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Rate of change, m = (y₂ - y₁)/(x₂ - x₁)
Next, we would determine the rate of change of the value of y with respect to the value of x for Space Explorer A;
Rate of change, m = (68,000 - 34,000)/(10 - 5)
Rate of change, m = 34,000/5
Rate of change, m = 6,800 miles per hour.
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Complete Question:
Scientists are preparing two satellites to be launched. The graph below represents the number of miles, y, that the satellite, Space Explorer A, flies in a hours. Find the rate of change.
I need help on question 30 and 32.
Refer to the attached images.
Tauri is 3 years older than Treasure. In 5 years the sum of their ages will be 63. How old is Tauri now?
Answer:
Tauri is currently 28 years old
Step-by-step explanation:
how do i solve this problem?
Option A correctly describes the area of the graph of the function.
How is an area of the graph calculated?
The area of a graph is the area under the function curve enclosed within the X-axis of the graph. Thus, it is the area of the figure formed by enclosing the function graph with the X-axis.
It is the integration of the function polynomial of the possible values of x that the function curve lies in.
Here the function equation is : [tex]f(x) = ( 25 - x^{2} )[/tex]
From the figure, we can see that the value of x varies from -5 to 5.
Initial x-value of integration = minimum value of x
= -5
Final x-value of integration = maximum value of x
= 5
Thus, the area of the graph can be correctly calculated as. :
[tex]Area = \int\limits^{maxX}_ {minX} \, f(x)dx[/tex]
[tex]Area = \int\limits^5_ {-5} \, (25-x^{2} )dx[/tex]
Hence, Option A correctly calculates the area of the graph given.
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Follow the steps below to find the value of x that makes A||B. offende 8x + 30 B 10x A Set the alternate exterior angles equal to each other. [ ? ]x = X + Enter
The value of x that makes A parallel to B is 15.
Describe Alternate Exterior Angles?Alternate exterior angles are a pair of angles that lie outside of two parallel lines and are on opposite sides of the transversal. In other words, they are a pair of non-adjacent angles that are formed when a transversal intersects two parallel lines. The angles are congruent, meaning that they have the same measure.
The term "alternate" refers to the fact that they are on opposite sides of the transversal and "exterior" indicates that they are outside of the parallel lines. Alternate exterior angles are important in geometry because they can be used to prove that two lines are parallel. If the alternate exterior angles are congruent, then the lines are parallel.
Since the alternate exterior angles are equal, we can set them equal to each other:
8x + 30 = 10x
Subtracting 8x from both sides, we get:
30 = 2x
Dividing both sides by 2, we get:
15 = x
So the value of x that makes A||B is 15.
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The tens digit in a dividend is less than the divisor. The divisor has one digit. Gracie says the quotient must have a 0 in the tens place. Is Gracie correct? Explain
Gracie's statement is not necessarily correct about the tens digit in a dividend is less than the divisor and the divisor has one digit. Gracie says the quotient must have a 0 in the tens place.
What is dividend?
In mathematics, a dividend is the number that is being divided in a division problem. For example, in the division problem 18 ÷ 3 = 6, 18 is the dividend.
Gracie's statement is not necessarily correct. Let's look at a couple of examples to see why.
Example 1:
Dividend = 25
Divisor = 4
In this case, the tens digit of the dividend (2) is less than the divisor (4). The quotient is 6 with a remainder of 1. There is no 0 in the quotient, so Gracie's statement is incorrect.
Example 2:
Dividend = 35
Divisor = 4
In this case, the tens digit of the dividend (3) is also less than the divisor (4). The quotient is 8 with a remainder of 3. Again, there is no 0 in the quotient, so Gracie's statement is still incorrect.
From these examples, we can see that Gracie's statement is not always true. It is possible for the tens digit of the dividend to be less than the divisor, and for the quotient to not have a 0 in the tens place.
The only time Gracie's statement would be true is if the remainder is 0 after dividing the dividend by the divisor. In that case, the quotient would indeed have a 0 in the tens place. However, this is not always the case.
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Simplify 1/cos x + 1/cos x -1
Answer:
-2cotxcscx
Step-by-step explanation:
Step 1: Find a common denominator
Step 2: Simplify
Write the ratio of corresponding sides for the similar triangles and reduce the ratio to lowest terms. 2 similar right triangles. Triangle 1 has side length 12, hypotenuse 15, and blank side. Triangle 2 has side length 4, hypotenuse 5, and blank side. a. StartFraction 4 Over 15 EndFraction = StartFraction 5 Over 12 EndFraction = StartFraction 4 Over 15 EndFraction b. StartFraction 4 Over 5 EndFraction = StartFraction 12 Over 15 EndFraction = StartFraction 4 Over 5 EndFraction c. StartFraction 4 Over 12 EndFraction = StartFraction 5 Over 15 EndFraction = StartFraction 1 Over 3 EndFraction d. StartFraction 5 Over 4 EndFraction = StartFraction 15 Over 12 EndFraction = StartFraction 5 Over 4 EndFraction
Answer:
Step-by-step explanation: To find the ratio of corresponding sides for the two similar triangles, we need to match up the corresponding sides of the triangles and write the ratios of their lengths.
For Triangle 1, we have:
Side length = 12
Hypotenuse = 15
We can use the Pythagorean theorem to find the length of the blank side:
(Blank side)^2 = (Hypotenuse)^2 - (Side length)^2
(Blank side)^2 = 15^2 - 12^2
(Blank side)^2 = 225 - 144
(Blank side)^2 = 81
Blank side = 9
So, for Triangle 1, we have the following ratios:
Side length : Hypotenuse = 12 : 15
Side length : Blank side = 12 : 9
Hypotenuse : Blank side = 15 : 9
For Triangle 2, we have:
Side length = 4
Hypotenuse = 5
We can use the Pythagorean theorem to find the length of the blank side:
(Blank side)^2 = (Hypotenuse)^2 - (Side length)^2
(Blank side)^2 = 5^2 - 4^2
(Blank side)^2 = 25 - 16
(Blank side)^2 = 9
Blank side = 3
So, for Triangle 2, we have the following ratios:
Side length : Hypotenuse = 4 : 5
Side length : Blank side = 4 : 3
Hypotenuse : Blank side = 5 : 3
Therefore, the ratio of corresponding sides for the two similar triangles is:
Side length : Side length = 12 : 4 = 3 : 1
Hypotenuse : Hypotenuse = 15 : 5 = 3 : 1
Blank side : Blank side = 9 : 3 = 3 : 1
Reducing each ratio to lowest terms, we get:
Side length : Side length = 3 : 1
Hypotenuse : Hypotenuse = 3 : 1
Blank side : Blank side = 3 : 1
So the correct answer is (b) StartFraction 4 Over 5 EndFraction = StartFraction 12 Over 15 EndFraction = StartFraction 4 Over 5 EndFraction.
Matt increased his typing speed from 32 to 38 words per minute. What is the percent of increase
Answer:
Matt's typing speed increased by 18.75%.
Step-by-step explanation:
To find the percent increase, we can use the following formula:
percent increase = (new value - old value) / old value x 100%
Using this formula, we can calculate the percent increase in Matt's typing speed as follows:
percent increase = (38 - 32) / 32 x 100%
percent increase = 6 / 32 x 100%
percent increase = 0.1875 x 100%
percent increase = 18.75%
Arrange the quantities from smallest to largest. 2.4 dag. , .21 kg , 190 cg
The quantities arranged from smallest to largest are 1.9 g, 24 g, and 210 g.
What is quantity?A quantity is a property or attribute of an object or phenomenon that can be measured or described numerically. Quantities can be either scalar, meaning they have a magnitude (or size) but no direction, or vector, meaning they have both a magnitude and a direction.
According to question:First, let's convert all the quantities to the same unit.
2.4 dag = 24 g (since 1 dag = 10 g)
0.21 kg = 210 g (since 1 kg = 1000 g)
190 cg = 1.9 g (since 1 cg = 0.01 g)
Now we can arrange the quantities from smallest to largest:
1.9 g
24 g
210 g
Therefore, the quantities arranged from smallest to largest are 1.9 g, 24 g, and 210 g.
Dag is a weight measuring unit that is rarely applied in daily life. Dag is an abbreviation for "dekagram", which is a metric unit of mass equal to 10 grams. One dekagram is equivalent to 0.01 kilograms or 0.3527 ounces.
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Find x
Given: x=AC, m
Answer:
x = 15.733
Step-by-step explanation:
Reflections; rotations and translations are transformations that change the what of a figure
Answer:
eflections, rotations, and translations are all transformations that change the position of a figure in the plane. These transformations change the orientation, location, or both of a figure without altering its size or shape. In other words, they change the position or location of the figure in the plane while maintaining the same size and shape.
How many solutions does this equation have? –8 + 3s + 17s = 20s + 8
Answer: There are no solutions
Step-by-step explanation:
Step 1: simplify both sides of the equation
-8 + 3s + 17s = 20s + 8
(3s+17s) + (-8) = 20s + 8 (combine like terms)
20s + -8 = 20s + 8
20s - 8 = 20s + 8
Step 2: Subtract 20s from both sides
20s - 8 - 20s = 20s + 8 - 20s
-8 = 8
Step 3: Add 8 to both sides
-8 + 8 = 8 + 8
0 = 16
Thus, there are no solutions
Write an equation to determine each unknown length. Then, solve the equation. Round your answer to the nearest tenth, if necessary.
X=___ Round to the nearest tenth (One number after the decimal)
Answer: 40
Step-by-step explanation:
a(2) + b(2) = c(2)
a(2) + 9(2) = 11(2)
121 - 81 = 40
A business analyst wants to determine if the prices of goods at a supermarket have changed significantly since the new owner of the company took over. She looks at the prices of ten items before the new owner took over compared to after the new owner started.
(Chart in photo below)
Based on the data in the table and using a significance level of 0.05, what is the correct P-value and conclusion?
A. With a t statistic of 1.4789 and a P-value of 0.173292, reject the no hypothesis that prices have not changed
B. With a T statistic of 1.4789 and a P value of 0.173292, fail to reject the null hypothesis. The prices have not changed.
C. With a T statistic of 0.7394 and a P value of 0.478505, reject the no hypothesis that prices have not changed.
D. With a T statistic of 0.7394 anda P value of 0.478505, fail to reject the no hypothesis that prices have not changed.
Please answer quickly, 100 points thank you !
Answer: Option A
Step-by-step explanation:
The t-statistic and P-value can be calculated using statistical software or a t-test calculator. Using a two-tailed t-test with a significance level of 0.05 and 8 degrees of freedom (n1 + n2 - 2), we obtain:
t = -1.4789
P-value = 0.173292
Therefore, the correct answer is A. With a t statistic of -1.4789 and a P-value of 0.173292, we fail to reject the null hypothesis that the prices have not changed significantly since the new owner took over. We cannot conclude that the prices have changed significantly.
With a T statistic of 0.7394 and a P value of 0.478505, fail to reject the no hypothesis that prices have not changed. The correct option is D.
What is null hypothesis?A null hypothesis is a type of statistical hypothesis that asserts that there is no statistical significance in a given set of observations.
Using sample data, hypothesis testing is used to assess the credibility of a hypothesis.
To determine if the prices of goods at a supermarket have significantly changed since the new owner took over, we can perform a two-sample t-test with the null hypothesis being that the mean difference in prices before and after the new owner took over is zero.
Using a significance level of 0.05, the critical t-value for a two-tailed test with 9 degrees of freedom is approximately 2.306.
To calculate the t-statistic, we first need to calculate the mean and standard deviation of the differences in prices:
Mean difference = (0.30 - 0.07) / 10 = 0.023
Standard deviation = 2.967
t-statistic = (0.023 - 0) / (2.967 / sqrt(10)) = 0.7394
The calculated t-value of 0.7394 is less than the critical t-value of 2.306, and the corresponding p-value is 0.4785. This means we fail to reject the null hypothesis that the mean difference in prices is zero.
Therefore, based on the given data and using a significance level of 0.05, the correct P-value and conclusion are:
Thus, the correct option is D.
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Mrs. Brown has constructed a pool that has a capacity of 800cm3 and a depth of 8cm what is the area of the upper surface?
Answer:
Step-by-step explanation:
To find the area of the upper surface of the pool, we need to know the dimensions of the pool. If we assume the pool has a rectangular shape, we can use the formula:
Area = length x width
We can find the length and width by using the given volume and depth. Since the volume of the pool is 800 cubic cm and the depth is 8cm, we can find the length and width using the formula:
Volume = length x width x depth
800 = length x width x 8
length x width = 800/8
length x width = 100
Now, we do not have enough information to determine the exact dimensions of the pool. However, we do know that the area of the upper surface of the pool is equal to the length times the width. Therefore, the area of the upper surface is 100 square cm.
In △ABC, AB=5, BC=4, and AC=3. The triangle is translated left 2 units and up 6 units to △A′B′C′. What is the length of segment A′C′?
A′C′=6
cap A prime cap c prime is equal to 6
A′C′=3
cap A prime cap c prime is equal to 3
A′C′=4
cap A prime cap c prime is equal to 4
A′C′=2
Answer:
3
Step-by-step explanation:
The entire triangle shifts 2 to the left and up 6, the lengths of the sides remain the same, therefore:
AC = A'C' = 3
9. The price of a Delta Air Lines ticket from New York to Orlando increased to $450. This is an 8% increase. What was the old fare to nearest cent?
Answer:
$416.66
Step-by-step explanation:
We take
450 divided by 108, then time 100 = $416.66
So, the old fare is $416.66
Solid metal support poles in the form of right cylinders are made out of metal with a
density of 6.6 g/cm³. This metal can be purchased for $0.38 per kilogram. Calculate
the cost of a utility pole with a diameter of 17.6 cm and a height of 790 cm. Round
your answer to the nearest cent.
The cost of the utility pole is $127.36.
which of the following is not an identity?
The trigonometry expression which is not an identity is 1+cos² x/sin²x + 1. The Option A is correct.
What is a trigonometric identities?Trigonometric Identities refers to equalities that involve trigonometry functions and hold true for all variables in the equation. There are numerous trigonometric identities involving the side length and angle of a triangle.
The primary trigonometry functions are sine, cosine, and tangent, while the other three functions are cotangent, secant, and cosecant. All six trig functions are used to derive the trigonometric identities.
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The ratio of the birth weight to the adult weight of a male black bear is 3:1000. The average birth weight is 12 ounces. Find the average adult weight. Your answer is probably in ounces. If there are 16 ounces in 1 pound, how many pounds does the adult bear weigh?
In response to the stated question, we may state that As a result, an expressions adult black bear weighs 250 pounds.
what is expression ?An expression in mathematics is a collection of numbers, variables, and mathematical (such as addition, reduction, multiplication, division, exponentiation, etc.) that express a quantity or value. Expressions might be as basic as "3 + 4" or as complicated as "(3x2 - 2) / (x + 1)". They may also contain functions like "sin(x)" or "log(y)". Expressions can also be evaluated by substituting values for the variables and performing the mathematical operations in the order specified. If x = 2, for example, the formula "3x + 5" equals 3(2) + 5 = 11. Expressions are commonly used in mathematics to describe real-world situations, construct equations, and simplify complicated mathematical topics.
Let B represent the birth weight of a male black bear and A represent the mature weight. The ratio of B to A is supplied to us as 3:1000. As a result, we may write:
B/A = 3/1000
The average birth weight (B) is also reported as 12 ounces. We may use these data to get the average adult weight (A):
[tex]B/A = 3/1000\\12/A = 3/1000\\A = 12/(3/1000)\\A = 4000[/tex]
A male black bear's typical mature weight is 4000 ounces.
To convert ounces to pounds, divide by 16 (since one pound contains 16 ounces):
[tex]4000/16 = 250[/tex]
As a result, an adult black bear weighs 250 pounds.
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!!!!!! ANSWER QUICK .Barbie is building a dream out and purshcansed a slanted roof that is 16 inches long on both sides. Her dream house is exactly 24 inches wide. How high should she build a roof the supports so that the roof fits the house perfectly. PLS TELL ME HOW TO SOLVE IT UNGRENTLY!!!!
Barbie should build a roof support that is 4√7 inches high to fit the slanted roof perfectly on her dream house.
Describe Pythagorean theorem?The Pythagorean theorem is a fundamental theorem in mathematics that describes the relationship between the three sides of a right-angled triangle. It is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery.
The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this can be expressed as:
a² + b² = c²
where a and b are the lengths of the two shorter sides of the triangle (called the legs), and c is the length of the hypotenuse.
The Pythagorean theorem has important implications in geometry, trigonometry, and other branches of mathematics. It is also widely used in a variety of real-world applications, such as architecture, engineering, and physics. For example, it can be used to calculate the distance between two points in a two-dimensional plane, the height of a building or a tree, or the length of a diagonal in a rectangle or a square.
To solve this problem, we need to use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the legs (the two shorter sides) is equal to the square of the length of the hypotenuse (the longest side).
In this case, the two legs of the right triangle are the height of the roof and half the width of the house (since the roof slants down from the center of the house). The hypotenuse is the length of the roof, which is 16 inches on both sides.
So, we can set up the equation:
(height)² + (12)² = (16)²
Simplifying and solving for the height:
(height)² = (16)² - (12)²
(height)² = 256 - 144
(height)² = 112
height = √112
Using a calculator or simplifying by hand, we find that:
height = 4√7 inches
Therefore, Barbie should build a roof support that is 4√7 inches high to fit the slanted roof perfectly on her dream house.
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