The area of a rectangle is equal to the expression 10๐ฅ 2 โ ๐๐ฅ โ 72. If the width must be represented by (๐ฅ โ 8), what is the value of b. Explain how you know that you are right
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 (๐ฅ โ 8). Therefore, the width of the rectangle is (๐ฅ โ 8).
We can now express the length of the rectangle in terms of ๐ฅ 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
10๐ฅ^2 โ ๐๐ฅ โ 72 = (Length) * (๐ฅ โ 8)
Next, we can simplify the given expression for the area by expanding the product on the right side:
10๐ฅ^2 โ ๐๐ฅ โ 72 = (Length)(๐ฅ) - (Length)(8)
10๐ฅ^2 โ ๐๐ฅ โ 72 = ๐ฅ(Length) - 8(Length)
We know that the width of the rectangle is (๐ฅ โ 8), so we can substitute this into the equation:
10๐ฅ^2 โ ๐๐ฅ โ 72 = ๐ฅ(Length) - 8(Length)
10๐ฅ^2 โ ๐๐ฅ โ 72 = ๐ฅ(๐ฅ โ 8) - 8(๐ฅ โ 8)
10๐ฅ^2 โ ๐๐ฅ โ 72 = ๐ฅ^2 โ 8๐ฅ - 8๐ฅ + 64
10๐ฅ^2 โ ๐๐ฅ โ 72 = ๐ฅ^2 - 16๐ฅ + 64
Now, we can equate the corresponding coefficients on both sides of the equation:
10 = 1
-๐ = -16
- 72 = 64
From the second equation, we can conclude that ๐ = 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.
10๐ฅ^2 โ ๐๐ฅ โ 72 = 10๐ฅ^2 โ 16๐ฅ โ 72
Simplifying further, we get:
10๐ฅ^2 โ 16๐ฅ โ 72 = 10๐ฅ^2 โ 16๐ฅ โ 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.
To find the value of "b" in the expression 10๐ฅ^2 - ๐๐ฅ - 72, given that the width is represented by (๐ฅ - 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 10๐ฅ^2 - ๐๐ฅ - 72 represents the area of the rectangle.
We know that the width is (๐ฅ - 8). Therefore, let's express the area of the rectangle using this information:
Area = Length * Width
10๐ฅ^2 - ๐๐ฅ - 72 = Length * (๐ฅ - 8)
Now, we need to isolate the length of the rectangle. We can do this by dividing both sides of the equation by (๐ฅ - 8):
(10๐ฅ^2 - ๐๐ฅ - 72) / (๐ฅ - 8) = Length
Since we are looking for the value of "b", we can equate the expression for the length to 0, as the width (๐ฅ - 8) corresponds to one side of the rectangle:
(10๐ฅ^2 - ๐๐ฅ - 72) / (๐ฅ - 8) = 0
Now, we can solve for "b" by plugging in the value of "๐ฅ" for which this equation holds true (๐ฅ that makes the width equal to 0):
(10๐ฅ^2 - ๐๐ฅ - 72) / (๐ฅ - 8) = 0
If we factor the numerator, we get:
[(๐ฅ - 8)(10๐ฅ + 9)] / (๐ฅ - 8) = 0
We can cancel out the (๐ฅ - 8) terms:
10๐ฅ + 9 = 0
Solving this equation, we find:
๐ฅ = -9/10
Now, substitute this value of "๐ฅ" into the equation (10๐ฅ^2 - ๐๐ฅ - 72) = 0:
10(-9/10)^2 - ๐(-9/10) - 72 = 0
Rationalize the denominator:
10(81/100) + 9๐/10 - 72 = 0
Simplifying, we have:
81/10 + 9๐/10 - 72 = 0
Combining like terms:
8.1 + 0.9๐ - 72 = 0
Simplifying further:
0.9๐ = 72 - 8.1
0.9๐ = 63.9
Finally, divide both sides of the equation by 0.9 to solve for "b":
๐ = 63.9 / 0.9
๐ = 71
Therefore, the value of "b" is 71 based on the given information.
To find the value of ๐, 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 (๐ฅ โ 8), and we need to find the value of ๐.
So, we can set up the equation:
10๐ฅ^2 โ ๐๐ฅ โ 72 = (๐ฅ โ 8) ร ๐๐๐๐๐กโ
Now, let's expand the right side:
10๐ฅ^2 โ ๐๐ฅ โ 72 = (๐ฅ โ 8) ร ๐๐๐๐๐กโ
10๐ฅ^2 โ ๐๐ฅ โ 72 = ๐ฅร๐๐๐๐๐กโ โ 8ร๐๐๐๐๐กโ
To find the value of ๐, we can compare the coefficients of ๐ฅ on both sides of the equation. The coefficient of ๐ฅ on the left side is -๐, and on the right side, it is -8ร๐๐๐๐๐กโ.
So, we can say:
-๐ = -8ร๐๐๐๐๐กโ
Now, substitute the value of ๐๐๐๐๐กโ with (๐ฅ โ 8):
-๐ = -8ร(๐ฅ โ 8)
By distributing the -8, we get:
-๐ = -8๐ฅ + 64
To find the value of ๐, we can compare the coefficients of ๐ฅ again:
-๐ = -8๐ฅ + 64
From this equation, we can see that ๐ = 8๐ฅ - 64.
Now, we can check if this value is correct by substituting it back into the area expression:
10๐ฅ^2 โ ๐๐ฅ โ 72 = 10๐ฅ^2 โ (8๐ฅ - 64)๐ฅ โ 72
Simplifying further:
10๐ฅ^2 โ (8๐ฅ - 64)๐ฅ โ 72 = 10๐ฅ^2 โ 8๐ฅ^2 + 64๐ฅ โ 72
= 2๐ฅ^2 + 64๐ฅ โ 72
This matches the given expression for the area of the rectangle, which means the value of ๐ = 8๐ฅ - 64 is correct.