To determine which system can be used to find the lengths of the legs of the sun shade, let's analyze each option step-by-step.
Given the equation:
[tex]\[
\frac{1}{2} x^2 = 64
\][/tex]
We need to check each system of equations provided to see which one matches this given equation.
Option 1:
[tex]\[
y = \frac{1}{2} x^2 + 64
\][/tex]
and
[tex]\[
y = 0
\][/tex]
- Substituting [tex]\(y = 0\)[/tex] into the first equation:
[tex]\[
0 = \frac{1}{2} x^2 + 64
\][/tex]
[tex]\[
\frac{1}{2} x^2 = -64
\][/tex]
This results in:
[tex]\[
x^2 = -128
\][/tex]
This is not correct since the right-hand side should be 64, not -64.
Option 2:
[tex]\[
y = \frac{1}{2} x^2
\][/tex]
and
[tex]\[
y = 64
\][/tex]
- Substituting [tex]\(y = 64\)[/tex] into the first equation:
[tex]\[
64 = \frac{1}{2} x^2
\][/tex]
Solving for [tex]\(x^2\)[/tex]:
[tex]\[
\frac{1}{2} x^2 = 64
\][/tex]
This directly matches the given equation:
[tex]\[
x^2 = 128
\][/tex]
Thus, [tex]\(x\)[/tex] will lead us to the lengths of the legs of the sun shade when solved correctly.
Option 3:
[tex]\[
y = \frac{1}{2} x^2 + 64
\][/tex]
and
[tex]\[
y = \frac{1}{2} x^2 - 64
\][/tex]
- These two equations represent different and additional terms, as follows:
1. [tex]\(y = \frac{1}{2} x^2 + 64\)[/tex]
2. [tex]\(y = \frac{1}{2} x^2 - 64\)[/tex]
Setting these equal would result in:
[tex]\[
\frac{1}{2} x^2 + 64 = \frac{1}{2} x^2 - 64
\][/tex]
[tex]\[
64 = -64
\][/tex]
This is a contradiction and does not match the given equation.
From the analysis, Option 2,
[tex]\[
y = \frac{1}{2} x^2
\][/tex]
and
[tex]\[
y = 64
\][/tex]
is the correct system that can be used to find the lengths of the legs of the sun shade.
