Certainly! Let's solve this step-by-step.
Given:
- The distance from the point on the ground to the base of the mast is [tex]\(24\)[/tex] meters.
- The angle of elevation from this point to the top of the mast is [tex]\(31.5^\circ\)[/tex].
We need to determine the height of the mast.
1. Understand the Problem Geometry:
- We have a right-angled triangle where the base (adjacent side) is the distance from the point on the ground to the base of the mast, which is [tex]\(24\)[/tex] meters.
- The angle of elevation from this point to the top of the mast is given as [tex]\(31.5^\circ\)[/tex].
- We need to find the height of the mast, which represents the opposite side of the right-angled triangle.
2. Trigonometric Function to Use:
- In a right-angled triangle, the tangent of an angle ([tex]\(\theta\)[/tex]) is given by the ratio of the length of the opposite side (height of mast, [tex]\(h\)[/tex]) to the length of the adjacent side (distance from base, [tex]\(d\)[/tex]).
- The formula is:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\][/tex]
3. Convert the Angle to Radians:
- Many calculators and mathematical functions require angles to be in radians. The conversion from degrees to radians is done using the formula:
[tex]\[
\text{angle in radians} = \text{angle in degrees} \times \frac{\pi}{180}
\][/tex]
- For [tex]\(31.5^\circ\)[/tex]:
[tex]\[
\text{angle in radians} \approx 31.5 \times \frac{\pi}{180} \approx 0.5498 \text{ radians}
\][/tex]
4. Calculate the Tangent of the Angle:
- Find the tangent of [tex]\(0.5498\)[/tex] radians.
- [tex]\( \tan(31.5^\circ) \approx 0.6118 \)[/tex]
5. Use the Tangent Function to Solve for the Height:
- Using the formula:
[tex]\[
\tan(31.5^\circ) = \frac{h}{24}
\][/tex]
Solving for [tex]\(h\)[/tex]:
[tex]\[
h = 24 \times \tan(31.5^\circ)
\][/tex]
[tex]\[
h \approx 24 \times 0.6118
\][/tex]
[tex]\[
h \approx 14.71 \text{ meters}
\][/tex]
Result:
- The height of the mast is approximately [tex]\(14.71\)[/tex] meters.
