Q: A fiber optic line runs 1.2 miles east and then 2.75 miles north. If the line costs $3500 per mile to install, how much could have been saved by installing the line diagonally across the field?

Total cost savings: $$3500 \times (1.2 + 2.75 - \sqrt{1.2^2$ + 2.75^2$}) = 3500 \times (3.95 - \sqrt{1.44 + 7.5625}) = 3500 \times (3.95 - \sqrt{9.0025}) = 3500 \times (3.95 - 3.00) = 3500 \times 0.95 = 3325$

Q: Given a right triangle with angle $$\theta$$, adjacent side $$= 8$$, opposite side $$= 15$$, and hypotenuse $$= 17$$, what is $$\sin(\theta)$$?

$$\sin(\theta) = \frac{15}{17}$$

Q: The proper angle for a ladder is about $$75^\circ$$ from the ground. If you have a 17-foot ladder, how far from the house should you place the base of the ladder? (Round to 1 decimal place.)

$$17 \cos(75^\circ) \approx 17 \times 0.2588 \approx 4.4$$ feet$

Q: The angle of elevation to the top of a building is $$10^\circ$$ from a point 2 miles away. What is the height of the building in feet? (1 mile = 5280 feet, round to 1 decimal place.)

$$h = 2 \times 5280 \times \tan(10^\circ) \approx 10560 \times 0.1763 \approx 1861.7$$ feet$

Question 1

A 13-foot ladder is placed 7 feet from the base of a house, as shown in the image. How high up the wall does the ladder reach? Round your answer to two decimal places.

Accepted Answer

$$\sqrt{13^2$ - 7^2$} = \sqrt{169 - 49} = \sqrt{120} \approx 10.95$$ feet$

Question 2

Given a right triangle with legs of 8 and 15, what is the length of the hypotenuse?

Accepted Answer

$$\sqrt{8^2$ + 15^2$} = \sqrt{64 + 225} = \sqrt{289} = 17$$

Question 3

A guy wire is to be stretched from the top of a 50-foot pole to a point 20 feet from the base. The wire must be 6 feet longer than the straight-line distance from the top to the ground point. How long should the wire be?

Accepted Answer

$$\sqrt{50^2$ + 20^2$} + 6 = \sqrt{2500 + 400} + 6 = \sqrt{2900} + 6 \approx 53.85 + 6 = 59.85$$ feet$

Question 4

Susan wants to buy a 46-inch TV. The diagonal is 46 inches and the length is 40.02 inches. What is the height of the TV?

Accepted Answer

$$\sqrt{46^2$ - 40.02^2$} = \sqrt{2116 - 1601.6} = \sqrt{514.4} \approx 22.68$$ inches$

Question 5

Jennifer is flying a kite with a 16-foot string, standing 8 feet from the tree, and holding the string 4 feet above the ground. How high above the ground is the kite? (Round up to the next foot.)

Accepted Answer

$$\sqrt{16^2$ - 8^2$} + 4 = \sqrt{256 - 64} + 4 = \sqrt{192} + 4 \approx 13.86 + 4 = 17.86$$ feet \rightarrow 18$$ feet$$

Question 6

A fiber optic line runs 1.2 miles east and then 2.75 miles north. If the line costs $3500 per mile to install, how much could have been saved by installing the line diagonally across the field?

Accepted Answer

Total cost savings: $$3500 	imes (1.2 + 2.75 - \sqrt{1.2^2$ + 2.75^2$}) = 3500 	imes (3.95 - \sqrt{1.44 + 7.5625}) = 3500 	imes (3.95 - \sqrt{9.0025}) = 3500 	imes (3.95 - 3.00) = 3500 	imes 0.95 = 3325$

Question 7

Given a right triangle with angle $$	heta$$, adjacent side $$= 8$$, opposite side $$= 15$$, and hypotenuse $$= 17$$, what is $$\sin(	heta)$$?

Accepted Answer

$$\sin(	heta) = \frac{15}{17}$$

Question 8

What is $$\cos(30^\circ)$$?

Accepted Answer

$$\cos(30^\circ) = \frac{\sqrt{3}}{2}$$

Question 9

What is $$	an(60^\circ)$$?

Accepted Answer

$$	an(60^\circ) = \sqrt{3}$$

Question 10

The proper angle for a ladder is about $$75^\circ$$ from the ground. If you have a 17-foot ladder, how far from the house should you place the base of the ladder? (Round to 1 decimal place.)

Accepted Answer

$$17 \cos(75^\circ) \approx 17 	imes 0.2588 \approx 4.4$$ feet$

Question 11

The angle of elevation to the top of a building is $$10^\circ$$ from a point 2 miles away. What is the height of the building in feet? (1 mile = 5280 feet, round to 1 decimal place.)

Accepted Answer

$$h = 2 	imes 5280 	imes 	an(10^\circ) \approx 10560 	imes 0.1763 \approx 1861.7$$ feet$

Question 12

You are designing a ramp with a height of 2 feet and an angle of $$5^\circ. $$How long will the ramp be from A to B?

Accepted Answer

$$	ext{Length} = \frac{2}{\sin(5^\circ)} \approx \frac{2}{0.0872} \approx 22.9$$ feet$

Question 13

A radio tower is 275 feet from a building. From a window, the angle of elevation to the top is $$39^\circ$$ and the angle of depression to the bottom is $$22^\circ. $$What is the height of the tower to 1 decimal place?

Accepted Answer

$$h = 275 	an(39^\circ) + 275 	an(22^\circ) \approx 275 	imes 0.8098 + 275 	imes 0.4040 \approx 222.7 + 111.1 = 333.8$$ feet$

Question 14

A 12-foot ladder is leaning against a building with the base 3 feet from the wall. What angle (to the nearest degree) does the ladder make with the ground? Is the ladder safe if the recommended angle is $$75^\circ or $$greater?

Accepted Answer

$$	heta = \arccos\left(\frac{3}{12}ight) = \arccos(0.25) \approx 75^\circ. $$Yes, the ladder is safe.$

Question 15

An amateur photographer is 403 meters from the base of a 586-meter mountain. What angle (to the nearest degree) should he point his camera from the horizontal to view the sunrise over the mountain?

Accepted Answer

$$	heta = \arctan\left(\frac{586}{403}ight) \approx \arctan(1.454) \approx 55^\circ$$

Step-by-Step Guidance for Trigonometry and Pythagorean Theorem Problems (College Algebra)

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