Question

Consider a rectangle that has a perimeter of 80\ \text{cm}" alt="80\ \text{cm}" align="absmiddle" class="latex-formula">. Write a function $A(l)$ that represents the area of the rectangle with length $l$.

Answer 1

Answer:

$\boxed{A(l)= 40l - l^2}$

Step-by-step explanation:

We know that a rectangle has four sides - two lengths and two widths. We also know that the lengths are equal and that the widths are equal. Therefore, to find the perimeter of a rectangle, we use:

$P = 2L + 2W$

However, this problem asks to find the area of the rectangle. The area can be found by multiplying length and width (A = L × W).

Using our formula above, we can solve for the length.

$80 = 2L + 2W$ (divide everything by 2)

$40 = L + W$ (subtract W in order to solve for L)

$40 - L = W$ (rearrange W to the left - standard equation notation)

$W = 40 - L$

The area of a rectangle is solved using the formula A = LW, so we just need to input our values and solve for A.

$A(l) = (40 - l)(l)$ (distribute the W throughout)

$A(l)= 40l - l^2$

This is your final function. Hope this helps!

Answer 2

Answer:

$A(L) =40L-L^2$

Step-by-step explanation:

The perimeter of a rectangle is modeled by the function: $P = 2L + 2W$

You are given that the perimeter is 80 cm, so substitute this value in for $P$

• $80=2L+2W$

You want the function $A(l)$ to represent the area of the rectangle in terms of the length $L$, so, therefore, you want to solve for the width of the rectangle (so you can still have that $L$ variable).

Solve for $W$ by first subtracting $2L$ from both sides of the equation.

• $80-2L = 2W$

Then divide both sides of the equation by 2 to isolate the variable $W$.

• $\frac{80-2L}{2}=W$

This can be simplified even further:

• $W=40-L$

The area of a rectangle is modeled by the function: $A=LW$

You have already solved for $W$, so substitute $40-L$ for it.

• $A=L(40-L)$

Use the distributive property to multiply everything inside the parentheses by $L$.

• $A=40L-L^2$

Since this function is in terms of $L$, you can write:

• $A(L) =40L-L^2$

This function represents the area of the rectangle with a perimeter of 80 cm and length $L$.