Question

relating perimeter to area
marta is making a flag with the given dimensions. the perimeter of the flag is 100 inches.
how much material, in square inches, is needed to make the flag?
square inches
if the material costs $0.15 per square inch, how much will marta spend on materials to make the flag?
$

Explanation:

Step1: Find the third - side length

Let the third - side length be $x$. The perimeter of a triangle is the sum of its side lengths. Given two side lengths are 18 inches and 41 inches, and perimeter $P = 100$ inches. So, $18 + 41+x=100$, which simplifies to $x = 100-(18 + 41)=41$ inches.

Step2: Use Heron's formula to find the area

First, find the semi - perimeter $s=\frac{100}{2}=50$ inches. Heron's formula for the area $A$ of a triangle with side lengths $a$, $b$, and $c$ is $A=\sqrt{s(s - a)(s - b)(s - c)}$. Here, $a = 18$, $b = 41$, $c = 41$. Then $A=\sqrt{50(50 - 18)(50 - 41)(50 - 41)}=\sqrt{50\times32\times9\times9}=\sqrt{25\times2\times32\times81}= \sqrt{25\times2\times2^5\times3^4}= \sqrt{5^2\times2^6\times3^4}=5\times2^3\times3^2=5\times8\times9 = 360$ square inches.

Step3: Calculate the cost

The material costs $0.15$ per square inch. So the total cost $C=0.15\times A$. Substitute $A = 360$ into the formula, we get $C=0.15\times360 = 54$.

Answer:

360
54