Question

a radio station tower was built in two sections. from a point 87 feet from the base of the tower, the angle of elevation of the top of the first section is $25^{circ}$, and the angle of elevation of the top of the second section is $40^{circ}$. to the nearest foot, what is the height of the top section of the tower?

Explanation:

Step1: Find height of total tower (first + top section)

We use the tangent function for the \(40^\circ\) angle. Let \(H\) be the total height (first + top section). \(\tan(40^\circ)=\frac{H}{87}\), so \(H = 87\times\tan(40^\circ)\).
\(\tan(40^\circ)\approx0.8391\), so \(H\approx87\times0.8391\approx73.0\) (rounded for now, but we'll keep more decimals for accuracy).

Step2: Find height of first section

Let \(h_1\) be the height of the first section. Using the tangent function for the \(25^\circ\) angle: \(\tan(25^\circ)=\frac{h_1}{87}\), so \(h_1 = 87\times\tan(25^\circ)\).
\(\tan(25^\circ)\approx0.4663\), so \(h_1\approx87\times0.4663\approx40.6\).

Step3: Find height of top section (\(x\))

The height of the top section \(x = H - h_1\). Substituting the values: \(x\approx87\times\tan(40^\circ)-87\times\tan(25^\circ)=87(\tan(40^\circ)-\tan(25^\circ))\).
Calculate \(\tan(40^\circ)-\tan(25^\circ)\approx0.8391 - 0.4663 = 0.3728\).
Then \(x\approx87\times0.3728\approx32.4336\approx32\) (to the nearest foot). Wait, wait, let's recalculate with more precise \(\tan\) values.
\(\tan(40^\circ)\approx0.8390996312\), \(\tan(25^\circ)\approx0.46630765815\).
\(87\times(0.8390996312 - 0.46630765815)=87\times0.37279197305\approx87\times0.3728\approx32.4336\), which rounds to 32? Wait, no, maybe I made a mistake. Wait, 87*0.3728: 80*0.3728=29.824, 7*0.3728=2.6096, total=29.824+2.6096=32.4336, so to the nearest foot, 32? Wait, but let's check again. Wait, maybe my initial approximation of H was wrong. Let's do it more accurately.
\(H = 87\times\tan(40^\circ)=87\times0.8390996312\approx87\times0.8391 = 87*0.8 + 87*0.0391 = 69.6 + 3.3917 = 72.9917\)
\(h_1 = 87\times\tan(25^\circ)=87\times0.46630765815 = 87*0.4 + 87*0.06630765815 = 34.8 + 5.76876625905 = 40.56876625905\)
Then \(x = 72.9917 - 40.56876625905 = 32.42293374095\), which is approximately 32? Wait, but maybe I miscalculated. Wait, let's use a calculator for 87*(tan(40)-tan(25)):
tan(40) ≈ 0.8390996312
tan(25) ≈ 0.4663076582
Difference: 0.8390996312 - 0.4663076582 = 0.372791973
87 * 0.372791973 = 87 * 0.372792 ≈ 32.4329, so to the nearest foot, 32? Wait, but maybe the answer is 32? Wait, no, wait, maybe I messed up the angles. Wait, the top section is the second section, so total height is top of second, first section is bottom section. So yes, x is top section, so H is total (first + top), h1 is first, so x = H - h1. So the calculation seems right. Wait, but let's check with another approach. Let's compute H and h1 more accurately.

H = 87 * tan(40°): tan(40°) is approximately 0.8390996312, so 87 * 0.8390996312 = 87 * 0.8390996312. Let's compute 87*0.8 = 69.6, 87*0.0390996312 = 87*0.03 = 2.61, 87*0.0090996312 ≈ 0.7916679144, so total 2.61 + 0.7916679144 = 3.4016679144, so total H = 69.6 + 3.4016679144 = 73.0016679144.

h1 = 87 * tan(25°): tan(25°) ≈ 0.46630765815, so 87*0.46630765815. 80*0.46630765815 = 37.304612652, 7*0.46630765815 = 3.26415360705, total = 37.304612652 + 3.26415360705 = 40.56876625905.

Then x = 73.0016679144 - 40.56876625905 = 32.43290165535, which rounds to 32? Wait, but maybe I made a mistake in the angle of elevation. Wait, the problem says "the angle of elevation of the top of the first section is 25°", so first section is lower, second is upper. So total height is top of second, first section is bottom. So yes, x is the height of the top section (second section), so the calculation is correct. So the answer should be 32? Wait, but let's check with a calculator. Let's compute 87*(tan(40)-tan(25)):

tan(40) ≈ 0.8390996312

tan(25) ≈ 0.4663076582

0.8390996312 - 0.4663076582 = 0.372791973

87 * 0.372791973 = 32.4329, which is approximately 32 when rounded to the nearest foot. Wait, but maybe the answer is 32? Wait, but let me check with another method. Let's use a calculator for 87*tan(40) - 87*tan(25):

87*tan(40) = 87*0.8390996312 = 73.00166791

87*tan(25) = 87*0.4663076582 = 40.56876626

73.00166791 - 40.56876626 = 32.43290165, which is 32 when rounded to the nearest foot. So the height of the top section is approximately 32 feet. Wait, but maybe I made a mistake in the angle values. Let me confirm tan(40) and tan(25). Tan(40°) is about 0.8391, tan(25°) is about 0.4663. So the difference is about 0.3728, times 87 is about 32.43, which rounds to 32. So the answer is 32.

Answer:

32