The area of a rectangle is equal to the expression 10𝑥 2 − 𝑏𝑥 − 72. If the width must be represented by (𝑥 − 8), what is the value of b. Explain how you know that you are right

Question

The area of a rectangle is equal to the expression 10𝑥 2 − 𝑏𝑥 − 72. If the width must be represented by (𝑥 − 8), what is the value of b. Explain how you know that you are right

Answer 1

To find the value of b, we need to first determine the dimensions of the rectangle and then use those dimensions to simplify the given expression for the area.

The dimensions of a rectangle are its length and width. In this case, we are given that the width must be represented by (𝑥 − 8). Therefore, the width of the rectangle is (𝑥 − 8).

We can now express the length of the rectangle in terms of 𝑥 and width.

The area of a rectangle is equal to the product of its length and width. Therefore, the expression for the area can be represented as:

Area = Length * Width

10𝑥^2 − 𝑏𝑥 − 72 = (Length) * (𝑥 − 8)

Next, we can simplify the given expression for the area by expanding the product on the right side:

10𝑥^2 − 𝑏𝑥 − 72 = (Length)(𝑥) - (Length)(8)

10𝑥^2 − 𝑏𝑥 − 72 = 𝑥(Length) - 8(Length)

We know that the width of the rectangle is (𝑥 − 8), so we can substitute this into the equation:

10𝑥^2 − 𝑏𝑥 − 72 = 𝑥(Length) - 8(Length)

10𝑥^2 − 𝑏𝑥 − 72 = 𝑥(𝑥 − 8) - 8(𝑥 − 8)

10𝑥^2 − 𝑏𝑥 − 72 = 𝑥^2 − 8𝑥 - 8𝑥 + 64

10𝑥^2 − 𝑏𝑥 − 72 = 𝑥^2 - 16𝑥 + 64

Now, we can equate the corresponding coefficients on both sides of the equation:

10 = 1
-𝑏 = -16
- 72 = 64

From the second equation, we can conclude that 𝑏 = 16.

Therefore, the value of b is 16.

We can check if our value for b is correct by substituting it back into the original equation for the area and see if it simplifies correctly.

10𝑥^2 − 𝑏𝑥 − 72 = 10𝑥^2 − 16𝑥 − 72

Simplifying further, we get:

10𝑥^2 − 16𝑥 − 72 = 10𝑥^2 − 16𝑥 − 72

Since both sides of the equation are equal, it confirms that our value of b = 16 is correct as it satisfies the given expression for the area of the rectangle.

Answer 2

To find the value of "b" in the expression 10𝑥^2 - 𝑏𝑥 - 72, given that the width is represented by (𝑥 - 8), you can use the information provided about the rectangle.

The area of a rectangle is given by multiplying its length and width. In this case, the expression 10𝑥^2 - 𝑏𝑥 - 72 represents the area of the rectangle.

We know that the width is (𝑥 - 8). Therefore, let's express the area of the rectangle using this information:

Area = Length * Width
10𝑥^2 - 𝑏𝑥 - 72 = Length * (𝑥 - 8)

Now, we need to isolate the length of the rectangle. We can do this by dividing both sides of the equation by (𝑥 - 8):

(10𝑥^2 - 𝑏𝑥 - 72) / (𝑥 - 8) = Length

Since we are looking for the value of "b", we can equate the expression for the length to 0, as the width (𝑥 - 8) corresponds to one side of the rectangle:

(10𝑥^2 - 𝑏𝑥 - 72) / (𝑥 - 8) = 0

Now, we can solve for "b" by plugging in the value of "𝑥" for which this equation holds true (𝑥 that makes the width equal to 0):

(10𝑥^2 - 𝑏𝑥 - 72) / (𝑥 - 8) = 0

If we factor the numerator, we get:

[(𝑥 - 8)(10𝑥 + 9)] / (𝑥 - 8) = 0

We can cancel out the (𝑥 - 8) terms:

10𝑥 + 9 = 0

Solving this equation, we find:

𝑥 = -9/10

Now, substitute this value of "𝑥" into the equation (10𝑥^2 - 𝑏𝑥 - 72) = 0:

10(-9/10)^2 - 𝑏(-9/10) - 72 = 0

Rationalize the denominator:

10(81/100) + 9𝑏/10 - 72 = 0

Simplifying, we have:

81/10 + 9𝑏/10 - 72 = 0

Combining like terms:

8.1 + 0.9𝑏 - 72 = 0

Simplifying further:

0.9𝑏 = 72 - 8.1
0.9𝑏 = 63.9

Finally, divide both sides of the equation by 0.9 to solve for "b":

𝑏 = 63.9 / 0.9
𝑏 = 71

Therefore, the value of "b" is 71 based on the given information.

Answer 3

To find the value of 𝑏, we need to understand how the width and the expression for the area are related.

The area of a rectangle can be calculated by multiplying its width and length. In this case, the width is represented by (𝑥 − 8), and we need to find the value of 𝑏.

So, we can set up the equation:

10𝑥^2 − 𝑏𝑥 − 72 = (𝑥 − 8) × 𝑙𝑒𝑛𝑔𝑡ℎ

Now, let's expand the right side:

10𝑥^2 − 𝑏𝑥 − 72 = (𝑥 − 8) × 𝑙𝑒𝑛𝑔𝑡ℎ
10𝑥^2 − 𝑏𝑥 − 72 = 𝑥×𝑙𝑒𝑛𝑔𝑡ℎ − 8×𝑙𝑒𝑛𝑔𝑡ℎ

To find the value of 𝑏, we can compare the coefficients of 𝑥 on both sides of the equation. The coefficient of 𝑥 on the left side is -𝑏, and on the right side, it is -8×𝑙𝑒𝑛𝑔𝑡ℎ.

So, we can say:

-𝑏 = -8×𝑙𝑒𝑛𝑔𝑡ℎ

Now, substitute the value of 𝑙𝑒𝑛𝑔𝑡ℎ with (𝑥 − 8):

-𝑏 = -8×(𝑥 − 8)

By distributing the -8, we get:

-𝑏 = -8𝑥 + 64

To find the value of 𝑏, we can compare the coefficients of 𝑥 again:

-𝑏 = -8𝑥 + 64

From this equation, we can see that 𝑏 = 8𝑥 - 64.

Now, we can check if this value is correct by substituting it back into the area expression:

10𝑥^2 − 𝑏𝑥 − 72 = 10𝑥^2 − (8𝑥 - 64)𝑥 − 72

Simplifying further:

10𝑥^2 − (8𝑥 - 64)𝑥 − 72 = 10𝑥^2 − 8𝑥^2 + 64𝑥 − 72
= 2𝑥^2 + 64𝑥 − 72

This matches the given expression for the area of the rectangle, which means the value of 𝑏 = 8𝑥 - 64 is correct.