Question

sun shades are sold in the shape of right isosceles triangles. if the equation represents one shade that shields 64 square feet of area, which system can be used to find the lengths of the legs of the sun shade?
\\( \frac { 1 } { 2 } x ^ { 2 } = 64 \\)
\\( y = \frac { 1 } { 2 } x ^ { 2 } + 64 \\) and \\( y = 0 \\)
\\( y = \frac { 1 } { 2 } x ^ { 2 } \\) and \\( y = 64 \\)
\\( y = \frac { 1 } { 2 } x ^ { 2 } + 64 \\) and \\( y = \frac { 1 } { 2 } x ^ { 2 } - 64 \\)

Explanation:

Step1: Recall the area formula for a right - isosceles triangle

The area formula for a right - isosceles triangle with leg length \(x\) is \(A=\frac{1}{2}x^{2}\). We know that \(A = 64\).

Step2: Set up the system of equations

We can set \(y=\frac{1}{2}x^{2}\) (the area formula as a function of \(x\)) and \(y = 64\) (the given area value). When we solve the system \(

$$\begin{cases}y=\frac{1}{2}x^{2}\\y = 64\end{cases}$$

\), we are essentially finding the value of \(x\) (the leg length) that satisfies the area condition.

Answer:

\(y=\frac{1}{2}x^{2}\) and \(y = 64\)