For what value of \( x \) is the area of the rectangle greater than the perimeter?
The base of the rectangle is \( x + 2 \).
The height of the rectangle is 7.
Answer (3)
if we call the area A, then A = 7 ( x + 2 ) = 7 x + 14 if we call the perimeter P, then P = 7 ∗ 2 + 2 ( x + 2 ) = 2 x + 18 P"> A > P 2x+18"> 7 x + 14 > 2 x + 18 subtracting 14 gives 2x+4"> 7 x > 2 x + 4 subtracting 2x gives 4"> 5 x > 4 dividing by 5 gives \frac{4}{5}"> x > 5 4 or 0.8"> x > 0.8
The value of x for which the area of the rectangle is greater than its perimeter is any x > 0.8. This is found by setting up the area and perimeter expressions, A = (x+2)(7) and P = 2(x+2+7), and solving the inequality A > P.
To find the value of x where the area of the rectangle is greater than its perimeter, we set up two expressions for comparison. The area (A) of the rectangle is given by A = (base)(height), which for this rectangle is A = (x + 2)(7). The perimeter (P) of the rectangle is given by P = 2(base + height), which simplifies to P = 2(x + 2 + 7). We want to find the value of x for which A > P.
First, calculate the area:
A = (x+2)(7)
A = 7x + 14
Now, calculate the perimeter:
P = 2(x+2+7)
P = 2x + 18
To find when the area is greater than the perimeter, we set the area expression greater than the perimeter expression:
7x + 14 > 2x + 18
Solving this inequality, we subtract 2x from both sides to get:
5x + 14 > 18
And then subtract 14 from both sides:
5x > 4
Dividing both sides by 5:
x > 0.8
Therefore, for any value of x greater than 0.8, the area of the rectangle will be greater than its perimeter.
The value of x must be greater than 0.8 for the area of the rectangle to be greater than the perimeter. This was determined by comparing the area and perimeter formulas and solving the inequality. Specifically, the inequality derived was 4"> 5 x > 4 , leading to the solution \frac{4}{5}"> x > 5 4 .
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