Question 1:
You are designing a 3D model of a building that is 100 feet tall with a sloped roof. The slope of the roof is 30 degrees from the horizontal plane. You want to create a 3D model of the building with the roof included. What is the length of the roof from one end to the other? Provide a step-by-step solution.
Solution:
To find the length of the roof, we need to know the height of the building and the angle of the slope. We can use trigonometry to calculate the length of the roof. First, we need to find the length of the side of the building that is perpendicular to the ground. We can use the Pythagorean theorem to find this value:
a^2 + b^2 = c^2 where a is the height of the building, b is half the length of the roof, and c is the hypotenuse.
a = 100 feet
b = (c / 2) * sin(30)
sin(30) = 0.5 (since sin(30) = opposite / hypotenuse, and we know the opposite is half of the length of the roof)
b = (c / 2) * 0.5 = c / 4
Using the Pythagorean theorem, we can solve for c:
100^2 + (c/4)^2 = c^2
10000 + c^2/16 = c^2
c^2 - c^2/16 = 10000
15c^2/16 = 10000
c^2 = 10000 * 16 / 15
c = 110.85 feet
Therefore, the length of the roof from one end to the other is approximately 110.85 feet.
Question 2:
You are designing a 3D model of a cylindrical tank that has a diameter of 6 meters and a height of 10 meters. The tank is made of steel with a thickness of 10 millimeters. Calculate the volume of steel used to manufacture the tank. Provide a step-by-step solution.
Solution:
To calculate the volume of steel used to manufacture the tank, we need to calculate the volume of the tank itself and subtract the volume of the empty space inside the tank.
The volume of the tank can be calculated using the formula for the volume of a cylinder:
V = πr^2h
where r is the radius of the cylinder (which is half the diameter), and h is the height of the cylinder.
r = 6 / 2 = 3 meters
h = 10 meters
V = π(3^2)(10) = 282.74 cubic meters
To calculate the volume of the empty space inside the tank, we need to subtract the volume of the cylinder that would fit inside the tank with a thickness of 10 millimeters. We can use the same formula for the volume of a cylinder, but with a slightly smaller radius:
r = 5.99 meters (since the thickness of the steel is 10 millimeters, or 0.01 meters, and we need to subtract this from the radius)
h = 10 meters
V_empty = π(5.99^2)(10) = 282.25 cubic meters
Finally, we can calculate the volume of steel used to manufacture the tank by subtracting the volume of the empty space from the volume of the tank:
V_steel = V - V_empty = 282.74 - 282.25 = 0.49 cubic meters
Therefore, the volume of steel used to manufacture the tank is 0.49 cubic meters.
Question 3:
You are designing a 3D model of a mechanical part that consists of two interlocking gears. The first gear has a diameter of 20 centimeters and 30 teeth, while the second gear has a diameter of 10 centimeters and 15 teeth. Calculate the gear ratio between the two gears. Provide a step-by-step solution.
Solution:
The gear ratio is the ratio of the number of teeth on the first gear to the number of teeth on the second gear. Since the gears are interlocking, they will rotate at the same speed, so the gear ratio will also represent the ratio of the diameters of the two gears.
Gear ratio = Number of teeth on the first gear / Number of teeth on the second gear
Number of teeth on the first gear = 30
Number of teeth on the second gear = 15
Gear ratio = 30 / 15 = 2
Therefore, the gear ratio between the two interlocking gears is 2:1.
Question 4:
You are designing a 3D model of a bridge that is supported by two towers. The bridge is 200 meters long, and the towers are 60 meters tall. The towers are positioned 50 meters apart from each other, and they are each 20 meters wide.
Calculate the volume of concrete needed to build the two towers.
Solution:
To calculate the volume of concrete needed to build the two towers, we need to calculate the volume of each tower and then add them together.
The volume of each tower can be calculated as the product of its height, width, and depth. We can assume that the towers are rectangular prisms with a square base and use the dimensions provided to calculate the volume:
Volume of each tower = Height x Width x Depth
Height = 60 meters
Width = 20 meters
Depth = Distance between the towers = 50 meters
Volume of each tower = 60 x 20 x 50 = 60,000 cubic meters
To find the total volume of concrete needed to build the two towers, we simply add the volumes of the two towers together:
Total volume of concrete = Volume of each tower x 2 = 60,000 x 2 = 120,000 cubic meters
Therefore, the volume of concrete needed to build the two towers is 120,000 cubic meters.