Question
If h= 16 meters and angle C = 40 degrees , then what is the length of side a? Round up
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GPT-4o mini
To find the length of side \( a \) in a triangle using the height \( h \) and angle \( C \), we can utilize the relationship with the sine function in trigonometry. Assuming angle \( C \) is the angle opposite to side \( a \) and the height \( h \) represents the altitude from \( C \) to side \( a \):
We can use the following formula to express the relationship:
\[
h = a \cdot \sin(C)
\]
We can rearrange this to solve for \( a \):
\[
a = \frac{h}{\sin(C)}
\]
Given \( h = 16 \) meters and \( C = 40^\circ \):
1. Calculate \( \sin(40^\circ) \).
Using a calculator, we find:
\[
\sin(40^\circ) \approx 0.6428
\]
2. Substitute the values into the equation for \( a \):
\[
a = \frac{16}{\sin(40^\circ)} \approx \frac{16}{0.6428} \approx 24.85
\]
3. Round up the result:
The length of side \( a \) rounded up is approximately:
\[
\boxed{25} \text{ meters}
\]
We can use the following formula to express the relationship:
\[
h = a \cdot \sin(C)
\]
We can rearrange this to solve for \( a \):
\[
a = \frac{h}{\sin(C)}
\]
Given \( h = 16 \) meters and \( C = 40^\circ \):
1. Calculate \( \sin(40^\circ) \).
Using a calculator, we find:
\[
\sin(40^\circ) \approx 0.6428
\]
2. Substitute the values into the equation for \( a \):
\[
a = \frac{16}{\sin(40^\circ)} \approx \frac{16}{0.6428} \approx 24.85
\]
3. Round up the result:
The length of side \( a \) rounded up is approximately:
\[
\boxed{25} \text{ meters}
\]
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