Question

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 of the triangle

Let the sides of the triangle be \(a = 18\) in, \(b = 41\) in, and the third - side be \(c\). The perimeter \(P=a + b + c\). Given \(P = 100\) in. So, \(c=P-(a + b)=100-(18 + 41)=100 - 59 = 41\) in.

Step2: Use Heron's formula to find the area of the triangle

First, find the semi - perimeter \(s=\frac{P}{2}=\frac{100}{2}=50\) in. Then, by Heron's formula \(A=\sqrt{s(s - a)(s - b)(s - c)}\), substituting \(a = 18\), \(b = 41\), \(c = 41\), and \(s = 50\) we get:
\[

$$\begin{align*} A&=\sqrt{50(50 - 18)(50 - 41)(50 - 41)}\\ &=\sqrt{50\times32\times9\times9}\\ &=\sqrt{25\times2\times32\times81}\\ &=\sqrt{25\times2\times2^5\times3^4}\\ &= 360 \end{align*}$$

\]

Step3: Calculate the cost of the material

The cost per square inch is \(\$0.15\). The total cost \(C=0.15\times A\). Substituting \(A = 360\) square inches, we get \(C=0.15\times360 = 54\).

Answer:

360
54