Find the volume of the rectangular prism.

The least common multiple is 2^4.3^4.5^2.7^11. The correct choice is option A.

Since the product of the two integers is 2^7.3^8.5^2.7^11 and their greatest common divisor is 23.34.5, then each of the two integers can be expressed as (2^a.3^b.5^c.7^d)(23.34.5) where a,b,c, and d are non-negative integers.

We know that the product of the two integers is 2^7.3^8.5^2.7^11, so (2^a.3^b.5^c.7^d)(23.34.5)(2^e.3^f.5^g.7^h)(23.34.5) = 2^7.3^8.5^2.7^11, where e,f,g, and h are non-negative integers.

Then, we have 2^(a+e).3^(b+f).5^(c+g).7^(d+h).(23.34.5)^2 = 2^7.3^8.5^2.7^11.

Comparing the exponents of the prime factors on both sides, we get:

a+e = 7, b+f = 8, c+g = 2, d+h = 11.

Since the least common multiple is the product of the highest power of each prime factor, we need to find the values of a,b,c,d,e,f,g,h that satisfy the equations above and maximize the exponents of the prime factors.

From the equation a+e = 7, the maximum value of a+e is 7, which is achieved when a = 4 and e = 3.

From the equation b+f = 8, the maximum value of b+f is 8, which is achieved when b = 4 and f = 4.

From the equation c+g = 2, the maximum value of c+g is 2, which is achieved when c = 0 and g = 2.

From the equation d+h = 11, the maximum value of d+h is 11, which is achieved when d = 0 and h = 11.

Therefore, the least common multiple is 2^4.3^4.5^2.7^11, which is option A.

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Answer:

100%

Step-by-step explanation:

There are 8 equally probable outcomes on the spinner, numbered from 1 to 8. Of these, the even numbers are 2, 4, 6, and there are 6 numbers less than 7, namely 1, 2, 3, 4, 5, 6.

To find the probability of the pointer stopping on an even number or a number less than 7, we need to add the probabilities of these two events occurring and subtract the probability of both events occurring at the same time, since this would lead to double counting:

P(even or less than 7) = P(even) + P(less than 7) - P(even and less than 7)

P(even) = 3/8, since there are 3 even numbers on the spinner out of 8 total outcomes.

P(less than 7) = 6/8, since there are 6 numbers less than 7 on the spinner out of 8 total outcomes.

P(even and less than 7) = 1/8, since only 4 satisfies both conditions (even and less than 7) out of 8 total outcomes.

Therefore, substituting these values, we get:

P(even or less than 7) = 3/8 + 6/8 - 1/8

P(even or less than 7) = 8/8 = 1

So the probability that the pointer will stop on an even number or a number less than 7 is 1 or 100%.

Hope this helps!

The amount of fencing, dirt and sod for the baseball field are: length of Fencing & 1410.5 ft. Area of the sod ≈ 118017.13ft² Area of the field covered with distance ≈ 7049.6ft²

Area of a circle = πr²

Circumference of a circle = 2πr

where r is the radius of the circle

The area of a Quarter of a circle is therefore;

Area of a circle/ 4

The perimeter of a Quarter of a Circle is;

The perimeter of a circle/4

Fencing = ¼ x 2 x π x 380 + 2 x 15 +2 x 380 + ¼ x 2 x π x 15

Fencing = 197.5π + 190π = 1410.5 feet.

Grass =

π/4 x (380 - 6)² + 87 ² - π/4 × (87 + 30)² + 2 x 380 x 15 + π/4 x 15² - (3/4) x π x 10² - 25π

= 31528π + 18969 = 118017.13

The area Covered by the sod is about 118017.13Sq ft.

Dirt = π/4 x 380 ² - π/4 x (380 - 6)² + π/4 (87 + 30)² - 87² + π100 = (18613π - 30276)/4

= 7049.6

Therefore, the area occupied by the dirt is about 7049.6 Sq ft.

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The length of the third side of this isosceles triangle is 2 units.

We have,

If two sides of an isosceles triangle are equal, then the third side must also be equal in length.

So,

If two sides of the triangle are 2 and 2, the length of the third side must also be 2.

Thus,

The length of the third side of this isosceles triangle is 2 units.

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Answer:

104m²

Step-by-step explanation:

area trapezium: ((Major base(bc)+ Minor base(ad))*height(ab))/2

area trapezium: [(16+10)*8]/2

(26*8)/2

208/2

104m²

Answer:

52.7 g

Step-by-step explanation:

We are given;

Initial mass of the element is 310 g

Rate of decay 8.9% per minute

Time for the decay 19 minutes

We are required to determine the amount of the element that will remain after 19 minutes.

We can use the formula;

New mass = Original mass × (1-r)^n

Where n is the time taken and r is the rate of decay.

Therefore;

Remaining mass = 310 g × (1-0.089)^19

                           = 52.748 g

                           = 52.7 g (to the nearest 10th)

Thus, the mass that will remain after 9 minutes will be 52.7 g

The value x may be expressed as (a - b)(ab + b) / (ab - 1)(ab + 1).

To simplify the given expression x = (a² + b²) / (a b + 1), start by multiplying both the numerator and the denominator by (a b - 1) as follows:

x = (a² + b²)(a b - 1) / (a b + 1)(a b - 1)

Expanding the numerator using the distributive property:

x = (a² b - a² + a b² - b²) / (a² b - a b + a b² - 1)

Rearranging the terms in the numerator:

x = (a² b + a b² - a² - b²) / (a² b - a b + a b² - 1)

Factoring the numerator:

x = [(a² b - a b) + (a b² - b²)] / (a²b - a b + a b² - 1)

x = [a b (a - b) + b²(a - b)] / (a b - 1)(a b + 1)

x = (a - b)(a b + b) / (a b - 1)(a b + 1)

Therefore, the simplified expression for x is (a - b)(ab + b) / (ab - 1)(ab + 1).

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The solid described by the given inequalities in spherical coordinates is a spherical cap. It is bounded by the spherical coordinates (2, 0, π), (2, π/4, π), (3, 0, 2π), and (3, π/4, 2π). The spherical cap can be visualized by connecting these points and plotting the points inside the boundaries.

Coordinates are the set of two or three numbers used to locate a point in space, in a two-dimensional plane or in a three-dimensional space. Coordinates are usually expressed as either latitude and longitude, or as x-y-z values. Coordinates are used to plot the location of points of interest on a map, or to plot the path of an object in motion.

This solid can be sketched as a spherical cap in spherical coordinates. The spherical cap is a portion of a sphere that is cut off by a plane. The boundary of the spherical cap is described by the inequalities given.

The spherical coordinates are defined by three parameters: rho, phi, and theta. The parameter rho is the radial distance from the origin, phi is the angle measured in the xy-plane from the positive x-axis, and theta is the angle measured from the positive z-axis.

In this case, the spherical cap is bounded by the inequalities 2 ≤ rho ≤ 3, 0 ≤ phi ≤ π/4, and π ≤ θ ≤ 2π. The spherical cap is defined as the portion of the sphere that lies between the two planes defined by these inequalities.

The solid is bounded by the following spherical coordinates: (2, 0, π), (2, π/4, π), (3, 0, 2π), and (3, π/4, 2π). The solid can be sketched by connecting these points and plotting the points inside the boundaries.

The spherical cap is a portion of a sphere that is bounded by two planes. The two planes intersect at the boundary of the solid, which is described by the inequalities given. The spherical cap is a portion of the sphere that is cut off by the planes and is bounded by the spherical coordinates given.

In conclusion, the solid described by the given inequalities in spherical coordinates is a spherical cap. It is bounded by the spherical coordinates (2, 0, π), (2, π/4, π), (3, 0, 2π), and (3, π/4, 2π). The spherical cap can be visualized by connecting these points and plotting the points inside the boundaries.

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a) the equation in terms of x and y is [tex]y = \sqrt(x) + 2.[/tex]

b) The positive orientation is the direction in which the parameter t increases, which corresponds to moving from left to right along the parabola. So the positive orientation is to the right.

a. To eliminate the parameter t, we can use the fact that [tex]x = (t+5)^2[/tex]. Solving for t, we get[tex]t = \sqrt(x) - 5.[/tex]Substituting this into the equation for y, we get[tex]y = \sqrt(x) - 5 + 7,[/tex] which simplifies to y = sqrt(x) + 2. Therefore, the equation in terms of x and y is [tex]y = \sqrt(x) + 2.[/tex]

b. The curve described by these parametric equations is a parabola that opens to the right. The positive orientation is the direction in which the parameter t increases, which corresponds to moving from left to right along the parabola. So the positive orientation is to the right.

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Answer:Since Tom and Kimberly are moving in opposite directions, they will cross paths at some point. Let's call the distance they will cover before they meet each other "x".

We can set up an equation to represent this:

x + (100 - x) = 100

Simplifying this equation, we get:

2x = 100 - x

Solving for x, we get:

x = 33.33 miles

This means that they will meet each other after traveling 33.33 miles from their respective homes. The time it takes to travel this distance can be calculated using the formula:

time = distance / speed

For Tom, the time taken to travel 33.33 miles at 65 mph is:

time = 33.33 / 65 = 0.5123 hours

Converting this to minutes, we get:

time = 0.5123 * 60 = 30.74 minutes

Similarly, for Kimberly, the time taken to travel 66.67 miles at 60 mph is:

time = 66.67 / 60 = 1.1111 hours

Converting this to minutes, we get:

time = 1.1111 * 60 = 66.67 minutes

Adding 20 minutes for loading and unloading at each home, the total time for each one-way trip is:

Tom: 30.74 + 20 + 20 = 70.74 minutes

Kimberly: 66.67 + 20 + 20 = 106.67 minutes

Since they are making five one-way trips, the total time for the move is:

Tom: 5 * 70.74 = 353.7 minutes

Kimberly: 5 * 106.67 = 533.35 minutes

To find out what time they will finish the move, we need to add the total time for the move to the time they started, which was 7 am. Let's convert the total time to hours:

Tom: 353.7 / 60 = 5.895 hours

Kimberly: 533.35 / 60 = 8.889 hours

Adding these times to 7 am, we get:

Tom: 7 am + 5.895 hours = 12:53 pm (rounded to the nearest minute)

Kimberly: 7 am + 8.889 hours = 3:53 pm (rounded to the nearest minute)

Therefore, they will finish the move at 12:53 pm and 3:53 pm, respectively.

the approximate value of[tex]e^{0.1243}[/tex] is 1.124. with three significant digits after the decimal.

The equation of a tangent line serves as the foundation for the linear approximation formula. We are aware that the derivative of a tangent drawn to the curve y = f(x) at the point x = an is given by its slope at that location. In other words, f'(a) is the slope of the tangent line. As a result, the linear approximation formula uses derivatives.

To approximate[tex]f(x) = e^x[/tex] at x = 0.1243 using linear approximation, we can use the formula:

[tex]f(x) = f(a) + f'(a)(x - a)[/tex]

For[tex]f(x) = e^x[/tex], we have [tex]f'(x) = e^x.[/tex] Since we're approximating at x = 0, a = 0. Thus,[tex]f(0) = e^0 = 1,[/tex]and f'(0) = e^0 = 1.

Using the linear approximation formula:

f(0.1243) ≈ 1 + 1(0.1243 - 0)

f(0.1243) ≈ 1 + 0.1243

f(0.1243) ≈ 1.124

So, the approximate value of[tex]e^{0.1243}[/tex] is 1.124.with three significant digits after the decimal.

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The probability that we reject 0 when it is true is equal to the chosen significance level (α).

The probability that we reject 0 when, in fact, 0 is true is known as the Type I error rate, or the false positive rate. In hypothesis testing, this probability is represented by the significance level, which is denoted by the Greek letter alpha (α). The significance level is a predetermined threshold, typically set at 0.05 or 5%. If the calculated p-value is less than the significance level (α), we reject the null hypothesis (0) even if it is true. So, the probability that we reject 0 when it is true is equal to the chosen significance level (α).

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(a) The hazard rate function for the random variable Y is λ. (b) An algorithm for generating the random variable Y from a uniform random variable in the interval (2,5) is y = -ln(1 - U) / λ. (c) The value for which the mean of the random variable Y is 5 is 1/5.

(a) For an exponentially distributed random variable, the hazard rate function is given by:

h(y) = fx(y)/[1 - Fx(y)]

where fx(y) is the PDF of Y and Fx(y) is the cumulative distribution function (CDF) of Y.

For,

Fx(y) = 1 - e^(-λy)

and

fx(y) = λe^(-λy)

So,

h(y) = λe^(-λy) / [1 - (1 - e^(-λy))] = λ

Therefore, the hazard rate function for the random variable Y is constant and equal to λ.

(b) Using the inverse transform method. CDF of Y is:

Fx(y) = 1 - e^(-λy)

Now,

1 - e^(-λy) = U

e^(-λy) = 1 - U

-λy = ln(1 - U)

y = -ln(1 - U) / λ

Generate value of U from uniform distribution on interval (0,1), and then transform U into Y.

(c) The mean of an exponentially distributed random variable with parameter λ is:

E(X) = 1/λ

Therefore, to choose a value for the parameter λ so that the mean of the random variable Y is 5:

E(Y) = E(X) = 1/λ = 5

Solving for λ, we get:

λ = 1/5

Therefore, we can choose the parameter λ = 1/5 so that the mean of the random variable Y is 5.

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A basis for the set of vectors in R2 on the line y = 19x is {(1, 19)}.

To find a basis for the set of vectors in R2 on the line y = 19x, we need to find a vector that lies on the line and can represent any other vector on the line through scalar multiplication.

1. Choose a point on the line y = 19x. Let's choose the point (1, 19) since when x = 1, y = 19(1) = 19.
2. Create a vector from the origin to the chosen point. The vector would be v = (1, 19).
3. Verify that this vector lies on the line. The equation of the line is y = 19x, and our vector v = (1, 19) satisfies this equation since 19 = 19(1).

So, a basis for the set of vectors in R2 on the line y = 19x is {(1, 19)}. Any other vector on the line can be represented as a scalar multiple of this basis vector.

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A partial order is a relation that is reflexive, antisymmetric, and transitive.

(a) To determine if R is a partial order on A, we need to check if it satisfies the following properties:
1. Reflexivity: Every element is related to itself.
2. Antisymmetry: If a is related to b and b is related to a, then a = b.
3. Transitivity: If a is related to b and b is related to c, then a is related to c.

A = {a, b, c}, R = {(a, a), (b, a), (b, b), (b, c), (c, c)}

1. Reflexivity: (a, a), (b, b), and (c, c) are in R. So, it is reflexive.
2. Antisymmetry: There are no pairs (a, b) and (b, a) with a ≠ b in R. So, it is antisymmetric.
3. Transitivity: We have (b, a) and (b, c) in R, but there is no (a, c) in R. Therefore, R is not transitive.

Since R is not transitive, R is not a partial order on A.

(b) The relation R on A = R (the set of real numbers) is not a partial order since it does not satisfy antisymmetry. For any two distinct real numbers x and y, either (x, y) or (y, x) (or both) will be in R. Therefore, R cannot be antisymmetric, and thus, it is not a partial order on R.

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We cannot definitively determine whether the interval (27.336, 33.064) covers the true mean based on the information provided. However, we can say that there is a 97% probability that the true mean falls within this interval. This is because the given interval is a 97% confidence interval, which means that if we were to take repeated samples of 97 students from the same population and construct 97% confidence intervals for each sample, approximately 97% of these intervals would contain the true mean.

Therefore, we cannot say for certain whether the true mean is within the given interval, but we can be highly confident that it is. Additionally, we should keep in mind that the College Board only analyzed a sample of 97 students, so there is some uncertainty and potential for sampling error in the estimation of the true mean.

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Therefore, the general solution is:

y(x) = c₁e^(-2x)cos(√6x) + c₁e^(-2x)sin(√6x)

or

y(x) = c₁e^(-2x)(cos(√6x) + sin(√6x))

where c₁ is an arbitrary constant and x is the independent variable.

The given differential equation is y'' - y' + 5y' - 5y = 0. To find the general solution, we first find the characteristic equation:

r² - r + 5r - 5 = 0

Simplifying, we get:

r² + 4r - 5 = 0

Using the quadratic formula, we get:

r = (-4 ± √(4² + 4(1)(5))) / 2
r = (-4 ± √36) / 2
r₁ = -2 - √6, r₂ = -2 + √6

Therefore, the general solution is:

y(x) = c₁e^(r₁x) + c₂e^(r₂x)

Substituting the values of r₁ and r₂, we get:

y(x) = c₁e^(-2-√6)x + c₂e^(-2+√6)x

Simplifying, we get:

y(x) = c₁e^(-2x)e^(-√6x) + c₂e^(-2x)e^(√6x)

Using Euler's formula, we can simplify further:

y(x) = c₁e^(-2x)(cos(√6x) - i sin(√6x)) + c₂e^(-2x)(cos(√6x) + i sin(√6x))

Separating the real and imaginary parts, we get:

y(x) = c₁e^(-2x)cos(√6x) + c₂e^(-2x)cos(√6x) + i(c₁e^(-2x)sin(√6x) - c₂e^(-2x)sin(√6x))

Since the differential equation is real-valued, the imaginary part must be zero. Therefore, we have:

c₁e^(-2x)sin(√6x) = c₂e^(-2x)sin(√6x)

Since sin(√6x) cannot be zero for all x, we must have:

c₁ = c₂ = c₃

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The orthogonal trajectories of the family of curves x² + 2y² = k² are given by the equation x² = K²y⁴.

To find the orthogonal trajectories of the family of curves x² + 2y² = k², follow these steps:

1. Write the given equation as a function: x² + 2y² = k².
2. Differentiate the equation implicitly with respect to x: 2x + 4y(dy/dx) = 0.
3. Solve for dy/dx: dy/dx = -2x / (4y) = -x / (2y).
4. Replace dy/dx with -dx/dy to obtain the orthogonal trajectory: -dx/dy = -x / (2y).
5. Simplify the equation: dx/dy = x / (2y).
6. Separate the variables: dx/x = 2dy/y.
7. Integrate both sides: ∫(1/x)dx = 2∫(1/y)dy.
8. Obtain the integrals: ln|x| = 2ln|y| + C.
9. Remove the natural logarithm by raising e to the power of both sides: |x| = [tex]|y|^2 * e^C[/tex].
10. Introduce a new constant K, where K = [tex]e^C: |x| = K|y|^2[/tex].
11. Eliminate the absolute values by squaring both sides: x² = K²y⁴.

The orthogonal trajectories of the family of curves x² + 2y² = k² are given by the equation x² = K²y⁴.

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Answer: d x+2x+2x=36

Step-by-step explanation:

Answer:

5 %

Step-by-step explanation:

Answer:

The probability of the 4th donation being Type O or Type B is:

P(Type O or B) = P(Type O) + P(Type B) = 0.49 + 0.20 = 0.69

The probability of the 4th donation being safe for someone with Type B blood is the probability that it is Type O or Type B, which is 0.69. Therefore, the probability that the 4th donation selected at random can be safely used in a blood transfusion on someone with Type B blood is:

P(safe for Type B) = 0.69

Answer: (0.69)³(0.31)

By choosing the annual contract and breaking it after 5 months, your parents would lose $574.00.

If you enter into an annual contract at $467.00/month and break it after 5 months, you would have paid:

= $467.00 x 5

= $2,335.00

Since breaking the annual contract incurs a penalty of 2 months' rent, your parents would need to pay an additional of:

= $467.00 x 2

= $934.00

If parents opted for the month-to-month contract at $539.00/month, the total cost for 5 months would be:

= $539.00 * 5 month

=  $2,695.00.

So, by choosing the annual contract and breaking it after 5 months, your parents would lose:

= $3,269.00 - $2,695.00

= $574.00.

Full question "Your parents are considering renting you an apartment instead of paying room and board at your college. The month-to-month contract is $539.00/month and the annual contract is $467.00/month. If you break the annual contract, there is a 2-month penalty. If you enter into an annual contract but decide to leave after 5 months, how much do your parents lose by not doing the month-to-month contract."

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Answer:

D has the following vertices

The probability P(A or B) is 3/5 or 0.6. Therefore, the probability of selecting a prime number or an odd number is 3/5 or 0.6.

To calculate the probability P(A or B), we first need to determine the number of outcomes for each event and the total number of outcomes in the experiment.
Event A: Obtaining a prime number.
Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The prime numbers between 0 and 9 are 2, 3, 5, and 7. So, there are 4 prime numbers in this range.
Event B: Obtaining an odd number.
Odd numbers are numbers that cannot be divided evenly by 2. The odd numbers between 0 and 9 are 1, 3, 5, 7, and 9. So, there are 5 odd numbers in this range.
Since 3, 5, and 7 are both prime and odd numbers, we must account for this overlap, so we subtract these three from the total.
Total number of outcomes (digits 0 through 9) = 10
Total outcomes of A or B = (prime numbers) + (odd numbers) - (overlap) = 4 + 5 - 3 = 6
Now, we calculate the probability P(A or B) as the ratio of the total outcomes of A or B to the total number of outcomes in the experiment:
P(A or B) = (Total outcomes of A or B) / (Total number of outcomes) = 6/10 = 3/5
So, the probability P(A or B) is 3/5 or 0.6.

To solve this problem, we need to first identify the prime numbers and odd numbers among the digits 0 through 9:
Prime numbers: 2, 3, 5, 7
Odd numbers: 1, 3, 5, 7, 9
We can see that the numbers 3, 5, and 7 are both prime and odd, so we need to be careful not to count them twice when calculating the probability of events A or B.
To find the probability of event A (obtaining a prime number), we count the number of prime numbers among the digits 0 through 9, which is 4. The probability of selecting a prime number is therefore 4/10 or 2/5.
To find the probability of event B (obtaining an odd number), we count the number of odd numbers among the digits 0 through 9, which is 5. The probability of selecting an odd number is therefore 5/10 or 1/2.
To find the probability of event A or B (obtaining a prime number or an odd number), we need to add the probabilities of the two events and then subtract the probability of selecting both a prime and an odd number (i.e., the probability of selecting 3, 5, or 7):
P(A or B) = P(A) + P(B) - P(A and B)
         = 2/5 + 1/2 - 3/10
         = 4/10 + 5/10 - 3/10
         = 6/10
         = 3/5
Therefore, the probability of selecting a prime number or an odd number is 3/5 or 0.6.

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Therefore, the domain of f(x) = 2ˣ is: All real numbers And the range of f(x) = 2ˣ is: y > 0.

We must look for the set of all possible values of x that do not result in the function being undefined in order to determine the domain of the function y = f(x). The usual examples are taking the square root of negative integers, dividing by 0, etc.

The given exponential function is f(x) = 2ˣ.

The domain of an exponential function is all real numbers, since any real number can be raised to a power.

The range of the function is all positive real numbers, since 2 raised to any power will always be positive and approach zero as x approaches negative infinity.

Therefore, the domain of f(x) = 2ˣ is: All real numbers

And the range of f(x) =2ˣ is: y > 0

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The probability of not landing on a yellow section when spinning the spinner once is 75%.

Option A is correct

The spinner has eight sections, two of which are yellow. Therefore, the probability of landing on a yellow section is:

P(yellow) = 2/8 = 1/4 = 0.25

To determine the probability of not landing on a yellow section, we can use the complement rule:

P(not yellow) = 1 - P(yellow)

P(not yellow) = 1 - 0.25

P(not yellow) = 0.75 or 75%

Therefore, the probability of not landing on a yellow section when spinning the spinner once is 75%.

Option A is correct

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The lateral surface area of the cone is 20π square units.

To find the lateral surface area of a right circular cone generated by revolving the region bounded by y = 3x/4, y = 3, and x = 0 about the y-axis, we need to follow these steps,

1. Find the height and slant height of the cone.
2. Use the formula for the lateral surface area of a cone: LSA = πr * l, where r is the radius and l is the slant height.

Find the height and slant height of the cone.
The equation of the line is y = 3x/4. We are given that y = 3, so we can solve for x:
3 = 3x/4
x = 4

Thus, the height (h) of the cone is 3, and the base radius (r) is 4. To find the slant height (l), we can use the Pythagorean theorem:
l² = h² + r²
l² = 3² + 4²
l² = 9 + 16
l² = 25
l = 5

Use the formula for the lateral surface area of a cone.
LSA = πr * l
LSA = π(4) * (5)
LSA = 20π

The lateral surface area of the cone is 20π square units.

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The equation representing the new price with the 28% markup is y = 1.28p.

The equation that represents the new price of a bag, y, given an original price, p, with a markup rate of 28% is:

y = 1.28p

This equation is derived as follows:

Convert the markup rate to a decimal by dividing by 100:

28% / 100 = 0.28

Add 1 to the decimal markup rate:

1 + 0.28 = 1.28

Multiply the original price by the result:

y = p × 1.28

So, the equation representing the new price with the 28% markup is y = 1.28p.

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Therefore, we need to prove the set of facts in option C: [tex]2^3 > 4^1, 3^3 > 4^2, and 4^3 > 4^3[/tex] (which is always true since any positive number raised to the power of 3 is greater than the same number raised to any power less than 3).

The theorem states that for any positive integer x less than 4, (x+1)³ > 4x.

To prove this theorem by exhaustion, we need to consider all possible values of x less than 4 and show that the inequality (x+1)³ > 4x holds for each of these values.

The possible values of x are 1, 2, and 3. Therefore, we need to prove the following three facts:

1³ > 4(0) (when x=1, the inequality becomes (1+1)³ > 4(1), which simplifies to 8 > 4, which is true)

2³ > 4(1) (when x=2, the inequality becomes (2+1)³ > 4(2), which simplifies to 27 > 8, which is true)

3³ > 4(2) (when x=3, the inequality becomes (3+1)³ > 4(3), which simplifies to 64 > 12, which is true)

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Each graph identified above are described below.

For the fundamental function h(x) = 2x:

f(x) = -h(  x) represents te x-axis graph of h(x). When C   is greater than 0, the f(x ) graph is always below the x-axis and approaches 0 as x approaches negative infinity. The graph of f( x) approaches negative infinity as x approaches positive infinity.

As a result, for C > 0, the f(x) graph is always declining and concave down.

g( x) = h(x - 0) moves the h(x) graph to the right by 0 units. When C is 0, the g(x) graph is always above the x-axis and approaches 0 as x approaches positive infinity. The graph of g( x) approaches positive infinity as x approaches negative infinity.

As a result, for C 0, the g(x) graph is constantly growing and concave up.

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