As our scientific and technical knowledge has increased, so has our need for ever-smaller and ever-larger units of measure. The prefixes used to denote some of these units of measure are listed in the table below.
These new units of measure are quickly working their way into our everyday lives. For example, it is quite easy to purchase a 1-terabyte hard drive. One terabyte (TB) is equal to 1012 bytes. As another example, many computers can do multiple operations in 1 nanosecond (ns). One nanosecond is equal to 10−9 s.
In the 1980s, it became possible to measure optical events in nanoseconds and picoseconds. Express 1 ps as a decimal.

A simplification of the fractions (-5/8) ÷ (-3/4) is equal to: B. -20/24.

The step in which Johnetta made her error include the following: A. step 1.

In Mathematics and Geometry, a fraction simply refers to a numerical quantity (numeral) which is not expressed as a whole number. This ultimately implies that, a fraction is simply a part of a whole number.

Based on the information provided above, the division can be calculated as follows;

Fraction = (-5/8) ÷ (-3/4)

Fraction = -5/8 × (-4/3)

Fraction = -20/24

For Johnetta, we have:

Fraction = 7/9 ÷ 3/8

Fraction = 7/9 × 8/3

Fraction = 56/27

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Answer:

(a) The product xy is negative because one of the factors (x) is negative and the other factor (y) is positive. Therefore, the sign of xy is negative.

(b) The expression x^2y is also negative because x^2 is positive (the square of any real number is positive) and y is positive, so their product is positive. But since x is negative, the overall product is negative.

(c) The expression x/y + x can be written as (x/x)y + x, which simplifies to y + x. Since y is positive and x is negative, the sum y + x could be either positive or negative, depending on which absolute value is greater. If |y| > |x|, then y + x is positive. If |x| > |y|, then y + x is negative.

(d) The expression y-x is positive because y is greater than x and y is positive while x is negative. So the difference y-x is positive.

Step-by-step explanation:

sin (pi/4)  = sqrt(2) / 2 = .707

  you either just know this after a time or you can use a calculator (in RADIAN mode)

Answer:

sin(π/4) = sin(45°) = 0.707106781186548 ≈ 0.707 (rounded to 3 decimal places)

Step-by-step explanation:

here are the steps:

Convert π/4 to degrees: π/4 * (180/π) = 45°

sin(45°) = opposite/hypotenuse

sin(45°) = opposite/hypotenuse = adjacent/hypotenuse = 1/√2

sin(45°) = 1/√2 * √2/√2 = √2/2

Therefore, sin(π/4) = sin(45°) = √2/2 ≈ 0.707 (rounded to 3 decimal places).

Answer:

The formula for the surface area of a cylinder is:

S = 2πrh + 2πr^2

where S is the total surface area, r is the radius of the base, and h is the height.

For one lipstick, the surface area is:

S = 2π(0.6)(2.4) + 2π(0.6)^2

S = 2.88π + 0.72π

S = 3.6π

To wrap 4 lipsticks, we need to multiply this surface area by 4:

S = 4(3.6π)

S = 14.4π

Therefore, the makeup artist will need approximately 14.4π square inches of gift wrap to wrap 4 lipsticks.

Answer:

Step-by-step explanation:

Total surface area of a cylinder = 2πr(r + h)

2 π (.6) (.6 + 2.4)

2 π .6 (3)

1.2π (3)

3.6 π  square inches

Mulitply by four for four lipsticks:

3.6π × 4 = 14.4π sq inches

The indefinite integral when evaluated has a solution of 2cot(2x⁵ - 5x) + C

From the question, we have the following parameters that can be used in our computation:

The integral of (10 - 20x⁴)csc²(2x⁵ - 5x)

³

This can be expressed as

∫(10 - 20x⁴)csc²(2x⁵ - 5x) dx

The above is a complex expression

So, we make use of a graphing tool to evaluate the solution

Using a graphing tool, we have

Step 1:

[tex]\frac{4\cos(10x)\sin(4x^5) - 4\sin(10x)\cos(4x^5)}{sin^2(4x^5) - 2\sin(10x)\sin(4x^5) + cos^2(4x5) - 2\cos(10x)\cos(4x5)+ \sin^2(10x) + \cos^2(10x)} + C[/tex]

When simplified, we get

2cot(2x⁵ - 5x) + C

hence, the solution is 2cot(2x⁵ - 5x) + C

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The mean of the given set of numbers is 21.8. and median is 25.

In mathematics and statistics, the mean is a measure of central tendency that represents the average value of a set of numbers. It is calculated by adding up all the numbers in the set and then dividing by the total number of values. The mean is also known as the arithmetic mean or average.

To find the mean and median of the given set of numbers, we can follow these steps:

Arrange the numbers in order from smallest to largest: 2, 5, 25, 35, 42.

To find the mean, add up all the numbers and divide by the total number of numbers:

Mean = (2 + 5 + 25 + 35 + 42) / 5

= 109 / 5

= 21.8

Therefore, the mean of the given set of numbers is 21.8.

To find the median, we need to find the middle number in the ordered list. If there is an odd number of values, the median is the middle number. If there is an even number of values, the median is the average of the two middle numbers.

In this case, there are 5 numbers, so the median is the middle number, which is 25.

Therefore, the median of the given set of numbers is 25.

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Answer: 20a +5,600 < 21,000

move the content to the right

20a < 21,000 -5600

calculate

20a<15400

20a <15400
divide both sides of the inequality by 20

a<770

The evaluation of the functions using piecewise function gives:

f(-9) =  8

f(0) =  2

f(6) =  7

f(-6) =  4

f(3) =  4.5

f(9) = -6

To evaluate the functions using piecewise function, we have to the condition they satisfy. That is:

For f(-9), x = -9. Thus, x≤ -6. So use f(x) = (-4/3)x - 4 to evaluate f(-9). That is:

f(-9) =  (-4/3)*(-9) - 4

f(-9) =  8

For f(0), x = 0. Thus, -6 x ≤ 6. So  use f(x) = (5/6)x + 2 to evaluate f(0). That is:

f(0) = (5/6)*0 + 2

f(0) =  2

For f(6), x = 6. Thus, -6 x ≤ 6. So use f(x) = (5/6)x + 2 to evaluate f(6). That is:

f(6) = (5/6)*6 + 2

f(6) =  7

For f(-6), f(x) = (-4/3)x - 4:

f(-6) = (-4/3)*(-6) - 4

f(-6) =  4

For f(3), f(x) = (5/6)x + 2:

f(3) = (5/6)*3 + 2

f(3) =  4.5

For f(9), x = 9. Thus, x > 6. Use f(x) = -2x + 12:

f(9) = -2*9 + 12

f(9) = -6

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The angle of elevation of the ladder is approximately 71.6 degrees.

To find the angle of elevation of the ladder, we can use trigonometry. The ladder forms a right triangle with the ground and the tree.

The base of the triangle is 4 feet, the height is 12 feet, and the hypotenuse is the length of the ladder.

Using the trigonometric function tangent (tan), we can write:

tan(angle) = opposite/adjacent

In this case, the opposite side is the height of the tree (12 feet) and the adjacent side is the base of the triangle (4 feet).

Therefore, we can calculate the angle of elevation as follows:

tan(angle) = 12/4

angle = arctan(12/4)

Using a calculator or a trigonometric table, we can find that arctan(12/4) is approximately 71.6 degrees.

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Answer:

Volume = (1/2)(7)(24)(22) =

1,848 cubic meters

Surface area = 2(1/2)(7)(24) + 7(22) + 22(24) + 22(25) = 1,400 square meters

Dimensions of the rectangle are 6 feet and 12 feet.

Rectangle is a quadrilateral whose each pair of opposite sides are equal in length and each two consecutive sides are in right angle to each other. In rectangle one pair of equal opposite sides is called Length and another one is called Width.  

If length of a rectangle is L and width of rectangle is W then the perimeter of that rectangle will be = 2(L+W)

Let W be the rectangle's width.

Here according to question the rectangle is twice as long as it is wide.

So, length of rectangle = 2W

Perimeter will be = 2(W+2W) = 2*(3W) = 6W

So, according to question,

6W = 36

W = 36/6 = 6

Thus the width of the rectangle is = 6 feet.

Then the length = 2*6 = 12 feet.

Hence dimensions of the rectangle are 12 feet and 6 feet respectively.

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The variables I₁ and I₂ using the matrix algebra and using the Cramer's rule are I₁ = 1 and I₂ = 1

From the question, we have the following parameters that can be used in our computation:

15I₁ + 5I₂ = 20

25I₁ + 5I₂ - 30 = 0

Rewrite as

15I₁ + 5I₂ = 20

25I₁ + 5I₂ = 30

Rewrite as

I₁     I₂

15    5    20

25   5    30

From the question, the matrix form is

AI = b

Ths matrix A from the above is

[tex]A = \left[\begin{array}{cc}15&5&25&5\end{array}\right][/tex]

Ths matrix B from the above is

[tex]B = \left[\begin{array}{c}20&30\end{array}\right][/tex]

And, we have the matrix I to be

[tex]I = \left[\begin{array}{c}I_1&I_2\end{array}\right][/tex]

Start by calculating the inverse of A from

[tex]A = \left[\begin{array}{cc}15&5&25&5\end{array}\right][/tex]

So, we have:

|A| = 15 * 5 - 5 * 25

|A| = -50

The inverse is

[tex]A^{-1} = -\frac{1}{50}\left[\begin{array}{cc}5&-5&-25&15\end{array}\right][/tex]

Recall that

AI = b

So, we have

[tex]I = -\frac{1}{50}\left[\begin{array}{cc}5&-5&-25&15\end{array}\right] * \left[\begin{array}{c}20&30\end{array}\right][/tex]

Evaluate the products

[tex]I = -\frac{1}{50}\left[\begin{array}{c}5 * 20 + -5 * 30&-25 * 20 + 15 *30\end{array}\right][/tex]

[tex]I = -\frac{1}{50}\left[\begin{array}{c}-50&-50\end{array}\right][/tex]

Evaluate

[tex]I = \left[\begin{array}{c}1&1\end{array}\right][/tex]

Recall that

[tex]I = \left[\begin{array}{c}I_1&I_2\end{array}\right][/tex]

So, we have

I₁ = 1 and I₂ = 1

Recall that the determinant of matrix A calculated in (a) is

|A| = -50

Replace the first column in A with b

So, we have

[tex]AI_1 = \left[\begin{array}{cc}20&5&30&5\end{array}\right][/tex]

Calculate the determinant

DI₁ = 20 * 5 - 30 * 5

DI₁ = -50

Replace the second column in A with b

So, we have

[tex]AI_2 = \left[\begin{array}{cc}15&20&25&30\end{array}\right][/tex]

Calculate the determinant

DI₂ = 15 * 30 - 20 * 25

DI₂ = -50

So, we have

I₁ = DI₁ / |A| = -50/-50 = 1

I₂ = DI₂ / |A| = -50/-50 = 1

So, we have

I₁ = 1 and I₂ = 1

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Therefore, the degree of the given monomial is 11.

In algebra, a monomial is an expression consisting of a single term. A term can be a constant, a variable, or the product of a constant and one or more variables. In other words, a monomial is a polynomial with only one term.  The degree of a monomial is the sum of the exponents of its variables.

Here,

The degree of a monomial is the sum of the exponents of its variables.

For the monomial -3q⁴rs⁶, the degree would be:

4 + 1 + 6 = 11

Therefore, the degree of the monomial -3q⁴rs⁶ is 11.

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Answer:

Draw a picture of Angkor Wat

The x-coordinates of all the points on the graph of f(2x) are half the x-coordinates of the corresponding points on the graph of f(x), which results in a horizontal compression or shrinking of the graph.

What is a graph?

In computer science and mathematics, a graph is a collection of vertices (also known as nodes or points) connected by edges (also known as links or lines).

The transformation f(2x) stretches the graph of f(x) horizontally by a factor of 1/2. In other words, the graph of f(2x) will be compressed horizontally compared to the graph of f(x).

To see why this is the case, consider the effect of replacing x with 2x in the expression for f(x). If we think of f(x) as a set of points in the xy-plane, then each point (x,y) on the graph of f(x) is transformed into the point (2x,y) on the graph of f(2x).

Therefore, this means that the x-coordinates of all the points on the graph of f(2x) are half the x-coordinates of the corresponding points on the graph of f(x), which results in a horizontal compression or shrinking of the graph.

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The volume of the suitcase is 35/8 cubic feet or approximately 4.375 cubic feet.

A rectangular prism is a three-dimensional solid object that has six faces, where each face is a rectangle. It is also called a rectangular parallelepiped. A rectangular prism is a type of prism because it has a constant cross section along its length.

A rectangular prism's volume V is determined by:

V = l × w × h

where l denotes the prism's length, w its width, and h its height. Here are the facts:

l = 2 3 ft

w = 1 1/2 ft

h = 1 1/4 ft

When we enter these values into the formula, we obtain:

V = (2 3 ft) × (1 1/2 ft) × (1 1/4 ft)

= (7/3 ft) × (3/2 ft) × (5/4 ft)

= 35/8 ft³

Therefore, the volume of the suitcase is 35/8 cubic feet or approximately 4.375 cubic feet.

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Answer:

Linear

Step-by-step explanation:

The table represents a consistent function and is therefore linear.

Answer:

To find the value of f(-1), we need to substitute -1 for x in the given function:

f(x) = 3x² + 1 - 1

f(-1) = 3(-1)² + 1 - 1

f(-1) = 3(1) + 0

f(-1) = 3

Therefore, f(-1) is equal to 3.

The experimental probability of tossing a 3 is 22% ( option D).

Experimental probability, which is also known as Empirical probability, is based on actual experiments and adequate recordings of the occurring of events. An experiment can be repeated a fixed number of times and each repetition is known as a trial. The formula for the experimental probability is defined by;

Probability of an Event P(E) = Number of times an event occurs / Total number of events

From the table it is visible that the occurrence of tossing 3 is 11 times.

So by the definition of experimental probability the possibility is 11/50

As the table given is the result for tossing a number cube 50 times.

In 50 times the possibility is 11

In 1 time the possibility is 11/50

In 100 times the possibility is (11×100)/50

                                              = 22

Hence, the experimental probability of tossing a 3 is 22%.

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Answer: 5,8,6,1

Step-by-step explanation:

math

The effective annual yield of the investment is 0.1501

To find the effective annual yield, we need to use the formula:

(1 + r/n)^n - 1

where r is the annual interest rate and n is the number of times the interest is compounded per year.

In this case, r = 0.14 (14% expressed as a decimal), and n = 52 (since interest is compounded weekly).

Plugging these values into the formula, we get:

(1 + 0.14/52)^52 - 1 = 0.1501

Therefore, the effective annual yield is 0.1501, which is equivalent to 15.01% (rounded to two decimal places).

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Answer:

Step-by-step explanation:

(a) The vertex is (6, 144),(b) When the fireworks are launched at time x = 0 and return to the earth at time x = 12 seconds, respectively, these times are denoted by zeros in the function,.(c) this is the quadratic function that represents the situation f(x) = -4 + 144.

a. To find the zeros and vertex of the quadratic function, we need to first write it in the standard form:

[tex]f(x) = ax^2 + bx + c[/tex]

where a, b, and c are constants.

The vertex of the function is given by:

x = -b/2a

Since the quadratic function is symmetrical around the vertex, we know that the time it takes to reach the maximum height is halfway between the launch time and the time it hits the ground again. So, the time to reach maximum height is (4 + 8)/2 = 6 seconds.

Therefore, we can set up a system of equations using the information given:

f(4) = 0 (the firework is launched from the ground)

f(6) = 256 (the firework reaches its maximum height)

f(12) = 0 (the firework hits the ground again)

Plugging in the values of x and f(x), we get:

16a + 4b + c = 0

36a + 6b + c = 256

144a + 12b + c = 0

Solving this system of equations, we get:

a = -4

b = 48

c = 0

Therefore, the quadratic function that represents the situation is:

[tex]f(x) = -x^2 + 48x[/tex]

The zeros of the function can be found by setting f(x) = 0:

[tex]f(x) = -4x^2 + 48x[/tex]

0 = x(-4x + 48)

x = 0 (the firework is launched from the ground)

x = 12 (the firework hits the ground again)

The vertex can be found using the formula:

x = -b/2a = -48/(-8) = 6

So the vertex is (6, 144).

b. Using the zeros and vertex, we can sketch a graph of f(x):

The quadratic function's graph

Considering the function's primary characteristics in light of the circumstances:

The peak height of the fireworks, which occurs at x = 6 seconds and f(6) = 256 feet, is represented by the function's vertex.

The firework is at ground level at x = 0 (launch time) and x = 12 seconds (when it reaches the ground again), which are represented by zeros in the function.

The graph's form suggests that the firework rises before falling again, which is consistent with the scenario given.

c. The quadratic function that represents the situation is:

[tex]f(x) = -4x^2 + 48x[/tex]

This function can be simplified by factoring out -4:

[tex]f(x) = -4( x^2- 12x)[/tex]

Completing the square:

[tex]f(x) = -4( x^2- 12x + 36 - 36)[/tex]

[tex]f(x) = -4( (x^2-6)- 36)[/tex]

[tex]f(x) = -4(x^2-6) + 144[/tex]

This form of the function shows that the vertex is (6, 144), and that the maximum height is 144 feet.

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The cosine of angle B is: cos(B) = AC / AB = 20/29

The Pythagorean Theorem, a well-known geometric theorem that states that the sum of the squares of the legs of a right-angled triangle is equal to the square of the hypotenuse (the opposite side of the right angle) - that is, in familiar algebraic notation. , a2 + b2 = c2 .

In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to its hypotenuse. In this case, angle B is the angle we are interested in, and the adjacent side is side AC. Using the Pythagorean theorem, we find the length of side AC:

AC² = AB²+ BC²

AC² = 29² - 21²

AC² = 400

AC = 20

Therefore, the cosine of angle B is:

cos(B) = AC / AB = 20/29

Other trigonometric ratios of angle B can be found using the following formulas:

sin(B) = BC / AB = 21/29

tan(B) = BC / AC = 21/20

csc(B) = AB / BC = 29 / 21

sec(B) = AB / AC = 29/20

bed (B) = AC / BC = 20 / 21  

Thus, the trigonometric ratios of angle B are:

sin(B) = 21/29

cos(B) = 20/29

tan(B) = 21/20

csc(B) = 29/21

sec(B) = 29/20

crib(B) = 20/21

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Answer: 924 paths

Step-by-step explanation:

Since there is no shorter path, we know that the path must consist of 6 blocks to the north and 6 blocks to the east. Thus, the problem is equivalent to finding the number of ways to arrange 6 N's (for north) and 6 E's (for east) in a sequence such that no two N's are adjacent and no two E's are adjacent.

Let's denote N by 1 and E by 0. Then the problem is equivalent to finding the number of 12-digit binary sequences (i.e., sequences consisting of 0's and 1's) such that there are no consecutive 1's or consecutive 0's.

We can solve this problem using dynamic programming. Let F(n,0) be the number of n-digit binary sequences that end in 0 and have no consecutive 0's, and let F(n,1) be the number of n-digit binary sequences that end in 1 and have no consecutive 1's. Then we have the following recurrence relations:

F(n,0) = F(n-1,0) + F(n-1,1)

F(n,1) = F(n-1,0)

with initial values F(1,0) = 1 and F(1,1) = 1.

Using these recurrence relations, we can compute F(6,0) and F(6,1), and the total number of valid sequences is F(6,0) + F(6,1) = 132. Therefore, there are 132 different paths of length 12 blocks from your apartment to the math class.

the actual area of the base of the fountain, in square feet, after it is built is approximately 1.3 square feet (rounded to the nearest tenth of a square foot).

How to solve the question?

To find the actual area of the base of the fountain, we need to convert the measurements from the blueprint to the actual measurements.

First, we need to find the radius of the circular base. The diameter of the base is given as 40 centimeters on the blueprint, so the radius is half of that, or 20 centimeters.

Next, we need to convert the scale of the blueprint from centimeters to feet. The scale is given as three centimeters to four feet, which can be simplified to a ratio of 3:4. To convert from centimeters to feet, we need to multiply by a conversion factor of 1 foot/30.48 centimeters, since there are 30.48 centimeters in a foot.

So, to find the actual radius of the circular base in feet, we multiply the blueprint radius (20 centimeters) by the conversion factor:

20 centimeters * (1 foot/30.48 centimeters) = 0.656168 feet

Now that we have the actual radius of the circular base, we can find the actual area of the base. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius. Plugging in the actual radius we just found, we get:

A = π(0.656168 feet)^2 = 1.34977 square feet

Therefore, the actual area of the base of the fountain, in square feet, after it is built is approximately 1.3 square feet (rounded to the nearest tenth of a square foot).

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Answer:

Option D) 400π cubic units.

Step-by-step explanation:

GIVEN :

TO FIND :

USING FORMULA :

[tex]\quad\star{\underline{\boxed{\sf{V_{(Cylinder)} = \pi{r}^{2}h}}}}[/tex]

SOLUTION :

Substituting all the given values in the formula to find the volume of cylinder :

[tex]\quad{\sf{\dashrightarrow{V_{(Cylinder)} = \pi{r}^{2}h}}}[/tex]

[tex]\quad{\sf{\dashrightarrow{V_{(Cylinder)} = \pi{(5)}^{2}16}}}[/tex]

[tex]\quad{\sf{\dashrightarrow{V_{(Cylinder)} = \pi{(5 \times 5)}16}}}[/tex]

[tex]\quad{\sf{\dashrightarrow{V_{(Cylinder)} = \pi{(25)} \times 16}}}[/tex]

[tex]\quad{\sf{\dashrightarrow{V_{(Cylinder)} = \pi \times 25\times 16}}}[/tex]

[tex]\quad{\sf{\dashrightarrow{V_{(Cylinder)} = \pi \times 400}}}[/tex]

[tex]\quad{\sf{\dashrightarrow{V_{(Cylinder)} = 400 \pi}}}[/tex]

[tex]\quad\star{\underline{\boxed{\sf{\pink{V_{(Cylinder)} = 400 \pi \: {units}^{3}}}}}}[/tex]

Hence, the volume of cylinder is 400π cubic units.

To find the value of a6, we need to first determine the value of a2, a3, a4, and a5 using the given recurrence relation:

a1 = 6

a2 = 5a1 = 5(6) = 30

a3 = 5a2 = 5(30) = 150

a4 = 5a3 = 5(150) = 750

a5 = 5a4 = 5(750) = 3750

Now, we can find a6 using the recurrence relation:

a6 = 5a5 = 5(3750) = 18750

Therefore, the value of a6 is 18750.

The model is 64.62 inches tall. The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

Any real number may be explained using the four basic operations, also referred to as "arithmetic operations." In mathematics, operations like division, multiplication, addition, and subtraction come first, followed by operations like quotient, product, sum, and difference.

We are given that height of tower is 1454 feet and the scale is given as follows:

2 inches = 45 feet

Now, using the division operation, we get

⇒ Height of the model = 1,454 ÷ 45

⇒ Height of the model = 32.31

Now, using the multiplication operation, we get

⇒ Height of the model = 32.31 * 2

⇒ Height of the model = 64.62 inches

Hence, the model is 64.62 inches tall.

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The correct question has been attached below.