Exercise 24 Page 567

We are given the following diagram. Let's find the measure of SB! We are told that F, the highest point the fireworks reach, is about 0.2 miles above sea level. We are also given that our eyes E are about 0.01 miles above the water. We can use these measurements and the Segment Addition Postulate, along with the fact that the radius of the Earth is 4000 miles, to find the lengths of FC and CE.

Segment Addition Postulate	Lengths
FS+SC=FC	0.2+ 4000=4000.2
CB+BE=CE	4000+0.01=4000.01

Therefore, the measures of FC and CE are 4000.2 and 4000.01, respectively. Additionally, it is given that FE is tangent to Earth at point T. From here, since the measure of any arc is the measure of its central angle, to find mSB we will find m∠ ECF. Note that m ∠ ECF is equal to the sum of m∠ TCF and m ∠ ECT, so we will first find them. To do so, because we know the length of the adjacent sides and the hypotenuse for m∠ TCF and m ∠ ECT, we will use the inverse cosine function.

Angles	Inverse Cosine Functions	Results
m∠ TCF	cos^(-1)(TC/FC)=cos^(-1)(4000/4000.2)	≈ 0.573 ^(∘)
m ∠ ECT	cos^(-1)(TC/CE)=cos^(-1)(4000/4000.01)	≈ 0.128^(∘)

Since we found m∠ TCF and m ∠ ECT, we will find m∠ ECF by the Angle Addition Postulate.Then we will round our answer to the nearest tenth. As a result, since m ∠ ECF is equal to the measure of SB, mSB is 0.7^(∘).