Chapter 10

From a point on level ground 30 yards from the base of a building, the angle of elevation to the top of the building is \(38.7^{\circ}\). Approximate the height of the building to the nearest foot.

What is the cost of concrete for a walkway that is 15 feet long, 8 feet wide, and 9 inches deep if the concrete costs \(\$ 30\) per cubic yard?

A rectangular floor measures 20 feet by 25 feet. What will it cost to carpet the floor if the carpet costs \(\$ 28\) per square meter? (Hint: \(1 \mathrm{ft} \approx 0.3 \mathrm{~m}\) )

Use an algebraic equation to determine each rectangle's dimensions. A rectangular field is five times as long as it is wide. If the perimeter of the field is 288 yards, what are the field's dimensions?

Use the Pythagorean Theorem to solve. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. The base of a 20-foot ladder is 15 feet from the house. How far up the house does the ladder reach?

The Statue of Liberty is approximately 305 feet tall. If the angle of elevation of a ship to the top of the statue is \(23.7^{\circ}\), how far, to the nearest foot, is the ship from the statue's base?

A furnace is designed to heat 10,000 cubic feet. Will this furnace be adequate for a 1400 -square-foot house with a 9 -foot ceiling?

A rectangular kitchen floor measures 12 feet by 15 feet. A stove on the floor has a rectangular base measuring 3 feet by 4 feet, and a refrigerator covers a rectangular area of the floor measuring 4 feet by 5 feet. How many square feet of tile will be needed to cover the kitchen floor not counting the area used by the stove and the refrigerator?

Use an algebraic equation to determine each rectangle's dimensions. An American football field is a rectangle with a perimeter of 1040 feet. The length is 200 feet more than the width. Find the width and length of the rectangular field.

Use the Pythagorean Theorem to solve. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. A flagpole has a height of 16 yards. It will be supported by three cables, each of which is attached to the flagpole at a point 4 yards below the top of the pole and attached to the ground at a point that is 9 yards from the base of the pole. Find the total number of yards of cable that will be required.

A 200 -foot cliff drops vertically into the ocean. If the angle of elevation of a ship to the top of the cliff is \(22.3^{\circ}\), how far off shore, to the nearest foot, is the ship?

A water reservoir is shaped like a rectangular solid with a base that is 50 yards by 30 yards, and a vertical height of 20 yards. At the start of a three- month period of no rain, the reservoir was completely full. At the end of this period, the height of the water was down to 6 yards. How much water was used in the three-month period?

A rectangular room measures 12 feet by 15 feet. The entire room is to be covered with rectangular tiles that measure 3 inches by 2 inches. If the tiles are sold at ten for \(\$ 1.20\), what will it cost to tile the room?

An American football field is a rectangle with a perimeter of 1040 feet. The length is 200 feet more than the width. Find the width and length of the rectangular field.A basketball court is a rectangle with a perimeter of 86 meters. The length is 13 meters more than the width. Find the width and length of the basketball court.

Use the Pythagorean Theorem to solve. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. A flagpole has a height of 10 yards. It will be supported by three cables, each of which is attached to the flagpole at a point 4 yards below the top of the pole and attached to the ground at a point that is 8 yards from the base of the pole. Find the total number of yards of cable that will be required.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Objects that appear to be quite different can be topologically equivalent.

A tower that is 125 feet tall casts a shadow 172 feet long. Find the angle of elevation of the Sun to the nearest degree.

The Great Pyramid outside Cairo, Egypt, has a square base measuring 756 feet on a side and a height of 480 feet. a. What is the volume of the Great Pyramid, in cubic yards? b. The stones used to build the Great Pyramid were limestone blocks with an average volume of \(1.5\) cubic yards. How many of these blocks were needed to construct the Great Pyramid?

Use the Pythagorean Theorem to solve. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. A rectangular garden bed measures 5 feet by 12 feet. A water faucet is located at one corner of the garden bed. A hose will be connected to the water faucet. The hose must be long enough to reach the opposite corner of the garden bed when stretched straight. Find the required length of hose.

The Washington Monument is 555 feet high. If you stand one quarter of a mile, or 1320 feet, from the base of the monument and look to the top, find the angle of elevation to the nearest degree.

Although the Eiffel Tower in Paris is not a solid pyramid, its shape approximates that of a pyramid with a square base measuring 120 feet on a side and a height of 980 feet. If it were a solid pyramid, what would be the Eiffel Tower's volume, in cubic yards?

Taxpayers with an office in their home may deduct a percentage of their home- related expenses. This percentage is based on the ratio of the office's area to the area of the home. A taxpayer with an office in a 2200 -square-foot home maintains a 20 foot by 16 foot office. If the yearly utility bills for the home come to \(\$ 4800\), how much of this is deductible?

Use the Pythagorean Theorem to solve. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. A rocket ascends vertically after being launched from a location that is midway between two ground-based tracking stations. When the rocket reaches an altitude of 4 kilometers, it is 5 kilometers from each of the tracking stations. Assuming that this is a locale where the terrain is flat, how far apart are the two tracking stations?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Euclidean geometry serves as an excellent model for describing the diverse forms that arise in nature.

You are about to sue your contractor who promised to install a water tank that holds 500 gallons of water. You know that 500 gallons is the capacity of a tank that holds 67 cubic feet. The cylindrical tank has a radius of 3 feet and a height of 2 feet 4 inches. Does the evidence indicate you can win the case against the contractor if it goes to court?

This activity is suggested for two or three people. Research some of the practical applications of topology. Present the results of your research in a seminar to the entire class. You may also want to include a discussion of some of topology's more unusal figures, such as the Klein bottle and the Möbius strip.

A police helicopter is flying at 800 feet. A stolen car is sighted at an angle of depression of \(72^{\circ}\). Find the distance of the stolen car, to the nearest foot, from a point directly below the helicopter.

Two cylindrical cans of soup sell for the same price. One can has a diameter of 6 inches and a height of 5 inches. The other has a diameter of 5 inches and a height of 6 inches. Which can contains more soup and, therefore, is the better buy?

Use the Pythagorean Theorem to solve. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. This problem appeared on a high school exit exam: Alex is building a ramp for a bike competition. He has two rectangular boards. One board is six meters and the other is five meters long. If the ramp has to form a right triangle, what should its height be? Students were asked to select the correct answer from the following options: 3 meters; 4 meters; \(3.3\) meters; \(7.8\) meters. a. Among the available choices, which option best expresses the ramp's height? How many feet, to the nearest tenth of a foot, is this? Does a bike competition that requires riders to jump off these heights seem realistic? (ouch!) b. Express the ramp's height to the nearest hundredth of a meter. By how many centimeters does this differ from the "correct" answer on the test? How many inches, to the nearest half inch, is this? Is it likely that a carpenter with a tape measure would make this error? c. According to the problem, Alex has boards that measure 5 meters and 6 meters. A 6 -meter board? How many feet, to the nearest tenth of a foot, is this? When was the last time you found a board of this length at Home Depot?

Research non-Euclidean geometry and plan a seminar based on your group's research. Each group member should research one of the following five areas: a. Present an overview of the history of the people who developed non- Euclidean geometry. Who first used the term and why did he never publish his work? b. Present an overview of the connection between Saccheri quadrilaterals and non-Euclidean geometry. Describe the work of Girolamo Saccheri. c. Describe how Albert Einstein applied the ideas of Gauss and Riemann. Discuss the notion of curved space and a fourth dimension. d. Present examples of the work of M. C. Escher that provide ways of visualizing hyperbolic and elliptic geometry. e. Describe how non-Euclidean geometry changed the direction of subsequent research in mathematics. After all research has been completed, the group should plan the order in which each group member will speak. Each person should plan on taking about five minutes for his or her portion of the presentation.

A wheelchair ramp is to be built beside the steps to the campus library. Find the angle of elevation of the 23 -foot ramp, to the nearest tenth of a degree, if its final height is 6 feet.

A circular backyard pool has a diameter of 24 feet and is 4 feet deep. One cubic foot of water has a capacity of approximately \(7.48\) gallons. If water costs \(\$ 2\) per thousand gallons, how much, to the nearest dollar, will it cost to fill the pool?

A school playground is in the shape of a rectangle 400 feet long and 200 feet wide. If fencing costs \(\$ 14\) per yard, what will it cost to place fencing around the playground?

If the measures of two angles of a triangle are known, explain how to find the measure of the third angle.

A kite flies at a height of 30 feet when 65 feet of string is out. If the string is in a straight line, find the angle that it makes with the ground. Round to the nearest tenth of a degree.

A rectangular field is 70 feet long and 30 feet wide. If fencing costs \(\$ 8\) per yard, how much will it cost to enclose the field?

Can a triangle contain two right angles? Explain your answer.

If you are given the lengths of the sides of a right triangle, describe how to find the sine of either acute angle.

Explain the following analogy: In terms of formulas used to compute volume, a pyramid is to a rectangular solid just as a cone is to a cylinder.

In Exercises 49-50, express the required calculation in terms of \(\pi\) and then round to the nearest tenth. How much fencing is required to enclose a circular garden whose radius is 20 meters?

What general assumption did Euclid make about a point and a line in order to prove that the sum of the measures of the angles of a triangle is \(180^{\circ}\) ?

Describe one similarity and one difference between the sine ratio and the cosine ratio in terms of the sides of a right triangle.

Explain why a cylinder is not a polyhedron.

Express the required calculation in terms of \(\pi\) and then round to the nearest tenth. A circular rug is 6 feet in diameter. How many feet of fringe is required to edge this rug?

What are similar triangles?

If one of the acute angles of a right triangle is \(37^{\circ}\), explain why the sine ratio does not increase as the size of the triangle increases.

How many plants spaced every 6 inches are needed to surround a circular garden with a 30 -foot radius?

If the ratio of the corresponding sides of two similar triangles is 1 to \(1\left(\frac{1}{1}\right)\), what must be true about the triangles?

If the measure of one of the acute angles and the hypotenuse of a right triangle are known, describe how to find the measure of the remaining parts of the triangle.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When completely full, a cylindrical soup can with a diameter of 3 inches and a height of 4 inches holds more soup than a cylindrical can with a diameter of 4 inches and a height of 3 inches.