An architect wants to draw a rectangle with a diagonal of 25inches. The length of the rectangle is to be 10 inches more than twice the width. What dimensions should she make the rectangle?

Answer:

students need to produce a prime factorization of 48. Donna states that the first factors in the tree should be 6 and 8. Larry states that the first factors in the tree should be 4 and 12. Trish states that the initial factors of 48 do not affect the prime factorization. Explain why Trish is correct.Trish is correct because the initial factors of 48 do not affect the prime factorization. Prime factorization is the process of breaking down a composite number into its prime factors. The prime factors of a number are those factors that are prime numbers. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3, which means that 48 can be expressed as a product of these prime factors. The order in which we choose the initial factors to start the prime factorization does not affect the result, as long as we continue to break down the resulting factors into their prime factors until we cannot break them down any further. Therefore, Trish is correct that the initial factors of 48 do not affect the prime factorization.

Trish is correct because the prime factorization of 48 will be the same regardless of which factors are chosen as the initial factors in the tree.

To see why, let's look at the prime factorization of 48:

48 = 2 × 2 × 2 × 2 × 3

No matter which factors are chosen as the initial factors, the same prime factors will eventually be found.

For example, if Donna's method is used, we could start with 6 and 8:

6 = 2 × 3

8 = 2 × 2 × 2

Then we could continue to factor each of these numbers until we reach prime factors:

6 = 2 × 3

8 = 2 × 2 × 2

= 2 × 2 × 2 × 3

= 2³ × 3

Now we have found all of the prime factors of 6 and 8, and we can combine them to get the prime factorization of 48:

48 = 2 × 2 × 2 × 2 × 3

If Larry's method is used, we could start with 4 and 12:

4 = 2 × 2

12 = 2 × 2 × 3

Then we could continue to factor each of these numbers until we reach prime factors:

4 = 2 × 2

12 = 2 × 2 × 3

= 2² × 3

Now we have found all of the prime factors of 4 and 12, and we can combine them to get the prime factorization of 48:

48 = 2 × 2 × 2 × 2 × 3

As we can see, the same prime factors are found regardless of which factors are chosen as the initial factors in the tree. Therefore, Trish is correct that the initial factors of 48 do not affect the prime factorization.

A equals (x+3) and b equals (x-3) the statement that is true for the two division problems is that a equals factor (x+3) and b equals (x-3).

For the first division problem, [tex](x^2-3x-18)[/tex]÷(x-6), the quotient is (x+3). To solve for the quotient, you first need to factor the numerator, which is (x-6)(x+3). Then, you need to divide the numerator by the denominator, which is (x-6). The quotient is (x+3). For the second division problem, [tex](x^3-x^2-5x-3)/(x^2+2x+1)[/tex], the quotient is (x-3). To solve for the quotient, you first need to factor the numerator, which is (x+1)(x-3). Then, you need to divide the numerator by the denominator, which is [tex](x^2+2x+1)[/tex]. The quotient is (x-3). Therefore, the statement that is true for the two division problems is that a equals (x+3) and b equals (x-3).

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A. The derivative of f'(x) = 26x + 5is 26x + 5

B. Slope at x = 2 is 57

Slope at x = 3 is 83

c. The equation of the tangent line at x = 3 is y - 122 = 83(x - 3), or y = 83x - 179.

D. The value of x where the tangent line is horizontal is x = -5/26.

(A) f'(x) = 26x + 5.

It should be noted that to find the derivative of f(x), we apply the power rule and the constant multiple rule:

f'(x) = d/dx (13x²+5x)

= d/dx (13x²) + d/dx (5x)

= 26x + 5

(B) ain this case, to find the slope of the graph of f(x) at x = 2 and x = 3, we plug these values into the derivative:

Slope at x = 2: f'(2) = 26(2) + 5 = 57

Slope at x = 3: f'(3) = 26(3) + 5 = 83

(C) Based on the information, to find the equation for the tangent line at x = 2 and x = 3, we use the point-slope form of a line:

Tangent line at x = 2:

We know the slope is 57, and the point (2, f(2)) is on the line.

Plugging in x = 2 to f(x) gives us f(2) = 13(2)² + 5(2) = 58.

So the equation of the tangent line at x = 2 is y - 58 = 57(x - 2), or y = 57x - 56.

Tangent line at x = 3:

We know the slope is 83, and the point (3, f(3)) is on the line.

Plugging in x = 3 to f(x) gives us f(3) = 13(3)² + 5(3) = 122.

So the equation of the tangent line at x = 3 is y - 122 = 83(x - 3), or y = 83x - 179.

D. In order to find where the tangent line is horizontal, we need to find where the slope is equal to zero. Setting f'(x) = 0 and solving for x gives:

f'(x) = 26x + 5 = 0

x = -5/26

Therefore, the only value of x where the tangent line is horizontal is x = -5/26.

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Answer:

hope this helps

Step-by-step explanation:

(a) Using the product rule for h(x) = f(x)g(x), h'(4) = 24

(b) Using the quotient rule, for h(x) = g(x) / (f(x) + g(x)), h'(4) = -33/49.

(c) Using chain rule for h(x) = f(g(x)), h'(4) = 80.

We can use the product rule, quotient rule, and chain rule to find the derivatives of the functions h(x).

(a) h(x) = f(x)g(x)

Using the product rule, we have:

h'(x) = f'(x)g(x) + f(x)g'(x)

At x = 4, we have:

h'(4) = f'(4)g(4) + f(4)g'(4) = 6(5) + 2(-3) = 24

Therefore, h'(4) = 24.

(b) h(x) = g(x) / (f(x) + g(x))

Using the quotient rule, we have:

h'(x) = [g'(x)(f(x) + g(x)) - g(x)f'(x)] / (f(x) + g(x))^2

At x = 4, we have:

h'(4) = [g'(4)(f(4) + g(4)) - g(4)f'(4)] / (f(4) + g(4))^2

= [(-3)(2 + 5) - 5(6)] / (2 + 5)^2

= -33 / 49

Therefore, h'(4) = -33/49.

(c) h(x) = f(g(x))

Using the chain rule, we have:

h'(x) = f'(g(x))g'(x)

At x = 4, we have:

h'(4) = f'(g(4))g'(4)

= f'(5)(10)

= 8(10)

= 80

Therefore, h'(4) = 80.

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Using the derivative of the velocity, the average value of the position function from t = 2 seconds to t = 5 seconds is 14.5.

Given:

Acceleration function, a(t) = 2sin(t) + 1

Initial velocity, v(0) = 0

Initial position, s(0) = 3

To find:

Velocity function, v(t)

Position function, s(t)

Average value of the position function from t = 2 seconds to t = 5 seconds

Solution:

We know that acceleration is the derivative of velocity, and velocity is the derivative of position. So we can find the velocity and position functions by integrating the acceleration function.

Velocity function:

[tex]v(t) = \int a(t) dt\\v(t) = \int (2sin(t) + 1) dt\\v(t) = -2cos(t) + t + C1[/tex]

We know that the initial velocity, v(0) = 0. Substituting this value in the above equation, we get:

[tex]0 = -2cos(0) + 0 + C1\\C1 = 2[/tex]

Therefore, the velocity function is:

[tex]v(t) = -2cos(t) + t + 2[/tex]

Position function:

[tex]s(t) = \int v(t) dt\\s(t) = \int (-2cos(t) + t + 2) dt\\s(t) = 2sin(t) + \frac{1}{2} t^2 + 2t + C2[/tex]

We know that the initial position, s(0) = 3. Substituting this value in the above equation, we get:

[tex]3 = 2sin(0) + 0 + 0 + C2\\C2 = 3\\[/tex]

Therefore, the position function is:

[tex]s(t) = 2sin(t) + \frac{1}{2} t^2 + 2t + 3[/tex]

Average value of the position function from t = 2 seconds to t = 5 seconds:

We can find the average value of the position function using the following formula:

[tex]Avg = (1/(b-a)) * \int(a,b) f(t) dt[/tex]

Here, a = 2 and b = 5. So, substituting the values in the above formula, we get:

[tex]Avg = (1/(5-2)) * \int(2,5) (2sin(t) + \frac{1}{2} t^2 + 2t + 3) dt\\Avg = \frac{1}{3} * [ -2cos(t) + 1/6 t^3 + t^2 + 2t ] \eval(2,5)\\[/tex]

[tex]Avg = \frac{1}{3} * [ (-2cos(5) + 1/6 (5^3) + 5^2 + 25) - (-2cos(2) + 1/6 (2^3) + 2^2 + 22) ]\\Avg = 14.5[/tex]

Therefore, the average value of the position function from t = 2 seconds to t = 5 seconds is 14.5.

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Answer:

Step-by-step explanation:

As seen in earlier sections, the process of completing the square is a useful tool in finding noninteger values of quadratic equations, especially intercepts. When a quadratic equation of the

form f (x) = ax2

+ bx + c is put through the process of completing the square it yields an

equation of the form f (x) = a(x – h)2

+ k . The conversion of the equation to this form will

yield critical information about the equation’s characteristics before you begin to graph it.

1.) The value of h is the distance left (if negative) or right (if positive) the graph

translates from the standard position.

2.) The value of k is the distance up (if positive) or down (if negative) the graph

translates from the standard position.

3.) The values of h and k, when put together as an ordered pair, give the vertex i.e.

(h, k).

4.) The equation x = h is the formula for the axis of symmetry.

The following example demonstrates how to find the following critical information of the

equation:

a.) vertex

b.) axis of symmetry

c.) y intercept (if any)

d.) x intercepts (if any)

Example 1: Find the vertex, axis of symmetry, x-intercept(s), and y-intercept and gr

Step-by-step explanation:

Let's assume that the monthly income of the family is x.

From the problem statement, we know that the ratio of monthly income to the monthly saving is 9:2.

Therefore, we can write:

x/4320 = 9/2

To solve for x, we can cross-multiply:

2x = 9*4320

2x = 38,880

x = 19,440

So, the monthly income of the family is Rs 19,440.

To find the monthly expenditure, we can subtract the monthly savings from the monthly income:

Monthly expenditure = Monthly income - Monthly saving

Monthly expenditure = 19,440 - 4,320

Monthly expenditure = 15,120

Therefore, the monthly expenditure of the family is Rs 15,120

Using the equations, the particular solution is y_p(x)=-3x+1+2e⁻ˣ.

As we already know, homogeneous equations have zero on the right side of the equation. Thus, it is said that non-homogenous differential equations are those with a function on the right side of their equation.

We are given a non-homogeneous differential equation in the form:

y′′+3y′-3y=3xe^-1​

The differential equation is called non-homogeneous because it is known to have a non-zero right-hand side.

To solve this differential equation, we first need to find the complementary function, which is the solution to the corresponding homogeneous differential equation y′′+3y′-3y=0.

We then take the first and second derivatives of y_p(x) and substitute them into the differential equation, y′′+3y′-3y=3xe^-1​ and simplify.

This leads to the system of equations:

a + 3c = 3

a + 3b - 3c = 0

a + b + 3c = 1

Solving this system of equations, we find that a=-3, b=1, and c=2. Therefore, the particular solution is y_p(x)=-3x+1+2e⁻ˣ.

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This set of equations can be solved, and the results show that a=-3, b=1, and c=2. As a result, the specific answer is [tex]y_{p(x)} =-3x+1+2e^{-x}[/tex].

Homogeneous equations, as we already know, have "0" on the right side of the equation. Therefore, it is said that differential equations with a function on the right side of the equation are non-homogenous differential equations.

A non-homogeneous differential equation of the following shape is provided to us:

[tex]y^{''} +3y^{'} -3y=3xe^{-1}[/tex]

Since the right-hand side of the differential equation is known to be non-zero, it is referred to as non-homogeneous.

Finding the complementary function, which is the answer to the related homogeneous differential equation [tex]y^{''} +3y^{'} -3y=0[/tex], is the first step in solving this differential equation.

The differential equation [tex]y^{''} +3y^{'} -3y=3xe^{-1}[/tex] is then simplified by taking the first and second derivatives of [tex]y_{p(x)}[/tex] and substituting them into

the equation.

This results in the formulae system:

a + 3c = 3

a + 3b - 3c = 0

a + b + 3c = 1

This set of equations can be solved, and the results show that a=-3, b=1, and c=2. As a result, the specific answer is  [tex]y_{p(x)} =-3x+1+2e^{-x}[/tex].

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The area of the triangle is 3 square units whose vertices are (-3,1), (1,1) and (1,4). We will use distance formulae in this question.

To find the area of a triangle whose vertices are (-3,1), (1,1), and (1,4), we can use the formula for the area of a triangle:

Area = 1/2 * base * height

where the base and height are perpendicular and are formed by any two sides of the triangle.

To apply this formula, we can choose the line segment between (1,1) and (1,4) as the base of the triangle, since it is a vertical line and therefore has a length equal to the height of the triangle. The length of this line segment is 4 - 1 = 3 units.

Next, we need to find the length of the perpendicular segment from the point (-3,1) to the line containing the base. To do this, we can use the formula for the distance between a point and a line:

distance = [tex]|ax + by + c| / \sqrt{(a^2 + b^2)[/tex]

where a, b, and c are the coefficients of the equation of the line and x, y are the coordinates of the point.

In this case, the equation of the line containing the base is x = 1, so a = 1, b = 0, and c = -1. Plugging in the coordinates of (-3,1), we get:

distance = [tex]|1*(-3) + 0*(1) - 1| / \sqrt{(1^2 + 0^2)} = 2[/tex]

Therefore, the height of the triangle is 2 units.

Finally, we can plug these values into the formula for the area of a triangle to get:

Area = 1/2 * base * height = 1/2 * 3 * 2 = 3

So the area of the triangle is 3 square units.

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In the exponential decay, A) r = -0.0839 , B) r = -8.39% , C) Value=$11,800.

What is exponential decay?

The term "exponential decay" in mathematics refers to the process of a constant percentage rate reduction in an amount over time. It can be written as y=a(1-b)x, where x is the amount of time that has passed, an is the initial amount, b is the decay factor, and y is the final amount.

To find the annual rate of change between 1992 and 2006, we can use the formula:

r = [tex](V_2/V_1)^{1/n}-1[/tex]

where V1 is the initial value, V2 is the final value, and n is the number of years between the two values.

=>r = -0.0839

Therefore, the rate of change between 1992 and 2006 is -0.0839.

To express the rate of change in percentage form, we can multiply the result from part A by 100:

=>r = -0.0839 x 100

=> r = -8.39%

Therefore, the rate of change between 1992 and 2006 is a decrease of 8.39%.

To find the value of the car in the year 2009, we can assume that the value continues to drop at the same percentage rate as calculated in part A.

From 2006 to 2009, there are 3 years. So, using the formula for exponential decay, we have:

where V0 is the value in 2006, r is the rate of decrease, and n is the number of years between 2006 and 2009.

=>V = 11792.51

Therefore, the value of the car in the year 2009 would be approximately $11,800 (rounded to the nearest $50).

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Answer:

Look at the green dot

Step-by-step explanation:

Answer:

Apple Pie = 66.67%

Cherry Pie = 16.67%

Peach Pie = 16.67%

A possible distribution of pie percentages could be 66.67% apple, 16.67% cherry, and 16.67% peach.

Step-by-step explanation:

Let's assume that the bakery has 1 peach pie. Then, according to the problem statement, the bakery has 4 apple pies.

So, the total number of pies is 1 + 4 + 1 = 6.

To find the percentage of each type of pie, we need to divide the number of each type of pie by the total number of pies and multiply by 100%.

Percentage of apple pies: (4/6) x 100% = 66.67%

Percentage of cherry pies: (1/6) x 100% = 16.67%

Percentage of peach pies: (1/6) x 100% = 16.67%

Therefore, a possible distribution of pie percentages could be 66.67% apple, 16.67% cherry, and 16.67% peach.

It is important to note that this is just one possible distribution based on the information given in the problem. If we were given different information, such as the total number of pies, the percentages could be different.

Answer:

The answer is 1.5

Step-by-step explanation:

You just calculate in calculator and you will get the answer as 1.5

Answer:

3 hours

Step-by-step explanation:

1200/400=3

Answer: 63 US$

Step-by-step explanation:

The average movie ticket cost in Switzerland and Japan each of these​ countries is 46.97.

Let's assume that the average cost of a movie ticket in Japan is x, and the average cost of a movie ticket in Switzerland is y.

According to the problem statement, we can write two equations based on the given information:

3x + 2y = 78.5 ...(1)

2x + 3y = 74.1 ...(2)

We can solve these equations simultaneously to find the values of x and y. Here's how:

Multiply equation (1) by 2 and equation (2) by 3, then subtract equation (1) from equation (2):

(2x + 3y) - 2(3x + 2y) = 74.1 - 2(78.5)

Simplifying this equation, we get:

-y = -109.3

Therefore, y = 109.3.

Now substitute y = 109.3 into either equation (1) or (2) and solve for x:

3x + 2(109.3) = 78.5

Simplifying this equation, we get:

3x = -140.9

Therefore, x = -46.97.

However, we cannot have negative ticket prices.

Therefore, the average cost of a movie ticket is 46.97.

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Answer:

b. 1/4

Step-by-step explanation:

Probability of rolling even number = 3/6 = 1/2

Probability of tails = 1/2

Multiplying these two probabilities, we have 1/4.

Answer:

61∘

Step-by-step explanation:

Isosceles triangle property:

The triangle in the circle is an isosceles triangle, so that means the lateral sides of the triangle are equal, and so are the angles;

x = 90∘ - 29∘ = 61∘ (90∘, because the tangent and the radius form a right angle (that the b) part, I think))

53 is the answer

47.5 + 79.5= 127

180-127=53

Answer:

Step-by-step explanation:

The probability that the sample mean would be less than 21583 dollars if a sample of 399 persons is randomly selected is approximately 0.0826 or 8.26% (rounded to four decimal places).

What is probability?

Probability is a way to gauge how likely something is to happen. It is a number between 0 and 1, where 0 denotes an impossibility and 1 denotes a certainty that the occurrence will occur.

We can use the central limit theorem (CLT) to approximate the sampling distribution of the sample mean. According to CLT, if we have a large enough sample size (n≥30), the sampling distribution of the sample mean will be approximately normal, regardless of the underlying distribution of the population.

The mean of the sampling distribution of the sample mean is the same as the population mean, which is given as μ = 21699 dollars per annum. The standard deviation of the sampling distribution of the sample mean is equal to the standard error of the mean (SEM), which is calculated as follows:

SEM = σ/√n, where n is the sample size, and is the total standard deviation.

With the numbers from the problem substituted, we obtain:

SEM = 835/√399 = 41.767

Now, we need to find the probability that the sample mean would be less than 21583 dollars. We can standardize the sample mean using the standard normal distribution as follows:

z = (x - μ) / SEM, where the sample mean is x.

Substituting the values, we get:

z = (21583 - 21699) / 41.767 = -1.389

Using a standard normal distribution table, we can find that the area to the left of z=-1.389 is 0.0826.

Therefore, the probability that the sample mean would be less than 21583 dollars if a sample of 399 persons is randomly selected is approximately 0.0826 or 8.26% (rounded to four decimal places).

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Answer:

Without an image or a more detailed description of the diagram, it's difficult to provide an exact answer to this problem. However, we can use some logical reasoning to try to solve it.

Let's assume that the three lines of three numbers are arranged in a Tic-Tac-Toe grid, like this:

CSS

Copy code

A B C

D E F

G H I

We know that the product of the three numbers in each line is the same. Let's call this product "P". Then we can write:

CSS

Copy code

A * B * C = P

D * E * F = P

G * H * I = P

If we divide the second equation by the first equation, we get:

CSS

Copy code

(D * E * F) / (A * B * C) = 1

Since all the numbers are single-digit, this means that either D or F is equal to A, B, C, or 1. If D or F is equal to 1, then E is also equal to 1, which means that the entire middle row is filled with 1s, and that cannot be the case since all the numbers are different.

Therefore, we can assume that either D or F is equal to one of the numbers in the top row. Without further information, we cannot determine which one it is, but we know that the product of the numbers in the bottom row must be divisible by the product of the numbers in the top row. This means that the number in the circle containing the question mark must be a factor of this product, and it must be different from all the other numbers in the diagram.

Again, without more information, we cannot determine the exact number in the circle containing the question mark, but this logic should help narrow down the possibilities.

Step-by-step explanation:

The stars in order from the closest one to the farthest one are:

0.883, 0.886, 0.89, 1.25

so, the answer is A.

What is a sequence?

A sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms).

To put the given distances in order from the closest one to the farthest one, we need to arrange them in ascending order.

That means we need to start with the smallest value and move toward the largest value.

Looking at the given distances, we see that the smallest value is 0.883, followed by 0.886, then 0.89, and finally 1.25, which is the largest value.

Putting the given distances in ascending order, we get:

0.883, 0.886, 0.89, 1.25

Therefore, the stars in order from the closest one to the farthest one are:

0.883, 0.886, 0.89, 1.25

so, the answer is A.

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The two rays with the same endpoint in the figure below are AB and BC. Although they share a common endpoint (B), they do not form an angle since they are collinear and lie on the same line. Two non-collinear rays that share an endpoint create an angle. In this case, AB and BC lie on line AC and do not form an angle.

An angle is formed by two rays that originate from a common endpoint. In the given figure, AB and BC share the same endpoint (B), but they do not form an angle since they lie on the same line. A line is an infinite set of points that extends in both directions, while a ray is a portion of a line that starts at a particular point and extends infinitely in one direction. When two rays share a common endpoint, they form an angle only if they are not collinear, i.e., they do not lie on the same line. In this case, since AB and BC lie on line AC, they do not form an angle. Therefore, AB and BC are collinear and do not form an angle.

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Answer:

To find the sum of money that amounts to Rs- 3450 in 4 months at the rate of 4.5% per annum, we can use the formula for simple interest:

Simple Interest = (Principal * Rate * Time) / 100

Where,

Principal = the sum of money borrowed or invested

Rate = the rate of interest per annum

Time = the time period in years

Since the time period given is 4 months, we need to convert it to years by dividing it by 12:

Time = 4/12 years = 1/3 years

Now, we can plug in the given values and solve for Principal:

Simple Interest = (Principal * Rate * Time) / 100

3450 = (Principal * 4.5 * 1/3) / 100

3450 * 100 = Principal * 4.5 * 1/3

11500 = Principal * 1.5

Principal = 11500 / 1.5

Principal = 7666.67 (rounded off to two decimal places)

Therefore, the sum of money that amounts to Rs- 3450 in 4 months at the rate of 4.5% per annum is Rs- 7666.67.

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Answer:

9

Step-by-step explanation: each month adding 18 starting at 45 and currently at 207 means he gained 162 and divided by 18,9 months.

Following the basic principles and theory of simplifying fraction it is visible that this question follows the, the basic calculations involving the basic mathematics

Therefore the , let us take the given numerator that is 300 and then divide it using the denominator that is 25 ,

so, when we divide 300 /25 we get, the answer as12

because, when we divide both the numerator and denominator with 5 (because 25 and 300 are a divisible by 5 and are also multiples of )

we clearly see that 300 is divisible by 5 and the answer comes out to be 12.

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Answer: The answer is [tex]\frac{5}{2}[/tex].

Step-by-step explanation:

We are given

25a = 10[tex]a^{2}[/tex].

First, we divide both sides by a

25 = 10a

Then we divide both sides by 10

[tex]\frac{5}{2}[/tex] = a