The probability of rolling no 2's and no 5's in 10 rolls is approximately 0.107.
The probability of rolling no 2's and no 5's on a fair die can be calculated using the formula for the probability of independent events:
P(A and B) = P(A) x P(B)
where P(A) is the probability of event A occurring, and P(B) is the probability of event B occurring.
The probability of rolling any number other than 2 or 5 on a single roll of a fair die is 4/6, or 2/3. Since each roll is independent, the probability of rolling no 2's and no 5's in n rolls can be calculated as follows:
P(no 2's and no 5's in n rolls) = (4/6)^n
To round the answer to three decimal places, we can use a calculator. For example, if n = 10, the probability is:
P(no 2's and no 5's in 10 rolls) = [tex](4/6)^10[/tex] = 0.107
So the probability of rolling no 2's and no 5's in 10 rolls is approximately 0.107, rounded to three decimal places.
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A carpenter bought 90 nails. Each nail has a mass of 5.2 × 10 − 4 kilograms. What is the total mass, in kilograms, of the nails the carpenter bought?
To find the total mass of the nails the carpenter bought, we can simply multiply the number of nails by the mass of each nail:
Total mass = 90 x 5.2 x 10^-4 kg/nail
Total mass = 0.0468 kg
Therefore, the total mass of the nails the carpenter bought is 0.0468 kg.
the sum of -3 and it's opposite
Answer:
it equals 0
Step-by-step explanation:
3 + -3=0
I need an answer asap!! any answer will help :D
By multiplication method the cargo will take up 7× 10⁷ energy.
What is multiplication?In arithmetic, the multiplication or product of two numbers it represents the repeated addition of one number to another. It can be done in between numbers which can be whole numbers, natural numbers, integers, fractions, etc.
Dr. Nandi plans to transport 3.5×10⁵ energy prism that each take up 2 ×10² cubic feet of cargo space.
The total can be calculated by multiplication method.
Here the multiplication gives
3.5×10⁵ × 2 ×10²
= 3.5 × 2 ×10⁵⁺²
= 7× 10⁷ energy.
Hence, the cargo will take up 7 × 10⁷ energy.
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The volume of a sphere is 2,143.57 yd³. To the nearest yard, what is the radius of the sphere? Use 3.14 for .
The radius of the sphere is about yd.
The radius of the sphere is about 8 yards.
Calculating the radius of the sphereThe formula for the volume of a sphere is V = (4/3)πr³, where V is the volume, r is the radius, and π is the mathematical constant pi, approximately equal to 3.14.
We can rearrange the formula to solve for the radius:
r = ((3V)/(4π))^(1/3)
Substituting the given volume V = 2,143.57 yd³ and π = 3.14, we get:
r = ((3 x 2,143.57)/(4 x 3.14))^(1/3)
Evaluate
r ≈ 8.0
Rounding to the nearest yard, the radius of the sphere is approximately 8 yards.
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ted claims that the two shaded triangles must be congruent. is ted's claim correct? include all work
and/or reasoning necessary to either prove the triangles congruent or to disprove ted's claim.
Ted's claim is correct - the two shaded triangles must be congruent.
We can prove this using the following reasoning:
The two triangles share the same base, segment BC.
The two triangles have the same height, as the height of a triangle is measured as a perpendicular line from the base to the opposite vertex, and the perpendicular line from B to segment AD is the same length as the perpendicular line from C to segment AE.
Therefore, the two triangles have the same area.
If two triangles have the same area and share a common side (in this case, segment BC), they must be congruent by the Side-Area-Side (SAS) congruence theorem.
Therefore, we can conclude that the two shaded triangles are congruent.
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What is the equation of the graph below?
A. y = − (x − 3)^2 + 1
B. y = − (x + 3)^2 + 1
C. y = (x − 3)^2 − 1
D. y = (x + 3)^2 − 1
Answer:
D
Step-by-step explanation:
the equation of a parabola in vertex form is
y = a(x - h)² + k
where (h, k ) are the coordinates of the vertex and a is a multiplier
here (h, k ) = (- 3, - 1 ) , then
y = a(x - (- 3) )² + (- 1)
y = a(x + 3)² - 1
to find a substitute the coordinates of any other point on the graph into the equation.
using (- 2, 0 )
0 = a(- 2 + 3)² - 1
0 = a(1)² - 1
0 = a - 1 ( ad 1 to both sides )
1 = a
then
y = (x + 3)² - 1 ← equation of graph
The graph of a quadratic function has a vertex at the point (-8,-3). It passes through the point (-2,3). When written in vertex form, the function is f(x)=a(x-h)^2+k, where:
...where h = -8 and k = -3.
what price do farmers get for their watermelon crops? in the third week of july, a random sample of 36 farming regions gave a sample mean of x
If a random sample of 36 farming regions gave a sample mean of x = $6.88 per 100 pounds of watermelon, then the 90% "confidence-interval" is (6.3426 , 7.4174).
The sample mean of x is (x') = $6.88,
the sample standard-deviation (σ) = $1.96,
the sample size (n) is = 36,
The z value for the 90% confidence interval is = 1.645,
So, the margin of error(E) is = (z×σ)/√n = (1.645×1.96)/√36 ≈ 0.5374,
So, the interval will be = (x' ± E),
Substituting the values,
we get,
⇒ (6.88 - 0.5374 , 6.88 + 0.5374),
⇒ (6.3426 , 7.4174)
Therefore, the required 90% confidence interval is (6.3426 , 7.4174).
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The given question is incomplete, the complete question is
In the third week of July, a random sample of 36 farming regions gave a sample mean of x = $6.88 per 100 pounds of watermelon. Assume that σ is known to be $1.96 per 100 pounds.
Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop.
Factor.
2x² + 11x + 12
(x + [?])([ ]x + [])
Enter the number that belongs in the
green box.
Answer:
2x²+11x+12
sum=11x
product =24x²
factor 3x and 8x
2x²+8x+3x+12
2x(x+4)+3(x+4)
(2x+3)(x+4)
Simplify the expression: -9 + 4x - 3(x + 2)
Answer: x - 15
Given:
-9 + 4x - 3(x + 2)
Distribute the -3:
-9 + 4x - 3x - 6
Combine similar terms:
-15 + x
Answer:
[tex] \sf \: x - 15[/tex]
Step-by-step explanation:
Now we have to,
→ Simplify the given expression.
The expression is,
→ -9 + 4x - 3(x + 2)
Let's simplify the expression,
→ -9 + 4x - 3(x + 2)
→ -9 + 4x - 3(x) - 3(2)
→ -9 + 4x - 3x - 6
→ 4x - 3x - 9 - 6
→ (4x - 3x) + (-9 - 6)
→ (x) + (-15)
→ x - 15
Hence, the answer is x - 15.
Find the height of the basketball hoop using similarity ratios. Explain step by step.
The height of the basketball hoop is 6 units.
Step 1: Identify similar triangles.
Similar triangles are triangles with the same shape but not necessarily the same size. They have proportional sides and equal angles.
Step 2: Determine the corresponding sides and angles
Once you have identified the similar triangles, determine which sides and angles correspond with each other. The ratio of the corresponding sides in similar triangles will be equal.
Step 3: Set up a proportion
Now, set up a proportion using the corresponding sides of the similar triangles.
(side of first triangle) / (side of second triangle) = (height of first triangle) / (height of second triangle)
Step 4: Plug in the known values
Fill in the known values from the problem into the proportion. For example:
[tex](3 / 5) = (h / 10)[/tex]
Step 5: Solve for the unknown variable
Cross-multiply to solve for the unknown variable (h). In this example:
[tex]3 * 10 = 5 * h[/tex]
[tex]30 = 5h[/tex]
h = 6
So, the height of the basketball hoop is 6 units.
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is alice looking at a region of maximum brightness, minimum brightness, or neither? explain your reasoning.
As Alice looks at the color spectrum, she is looking at a region of maximum brightness.
Alice is looking at a region of maximum brightness because when we observe the color spectrum, we can see that the rainbow's center is the brightest, and the intensity of light at this location is the highest. Red color has the lowest energy, followed by orange, yellow, green, blue, indigo, and violet. It can be observed from the color spectrum that red has the lowest energy, followed by orange, yellow, green, blue, indigo, and violet.
White light can be separated into several colors in the color spectrum. The highest energy level is found in violet, whereas the lowest energy level is found in red. As Alice looks at the color spectrum, she is looking at a region of maximum brightness.
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a random variable x is characterized by a normal probability density function with known mean 20. there are two hypotheses for the variance. hypothesis h0 claims the variance is 16 while hypothesis h1 claims the variance is 25. we will choose between these hypotheses by observing three sample values x1, x2, and x3 and applying a threshold rule of the form reject h0 if x1 x2 x3 > t. for some scalar t. determine the value of t so that the probability of false rejection is 0.05. what is the corresponding probability of false acceptance? your answers should be in terms of the q function. 1
The value of the threshold t that gives a false rejection rate of 0.05 is found to be 5.991. The corresponding probability of false acceptance, or type II error, is computed using the power function of the test and found to be approximately 0.1485.
This is the probability of failing to reject the null hypothesis when the alternative hypothesis is true and the variance is 25.
To determine the value of t that gives a probability of false rejection of 0.05, we need to find the distribution of the test statistic under the null hypothesis H0: σ^2 = 16. The test statistic is:
T = (X1 - μ)² + (X2 - μ)² + (X3 - μ)² / (nσ²)
Under the null hypothesis, T follows a chi-squared distribution with 2 degrees of freedom (n-1). We can use this distribution to find the value of t such that the probability of false rejection is 0.05.
From the tables of the chi-squared distribution, we find that the critical value of T for a false rejection rate of 0.05 is 5.991.
Thus, we reject the null hypothesis if T > 5.991.
The probability of false acceptance, also known as type II error, is the probability of failing to reject the null hypothesis when it is actually false (i.e., when H1: σ^2 = 25 is true). This probability depends on the value of σ^2 and the threshold t.
To find the probability of false acceptance, we need to compute the power of the test, which is the probability of rejecting the null hypothesis when it is false. The power function is given by:
β(σ²) = P(T > t | σ² = 25)
The distribution of T under H1 is also a chi-squared distribution with 2 degrees of freedom, but with a different scale parameter:
T ~ χ^2(2, nσ^2/25)
Using the non-central chi-squared distribution, we can compute the power function:
β(σ²) = Q(√(n/25)(t - 3.2), 2, δ)
where Q is the complementary cumulative distribution function (CCDF) of the non-central chi-squared distribution, δ = √(n)(20-25)/5, and t is the threshold value.
For t = 5.991, we have:
[tex]δ = √(3)(20-25)/5 = -1.3416\\\\β(16) = Q(√(3/25)(5.991 - 3.2), 2, -1.3416) ≈ 0.1485\\β(25) = Q(√(3/25)(5.991 - 3.2), 2, 0) ≈ 0.4259[/tex]
Therefore, the probability of false acceptance is
P(accept H1 | H1 is false) = β(16) ≈ 0.1485
Note that this is the probability of failing to reject H0 when H1 is true and σ^2 = 25. It is not the probability of accepting H1 when H0 is true and σ^2 = 16, which is 1 - the probability of false rejection.
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Select the correct answer. The sum of two consecutive numbers is 157. This equation, where n is the first number, represents the situation: 2n + 1 = 157. What is the first number? A. 77 B. 78 C. 79 D. 80
With the help of given expression 2n + 1 = 157, the first number is 78.
What exactly are expressions?
In mathematics, an expression is a combination of numbers, symbols, and operators that represent a value. It can be a single term, or it can be a combination of terms connected by operators. For example, 2x + 5 is an expression with two terms connected by the operator +. Expressions can also include functions, variables, and constants.
Now,
Let's assume the first number be x=n. Then the next consecutive number will be x+1.
According to the given information, the sum of the two consecutive numbers is 157.
So, we can write the equation as:
x + (x+1) = 157
Simplifying the equation, we get:
2x + 1 = 157
Subtracting 1 from both sides, we get:
2x = 156
Dividing both sides by 2, we get:
x = 78
i.e. n=78
Therefore, the first number is 78.
So, the correct answer is B) 78.
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Two chords: help Find x
The value of x in the given figure on the basis of chords intersecting in a circle at an anle of 65° and the intercepted arcs being (4x) and (2x+10) is 20.
What is chord?
A chord is a line segment which joins any two points on the circle. If the line segment joins two end points and also passes through the circle then it would be the largest chord which is also called as the diameter. The chord divides the circle into two parts namely the major segment & minor segment.
Here two chords intersect inside the circle at 65° and the measure of the intercepted arcs are given as (4x)° and (2x+10)°
As per the thoerem of circles, the angle of intersection of chords is equal to half of the sum of the values of intercepted arcs.
Angle of intersection= [tex]\frac{1}{2}[/tex] (sum of intercepted arcs)
65° = [tex]\frac{1}{2}[/tex] (sum of (4x)° and (2x+10)°)
65° = [tex]\frac{1}{2}[/tex] ((4x)° + (2x+10)°)
65° = [tex]\frac{1}{2}[/tex] (4x + 2x+10)
65° = [tex]\frac{1}{2}[/tex] (6x+10)
2 (65°) = (6x+10)
130 = 6x + 10
120 = 6x
x = 20
The value of x = 20
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one of the two linear equations in a system is given. the system has exactly one solution. which equation could be the second equation in this system?
The second equation in the system could be 2y - 1/x = 7
We know that the given equation is
y = 1/x + 5
If the system has no solution, then the second equation must be inconsistent with the given equation. In other words, the two equations must represent two lines that do not intersect.
To find such an equation, we need to look for a linear equation that cannot be satisfied simultaneously with y = 1/x + 5. One such equation could be
2y - 1/x = 7
To see why this equation is inconsistent with y = 1/x + 5, let's try to solve the system formed by these two equations
y = 1/x + 5 (equation 1)
2y - 1/x = 7 (equation 2)
Multiplying equation 1 by 2, we get
2y = 2/x + 10
Substituting this into equation 2, we get
2/x + 10 - 1/x = 7
Simplifying this equation, we get
1/x = -3
But this equation has no solution, because there is no value of x that can make 1/x equal to -3. Therefore, the system formed by equations 1 and 2 has no solution.
The second equation is
2y - 1/x = 7
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The given question is incomplete, the complete question is:
y = 1/x + 5
One of the two equations in a linear system is given. The system has no solution. Which equation could be the second equation in this system?
Ben's dad is making a large pot of pasta sauce that calls for 3.5 kilograms of tomatoes. if he triples the amount of tomatoes for the recipe, how many grams of tomatoes will he use?
Answer:
10,500 grams
Step-by-step explanation:
3.5 x 3 = 10.5 kg
To convert kg to grams, multiply by 1000. To multiply by 1,000 move the decimal right 3 places.
10,500 grams
Helping in the name of Jesus.
part 2. design the interior girder supporting the ends of joists [which attach on both sides of the girder] from part 1. how many 2x12s does this girder need to span 12 feet?
A single 2 x 12 may suffice to span 12 feet for light residential loads, but it is crucial to verify this with local building codes and a structural engineer for your specific situation.
To design the interior girder supporting the ends of joists from part 1, we need to follow these steps:
Step 1: Determine the load on the girder
First, we need to find out the total load on the girder.
Assuming the joists are evenly spaced and have equal load, we can calculate the total load on the girder.
Step 2: Select the appropriate size of lumber for the girder
For this question, we are asked to use 2 x 12 lumber for the girder.
A 2 x 12 has a nominal size of 1.5 inches x 11.25 inches.
Step 3: Calculate the required number of 2 x 12s for the girder
To determine how many 2 x 12s are needed to span the 12 feet, we need to consider the strength and stiffness of the girder. To do this, we can refer to span tables or consult a structural engineer.
Based on typical span tables, a single 2x12 can span approximately 12 feet for light residential loads.
However, it is essential to consider local building codes and consult a structural engineer to ensure the correct number of 2 x 12s for your specific situation.
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The line of best fit is y=2x+1.5 where x represents the puppy’s age in weeks and y represents the puppy’s weight. What is the weight of the puppy when it is 15 weeks old?
To find the weight of the puppy when it is 15 weeks old, we need to substitute 15 for x in the equation of the line of best fit:
y = 2x + 1.5
y = 2(15) + 1.5
y = 31.5
Therefore, the weight of the puppy when it is 15 weeks old is 31.5 units, where the units depend on the units used to measure weight (e.g. pounds, kilograms, etc.).
pls help soon I need this asap
Answer:
QPS=110
m<2= 55
m<7=35
m<3=35
m<PQR=70
Step-by-step explanation:
the two lines interesting the rhombus were perfectly cut in half
Find the indefinite integral of:
∫ xsinx^2 (dx)
The indefinite integral of [tex]xsin(x^2)[/tex] with respect to x is [tex]-(1/2)cos(x^2) + C[/tex], where C is the constant of integration.
To evaluate the indefinite integral ∫ [tex]xsinx^2 (dx)[/tex], we used integration by substitution, which is a method for simplifying integrals by substituting a new variable for the original variable.
We can use integration by substitution to evaluate this integral.
Let u = [tex]x^2[/tex], then du/dx = 2x, or equivalently, dx = du/(2x).
Substituting, we have:
∫ [tex]xsinx^2 dx[/tex] = ∫ sin(u) * (du/2)
Now we can integrate with respect to u:
∫ sin(u) * (du/2) = -(1/2)cos(u) + C
We then integrated the resulting expression with respect to u, obtaining -(1/2)cos(u) + C, where C is the constant of integration.
∫ [tex]xsinx^2 dx = -(1/2)cos(x^2) + C[/tex]
Finally, we substituted back u = [tex]x^2[/tex], obtaining [tex]-(1/2)cos(x^2) + C[/tex] as the indefinite integral of [tex]xsinx^2 (dx)[/tex] with respect to x.
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g the size of bass caught in strawberry lake is normally distributed with a mean of 11 inches and a standard deviation of 3 inches. suppose you catch 4 fish. what is the probability the average size of the fish you caught is more than 13 inches?
The probability that the average size of the fish you caught is more than 13 inches is approximately 0.0918, or about 9.18%.
We can use the Central Limit Theorem to approximate the distribution of the sample mean. According to the theorem, the sample mean of a sufficiently large sample will be approximately normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, we have a sample size of 4, which may not be considered sufficiently large, but we can still use the approximation. Thus, the distribution of the sample mean can be approximated as:
mean = 11
standard deviation = 3 / sqrt(4) = 1.5
To find the probability that the average size of the fish you caught is more than 13 inches, we need to standardize the sample mean using the z-score formula:
z = (x - mean) / standard deviation
where x is the sample mean we want to find the probability for. Plugging in the values, we get:
z = (13 - 11) / 1.5 = 1.33
Now, we need to find the probability of getting a z-score greater than 1.33 in a standard normal distribution table or calculator. Using a calculator or statistical software, we find that this probability is approximately 0.0918.
Therefore, the probability that the average size of the fish you caught is more than 13 inches is approximately 0.0918, or about 9.18%.
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for the following problem write the simplest polynomial function with the given zeros: 2, -1, and -8
The simplest polynomial function with the given zeros 2, -1, and -8 is:
f(x) = x³ + 7x² - 6x - 16.
the simplest polynomial function with the given zeros: 2, -1, and -8?Given the roots or zeroes: 2, -1, and -8
Polynomial function f(x) = ?
If the given zeros are 2, -1, and -8, then the corresponding factors of the polynomial function are:
(x - 2), (x + 1), and (x + 8).
The simplest polynomial function with these zeros is the product of these factors:
(x - 2)(x + 1)(x + 8)
Expanding this product gives:
x³ + 7x² - 6x - 16
Hence;
The polynomial function is f(x) = x³ + 7x² - 6x - 16.
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Evaluate the definite integral. Use a graphing utility to verify your result.
The value of the definite integral is -28/3
How to evaluate the definite integralUsing the power rule of integration, we can find the antiderivative of t^2 - 5 as follows:
∫(t^2 - 5) dt = (1/3)t^3 - 5t + C
where C is the constant of integration.
To evaluate the definite integral, we substitute the limits of integration into this expression and take the difference:
∫^-1_1 (t^2 - 5) dt = [(1/3)(1^3) - 5(1)] - [(1/3)(-1^3) - 5(-1)]
= (1/3 - 5) - (1/3 + 5)
= (-14/3) (16/3)
= -28/3
Therefore, the value of the definite integral is -28/3.
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HELP ! find the unit price for a 16-oz, jar of peanut butter for $3.28
Step-by-step explanation:
$ 3.28 / 16 oz = $ .205 per ounce which rounds to 21 cents per ounce
How much yards is the white board the board has 54 inches
54 inches = 1 1
2
yards
Formula: divide the value in inches by 36 because 1 yard equals 36 inches.
So, 54 inches = 54
36
= 1 1
2
or 1.5 yards.
Conversion of 54 inches to other length, height & distance units
54 inches = 0.00137 kilometer
54 inches = 1.37 meters
54 inches = 137 centimeters
54 inches = 13.7 decimeters
54 inches = 1370 millimeters
54 inches = 1.372 × 1010 angstroms
54 inches = 0.000852 mile
54 inches = 3
4
fathom
54 inches = 4 1
2
feet
54 inches = 13 1
2
hands
54 inches = 12 fingers
54 inches = 0.429 bamboo
54 inches = 161 barleycorns
Earl selects 8 toy cars to bring to show and tell at his daycare. These cars make up 40% of Earl's total toy car collection. What is the total number of toy cars that Earl owns?
Enter the correct number in the box.
_____
Answer:
20 cars
Step-by-step explanation:
.4x = 8
x = 20 cars
To verify the answer, 40% of 20 is 8.
Find the area of the shaded region. Round your answer to the nearest hundredth.
Help ASAP please!!!
Check the picture below.
so hmmm the square inscribed in the circle, we can see it as two congruent triangles, whose base is 28 and height is 14. Now, if we get the whole area of the circle with radius of 14, and subtract the area of those triangles, what's leftover is the shaded area.
[tex]\stackrel{ \textit{\LARGE Areas} }{\stackrel{ circle }{\pi (14)^2}~~ - ~~\stackrel{ \textit{two triangles} }{2\left[\cfrac{1}{2}(\underset{b}{28})(\underset{h}{14}) \right]}}\implies 196\pi -392 ~~ \approx ~~ \text{\LARGE 223.75}[/tex]
What is the value of F
Explain
Hint: use your knowledge of either vertical angles or supplementary angles for this one. Set up an equation and solve for F.
Therefore, the value of F is approximately 18.92 degrees.
What is angle?An angle is a geometric figure formed by two rays that share a common endpoint, called the vertex of the angle. The rays are also known as the sides or legs of the angle. Angles are typically measured in degrees or radians and can be classified based on their measure as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (between 90 and 180 degrees), or straight (exactly 180 degrees). Angles are an important concept in geometry and have many real-world applications, including in physics, engineering, and architecture.
Here,
If we look at the diagram, we can see that angles d° and e° are vertical angles, which means they are equal in measure. Similarly, angles (9-86)° and 45° are supplementary angles, which means they add up to 180°. Using this information, we can set up an equation:
d = e (since they are vertical angles)
(9-86) + 45 + e + (6f) + 15 = 180 (since the sum of angles in a triangle is 180°)
Simplifying the equation, we get:
-32 + e + 6f + 60 = 180
e + 6f + 28 = 180
e + 6f = 152
Substituting d = e, we get:
d + 6f = 152
But we know that d + e + (9-86) = 180 (since the sum of angles in a straight line is 180°)
Substituting d = e, we get:
2d + (9-86) = 180
2d = 77
d = 38.5
Substituting d = e = 38.5 in the equation e + 6f = 152, we get:
38.5 + 6f = 152
6f = 113.5
f = 18.92
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a group of 286 students were surveyed about the courses they were taking at their college with the following results: 131 students said they were taking math. 151 students said they were taking english. 158 students said they were taking history. 67 students said they were taking math and english. 55 students said they were taking math and history. 113 students said they were taking english and history. 50 students said they were taking all three courses. how many students took math, english, or history
Using the principle of inclusion-exclusion, 59 students took math only, 21 students took English only, and 40 students took history only.
We can use the principle of inclusion-exclusion to find the number of students who took math, English, or history.
The total number of students who took each course is,
131 (Math) + 151 (English) + 158 (History) = 440
Next, we subtract the number of students taking two courses at a time, since these students have been counted twice in the above totals,
Math and English = 67
Math and History = 55
English and History = 113
We then add back the number of students taking all three courses, since these students have been subtracted twice,
All three courses = 50
Using these values, we can calculate the number of students taking only one course as follows,
Math only = Math - Math and English - Math and History + All three courses = 131 - 67 - 55 + 50 = 59
English only = English - Math and English - English and History + All three courses = 151 - 67 - 113 + 50 = 21
History only = History - Math and History - English and History + All three courses = 158 - 55 - 113 + 50 = 40
Therefore, 59 students took math only, 21 students took English only, and 40 students took history only.
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