You propose to build a space ship that allows the entire population of the United States (280 million) to leave earth. Your plans allocate an average space of 1,000(=10*10*10) cubic feet to each person. You space ship is going to have the shape of a sphere. The radius of you space ship is ?????? feet. The volume of a sphere of radius "r" equals (4/3)*pi*r^3
After due Congress decide to support your space ship if you provide a cabin with a skylight for each of the 280,000,000 passengers. Your figure that on average each passenger must have 100 square feet on the surface of your space ship. The radius of you new and improved space ship is ????? feet. The surface of a sphere of radius "r" equals 4*pi*r^2.
You tell Congress that the space ship they ask for would have a radius that is ?????? times bigger than what you suggested and the volume (and weight and cost) would be ?????? as large.
How would i start solving this? I am so stuck.
To solve this problem, we need to find the radius of the space ship in both scenarios – first without the skylights, and then with the skylights.
Let's start with the first part, where each person is allocated an average space of 1,000 cubic feet. We can use the formula for the volume of a sphere to calculate the radius:
Volume = (4/3) * π * r^3
Given that each person is allocated 1,000 cubic feet and there are 280 million people in the United States, the total volume needed for the population would be:
Total volume needed = 1,000 cubic feet/person * 280,000,000 people
Now, we can solve for the radius:
Total volume needed = (4/3) * π * r^3
1,000 * 280,000,000 = (4/3) * π * r^3
Dividing both sides by (4/3) * π:
1,000 * 280,000,000 / ((4/3) * π) = r^3
Taking the cube root of both sides gives us the radius:
r = (1,000 * 280,000,000 / ((4/3) * π))^(1/3)
Now, let's move on to the second part, where each passenger must have 100 square feet on the surface of the space ship. We can use the formula for the surface area of a sphere to calculate the new radius:
Surface area = 4 * π * r^2
Given that each passenger needs 100 square feet and there are 280 million passengers, the total surface area needed for the population would be:
Total surface area needed = 100 square feet/person * 280,000,000 people
Now, we can solve for the radius:
Total surface area needed = 4 * π * r^2
100 * 280,000,000 = 4 * π * r^2
Dividing both sides by 4 * π:
100 * 280,000,000 / (4 * π) = r^2
Taking the square root of both sides gives us the new radius:
r = √(100 * 280,000,000 / (4 * π))
Finally, to determine the ratio between the two radii, we divide the second radius by the first radius:
Ratio = new radius / original radius = (√(100 * 280,000,000 / (4 * π))) / ((1,000 * 280,000,000 / ((4/3) * π))^(1/3))
The volume, weight, and cost of the space ship would be proportional to the cube of the radius. So, to find the ratio of the volume, weight, and cost between the two scenarios, we calculate:
Ratio = (new radius / original radius)^3
I hope this explanation helps you solve the problem!