Question

HRC is shown where s=9, m= square root of 125, x=36.4
Select all of the equations that can be used to determine the value of


A
w=92+(125)2w=\sqrt{9^2+\left(\sqrt{125}\right)^2}w=
9
2
+(
125
​
)
2

​


B
tan⁡(x)=9w\tan\left(x\right)=\frac{9}{w}tan(x)=
w
9
​


C
cos⁡(53.6∘)=w125\cos\left(53.6^{\circ}\right)=\frac{w}{\sqrt{125}}cos(53.6
∘
)=
125
​

w
​


D
sin⁡(36.4∘)=w125\sin\left(36.4^{\circ}\right)=\frac{w}{\sqrt{125}}sin(36.4
∘
)=
125
​

w
​


E
cos⁡(y)=w125\cos\left(y\right)=\frac{w}{\sqrt{125}}cos(y)=
125
​

w
​


F
92+w2=(125)29^2+w^2=\left(\sqrt{125}\right)^29
2
+w
2
=(
125
​
)
2

Answer 1

To determine which equations can be used to find the value of \( w \) based on the provided information, we will analyze each option.

Given:
- \( s = 9 \)
- \( m = \sqrt{125} \)
- \( x = 36.4 \)

1. **Equation A**:
\[
w = \sqrt{9^2 + \left(\sqrt{125}\right)^2}
\]
This equation correctly uses the Pythagorean theorem to find \( w \). Since \( s = 9 \) and \( m = \sqrt{125} \), this can be simplified to
\[
w = \sqrt{81 + 125} = \sqrt{206}
\]
Thus, this equation can be used.

2. **Equation B**:
\[
\tan(x) = \frac{9}{w}
\]
Rearranging gives \( w = \frac{9}{\tan(x)} \), which calculates \( w \) directly from the tangent function involving angle \( x \). This is a valid equation and can be used.

3. **Equation C**:
\[
\cos(53.6^\circ) = \frac{w}{\sqrt{125}}
\]
Rearranging gives \( w = \sqrt{125} \cos(53.6^\circ) \). This is valid if we know \( \cos(53.6^\circ) \), thus it can also be used.

4. **Equation D**:
\[
\sin(36.4^\circ) = \frac{w}{\sqrt{125}}
\]
Rearranging gives \( w = \sqrt{125} \sin(36.4^\circ) \). This is valid as well, thus it can be used.

5. **Equation E**:
\[
\cos(y) = \frac{w}{\sqrt{125}}
\]
Similar to Equation C, it provides \( w \) in terms of some angle \( y \). This can also be used if we know \( \cos(y) \).

6. **Equation F**:
\[
92 + w^2 = (125)^2
\]
Rearranging gives \( w^2 = 15625 - 92 \). This is also valid and can be used.

### Conclusion
The equations that can be used to determine the value of \( w \) are:
- **A, B, C, D, E, F**.

All of them are valid equations that could help in finding \( w \) based on the provided relationships.