3. If c=37 in. and b = 35 in., find the length of a, sinϕ, cosϕ, and tanϕ.
length of a = sqrt(c^2 - b^2) = sqrt(37^2 - 35^2) = sqrt(1369 - 1225) = sqrt(144) = 12 in.
sinϕ = b/c = 35/37 ≈ 0.946
cosϕ = a/c = 12/37 ≈ 0.324
tanϕ = b/a = 35/12 ≈ 2.917
8. The distance from the top of the tree to the top of its shadow is √(30^2 - 16^2) = √(900 - 256) = √644 ≈ 25.37 feet.
9. The length of the ramp can be found using the Pythagorean theorem: √(4.5^2 + 6^2) = √(20.25 + 36) = √56.25 = 7.5 feet.
10. x = √(101^2 - 20^2) = √(10201 - 400) = √9801 = 99.
11. The new Pythagorean triple using (12, 35, 37) that contains 140 is (24, 70, 74).
12. The angle of elevation formed by the ramp and the ground is arctan(2.5/6) ≈ 21.8 degrees.
13. The horizontal distance can be found using the equation x/2.5 = tan(14°), so x = 2.5*tan(14°) ≈ 0.62 feet.
14. The approximate angle of elevation formed between the board and the lower side of the ditch can be found using arctan(3/8) ≈ 20.21 degrees.
17. The angle of depression from the cat to you can be found using arctan(12/18) ≈ 33.69 degrees.
18. Using the sine rule, c/sinC = a/sinA, you can find c = a*sinC/sinA = 22*sin(52°)/sin(90°) ≈ 17.1 miles.
19. The angle formed between the top of the pole and the rope is arccos(8/10) ≈ 36.87 degrees.
22. Using the Pythagorean theorem, the length of HG = √(85^2 - 13^2) = √(7225 - 169) = √7056 = 84 meters.
23. Using the Pythagorean theorem, the length of AO = √(108^2 - 45^2) = √(11664 - 2025) = √9639 ≈ 98 inches.
24. The inscribed angle JK can be found by halving the measure of the inscribed arc, so mJK = 36°/2 = 18°.
25. m∠KGH = m∠KIJ = 55°.
26. m∠KIJ = arc KJ - arc IJ = (15x - 10) - (5x - 10) = 10x, so m∠KIJ = 10(99) = 990°.
37. The area of the sector of a circle with θ = 225° and a radius of 18 inches is (225/360)*π*(18)^2 = 283.2π square inches.
38. The area of the remaining portion of the cake can be found by subtracting the area of the missing section from the total area of the cake, so (225/360)*π*(14/2)^2 - (225/360)*π*(14/2)^2 = 49.35 square inches.
39. The center of the circle is (-2, 7) and the radius is 17.
40. The equation of the circle is (x-6)^2 + (y+1)^2 = 64.
45. The volume of the cylinder is the same as the cone, so it is also 10π cubic units.
49. The length of the log can be found using the formula for the volume of a cylinder: V = πr^2h, so h = V/(πr^2) = 10000π/(π*16) ≈ 198.94 inches.
50. The volume of the sphere is (4/3)π(7)^3 ≈ 1436.75 cubic inches.
51. The amount of wallpaper that Allyson should purchase is the total area of the wall minus the area of the windows, so (10*22) - (3*6*4) = 220 - 72 = 148 square feet.
52. The exact surface area of the cylinder is 2πrh + 2πr^2 = 2π(3*8) + 2π(4) = 48π + 8π = 56π square inches. The approximate surface area is 56π ≈ 175.93 square inches.
53. The surface area of the pyramid is given by the formula A = (1/2)*b*p + 3*((1/2)*b*s) = (1/2)*3*2.6*3 + 3*((1/2)*3*(9)) ≈ 19.5 + 40.5 = 60 square meters.
54. The surface area of the cone is A = πr^2 + πr√(r^2 + h^2) = π*(6)^2 + π*6*√(6^2 + 8.5^2) = 36π + 51π ≈ 87π square feet.
55. The population density of Canada is 39,292,355 people / 3,855,100 square miles ≈ 10.19 people per square mile.
56. The population density of Europe is 743.5 million people / 3.9 million square miles ≈ 190.64 people per square mile, and the population density of the United States is 332.4 million people / 3.5 million square miles ≈ 95.26 people per square mile, so Europe has a higher population density.
57. The density of the toy is 35 grams / (8*6*9) cm³ = 35 / 432 ≈ 0.0813 grams per cubic centimeter.
58. The mass of the magnesium cylinder sample is π*(3/2)^2*8*1.78 ≈ 38.22 grams.
3. If c=37
in. and b = 35
in., find the length of a
, sinϕ
, cosϕ
, and tanϕ
.
length of a=
Write ratios as a fraction.
sinϕ=
cosϕ=
tanϕ=
8. A 30-foot-tall tree casts a shadow that is 16 feet long. What is the distance from the top of the tree to the top of it's shadow?
9. Tyler buys potting soil every January to plant his vegetable garden. This year he bought 15 25-pound bags of potting soil. He must push them up a 4.5-foot high ramp to his truck. The horizontal distance from the base of the ramp to the truck is 6 feet. Find the length of the ramp.
10. Given that (20, x, 101)
is a Pythagorean triple, and x<101,
what is the value of x
?(
11. Use the Pythagorean triple (12,35, 37) to find a new Pythagorean Triple that contains 140.
Write the Pythagorean triple in order from smallest to largest ( , , )
12. Jerry is building a skateboard ramp. The length of the ramp is 6 feet long and rises to a height of 2.5 feet. What is the approximate angle of elevation formed by the ramp and the ground. Round your answer to the nearest degree.
Write the equation to represent the problem. Use the equation you indicated above to solve for x
13. For a ramp with an angle of elevation of 14° and a height of 2.5 feet. What is the horizontal distance? Round your answer to the nearest tenth.
Write the equation to represent the problem. Use the equation indicated above to solve for the desired side. Round answer to the nearest tenth
14. Oscar uses a board to form a walking plank across a small ditch that has uneven sides. The board is 8 feet in length and stretches across the entire gap. The higher side is 3 feet above the other side. What is the approximate angle of elevation formed between the board and the lower side of the ditch?
Write the equation to represent the problem. Use the equation you indicated above to solve for x
17. A cat in an 18-foot tree looks down at you. You are standing 12 feet from the base of the tree. What is the angle of depression from the cat to you? Round your answer to the nearest whole number. Use the equation you indicated above to solve for x
18. If ∠A=52°
and a = 22
mi., then how long is side c
to the nearest tenth of a mile?
19. Ramon secured an 8 foot volleyball net pole to the ground with a 10-foot rope attached to the top and staked it in the ground, What is the angle formed between the top of the pole and the rope? Round your answer to the nearest degree(1 point)
Write the equation to represent the problem. Use the equation you indicated above to solve for x
22. If CH = 13 m and CF = 85 m, then what is the length of HG to the nearest meter?
23. If BO = 45 in. and BA = 108 in., then what is the length of AO to the nearest whole inch?
24. A triangle is inscribed inside a circle with diameter JL and the inscribed angle at K. The m∠J = 36°
. Using what you know about inscribed angles, find mJK
.
25. If HI≅IJ
and m∠KIJ=55°
, then what is m∠KGH
?
26. If arc KJ=15x−10
and arc IJ=5x−10
, then find the m∠KIJ
.
28. An inscribed circle of a triangle refers to which of the following?
29. The m∠B
is 113°, mBC
is 58°, and mCD
is 116°. Find the missing angle measures.
30. Find the measure of the arc or angle indicated. Assume that lines which appear tangent are tangent.
31. Find the measure of the arc or angle indicated. Assume that lines which appear tangent are tangent.
32. Find the measure of the arc or angle indicated. Assume that lines which appear tangent are tangent.
33. The circumference of Jupiter is approximately 272,946 miles. The circumference of Earth is approximately 24,901 miles. What is the scale factor of Jupiter to Earth rounded to the nearest whole number?
37. What is the area of the sector of a circle with θ=225°
and a radius of 18 inches? Express your answer in terms of π
.
38. A cake has a diameter of 14 in. If part of the cake is eaten, the missing section forms an angle of 225°
. What is the area of the remaining portion of the cake? Round your answer to the nearest tenth
39. Use the equation of the circle provided to identify the center and radius of the circle.
(x+2)2+(y−7)2 =289
40. A circle is drawn on a coordinate plane with the center point at (6, -1) and a radius of 8. Derive the equation of the circle from the given information. Fill in the missing information in the following equation.
( )2 + ( )2=
42. Select the two-dimensional cross-sections that a rectangular prism and a square pyramid have in common
45. A cone and a cylinder have the same base radius and the same height. If the volume of the cone is 10π cubic units, what is the volume of the cylinder?
47. Mei Li measures a cube and a square-based pyramid and finds that the pyramid has the same height and base area as the cube. She also calculates that the volume of the cube is 36 cubic meters. What is the volume of the pyramid?
49. You are examining a log that you find on the ground. The diameter of the log is 32 inches, but the length of the log is unknown. Find the log’s length if its volume is 10,000π
in³. Round your answer to the nearest hundredth.
50. Calculate the volume of a sphere that has a radius of 7 inches, approximate the value rounded to the nearest tenth.
51. Allyson is purchasing wallpaper to cover one wall in her bedroom. The wall is rectangular, with a height of 10 ft. and a length of 22 ft. The wall contains three rectangular windows measuring 6 ft. by 4 ft. each. Determine the amount of wallpaper that Katerina should purchase to cover the wall.
52. A cylinder with a base diameter length of 8 in. and a height of 10 in. Find the exact measure and approximate surface area to the nearest hundredth.The exact surface area =
in²
Approximate surface area =
in
53. A pyramid with a slant height of 9 m whose triangular base measures 3 m on each side. Each altitude of the base measures 2.6 m. Find the surface area of each figure.
54. A cone with radius 6 ft. and a slant height of 8.5. Find the surface area of each figure. giving both the exact value and the approximate rounded to the nearest hundredth(1 point)
The exact surface area =
ft²
Approximate surface area =
ft²
55. Canada’s population was estimated at 39,292,355 people, and its area is 3,855,100 square miles. Which of the following correctly calculates the population density of Canada?
56. In 2022, the population of Europe was approximately 743.5 million people, and the population of the United States was approximately 332.4 million people. Europe is about 3.9 million square miles, and the USA is about 3.5 million square miles in land area. Which answer correctly compares the population densities of the regions?
57. Matthew found a toy in the shape of a rectangular pyramid and is made out of plastic. The rectangular base has a length of 8 cm and a width of 6 cm. The height of the pyramid is 9 cm. Marcus weighs the toy and finds that it is 35 grams. Find the density of the toy to the nearest thousandth.
58. A piece of magnesium has a cylindrical shape with a height of 8 centimeters and a diameter of 3 centimeters. If the magnesium has a density of 1.78 g/cm³, what is the mass of the cylinder sample? Round your answer to the nearest hundredth.
1 answer