Byron’s house has a roof with dimensions shown. (Roof: 1m Height, 7m Hypotenuses, 12m length)
He knows that the average rainfall in his
suburb is 30cm per year. Byron would like
to install a cylindrical rainwater tank to hold the
water that runs off the roof. The tank is to be made of
moulded plastic but Byron wants to minimise the
area of moulded plastic required and hence the cost.
Can you help Byron answer the following questions?
1) What volume of water will fall on the roof per year?
2) How many litres of water does the tank need to hold?
3) How high will a tank that has base diameter 3 m need to be?
4) What is the surface area of moulded plastic required to build the tank in 3?
5) Suggest the dimensions of the tank so that the minimum amount of moulded plastic
is used.
1) 30cm * √48m * 12m = 24.94 m^3
...
3) π/4 * 3^2 * h = 24.94
Now the rest should not be too hard. Come back if you get stuck.
Sure, I'd be happy to help Byron with his rainwater tank questions!
1) To calculate the volume of water that will fall on the roof per year, we first need to find the area of the roof. The formula to find the area of a triangle is (base * height) / 2. So, the area of the roof is (12m * 1m) / 2 = 6 square meters.
Next, we multiply the area of the roof by the average rainfall per year: 6 square meters * 0.3 meters (converted from cm) = 1.8 cubic meters.
2) To convert cubic meters to liters, we multiply by 1000. So, the tank needs to hold 1.8 cubic meters * 1000 = 1800 liters of water.
3) To determine how high a tank with a base diameter of 3 meters needs to be, we can use the formula for the volume of a cylinder, which is pi * r^2 * h, where r is the radius and h is the height. We know the base diameter is 3 meters, which means the radius is 1.5 meters. Since Byron wants to hold 1800 liters of water, we need to solve for h.
The formula can be rearranged as h = V / (pi * r^2), where V is the volume. Plugging in the values, we get h = 1800 / (pi * (1.5)^2) ≈ 254.65 meters.
4) The surface area of the molded plastic required to build the tank is the sum of the lateral surface area and the base area of the cylinder. The lateral surface area of a cylinder is given by the formula 2 * pi * r * h, where r is the radius and h is the height. The base area is given by pi * r^2. Plugging in the values, the surface area is (2 * pi * 1.5 * 254.65) + (pi * 1.5^2) ≈ 2410.79 square meters.
5) To suggest dimensions of the tank that minimize the amount of molded plastic used, we need to consider the relationship between the height and base diameter of the tank. To minimize the surface area, we want to minimize the sum of the lateral surface area and the base area. One way to achieve this is by making the base diameter and height equal. So, I suggest building a tank with a base diameter of 3 meters and a height of 3 meters.
I hope that helps Byron with his rainwater tank planning!
1) To find the volume of water that will fall on the roof per year, we need to calculate the area of the roof and multiply it by the average rainfall.
The area of the roof can be calculated using the formula for the area of a triangle: (1/2) * base * height.
In this case, the base is 12m and the height is 1m, so the area of the roof is (1/2) * 12m * 1m = 6m².
To convert the average rainfall from cm to meters, we divide it by 100:
30cm / 100 = 0.3m.
So the volume of water that will fall on the roof per year is the product of the area and the average rainfall:
Volume = Area * Average rainfall = 6m² * 0.3m = 1.8 cubic meters.
2) To convert the volume of water from cubic meters to liters, we multiply it by 1000:
Volume in liters = 1.8 cubic meters * 1000 = 1800 liters.
Therefore, the tank needs to hold 1800 liters of water.
3) To find the height of a cylindrical tank with a base diameter of 3m, we can use the formula for the volume of a cylinder: V = πr²h, where r is the radius of the base and h is the height.
The radius of the base is half the diameter, so it is 3m / 2 = 1.5m.
We know that the volume of water the tank needs to hold is 1800 liters, which is equal to 1800 cubic decimeters (1 liter = 1 cubic decimeter).
Converting the volume from cubic decimeters to cubic meters, we divide by 1000:
1800 cubic decimeters / 1000 = 1.8 cubic meters.
Now we can rewrite the volume formula and solve for h:
1.8 = π(1.5)²h.
Simplifying the equation:
1.8 = 2.25πh.
Dividing both sides of the equation by 2.25π:
h = 1.8 / (2.25π).
Using a calculator, we can solve this equation to find the height of the tank.
4) To find the surface area of the molded plastic required to build the tank in the previous question, we need to find the lateral surface area of the cylinder.
The lateral surface area of a cylinder can be calculated using the formula: A = 2πrh, where r is the radius and h is the height.
We already found the radius in the previous question, which is 1.5m.
Now, using the height calculated in question 3, we can calculate the surface area:
Surface area = 2π(1.5)(h).
Substituting the value of h in the equation, we can find the surface area.
5) To suggest the dimensions of the tank that minimize the amount of molded plastic used, we need to consider that the surface area of the cylinder is minimized when the height and diameter are equal.
So, the suggested dimensions for the tank are a base diameter of 3m and a height of 3m.
To help Byron answer the questions, let's break down the problem step by step:
1) To find the volume of water that will fall on the roof per year, we need to calculate the area of the roof and multiply it by the average rainfall.
- The area of the roof can be calculated using the formula for the area of a triangle: A = 0.5 * base * height. In this case, the base is 12m and the height is 1m. Thus, A = 0.5 * 12m * 1m = 6 square meters.
- Since the average rainfall is given in centimeters, we need to convert it to meters by dividing by 100. So, the average rainfall in meters is 30cm / 100 = 0.3m.
- Now, we can calculate the volume of water using the formula for the volume of a cuboid: V = A * h, where A is the area of the roof and h is the average rainfall. Substituting the values, we get V = 6 square meters * 0.3m = 1.8 cubic meters.
Therefore, the volume of water that will fall on the roof per year is 1.8 cubic meters.
2) To calculate the number of liters the tank needs to hold, we need to convert the volume from cubic meters to liters.
- Since 1 cubic meter is equal to 1000 liters, we can multiply the volume in cubic meters by 1000 to get the volume in liters. In this case, the volume in liters would be 1.8 cubic meters * 1000 = 1800 liters.
Therefore, the tank needs to hold 1800 liters of water.
3) To find the height of the tank with a base diameter of 3m, we need to use the formula for the volume of a cylinder: V = π * r^2 * h, where V is the volume, π is a mathematical constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cylinder.
- Since the base diameter is given as 3m, the radius (r) would be half of that, which is 1.5m.
- By rearranging the formula, we can solve for h: h = V / (π * r^2). Substituting the values, we get h = 1800 liters / (3.14 * (1.5m)^2) = approximately 254.45 meters.
Therefore, the tank that has a base diameter of 3 meters needs to be approximately 254.45 meters high.
4) To find the surface area of moulded plastic required to build the tank, we need to calculate the lateral surface area of the cylinder.
- The lateral surface area of a cylinder can be calculated using the formula: A = 2 * π * r * h, where A is the surface area, π is the mathematical constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cylinder.
- Substituting the values, we get A = 2 * 3.14 * 1.5m * 254.45m = approximately 2396.76 square meters.
Therefore, the surface area of moulded plastic required to build the tank would be approximately 2396.76 square meters.
5) To suggest the dimensions of the tank that use the minimum amount of moulded plastic, we need to consider the relationship between the height and base diameter of the cylinder.
- To minimize the surface area, we want a tank with a smaller height and larger base diameter. However, the specific dimensions would depend on the desired volume of the tank.
For example, if we want a tank with a larger volume, we might choose a larger base diameter and a shorter height. Conversely, if we want a smaller volume, we might choose a smaller base diameter and a taller height.
Ultimately, the optimal dimensions for minimizing the amount of moulded plastic used would depend on the specific requirements and constraints of the situation.