A movie theater offers two options for membership. The first option costs $10 per ticket but an initial fee of $10. The second option costs $7.50 per ticket but an initial fee of $50. After how many tickets are both options at the same price?
Let x be the number of tickets bought.
For the first option, the total cost is 10x + 10.
For the second option, the total cost is 7.5x + 50.
Setting these two equations equal to each other, we get:
10x + 10 = 7.5x + 50
2.5x = 40
x = 16
Therefore, after 16 tickets, both options are at the same price.
Two pumps are being filled by two different pumps. The first vat has 4 gallons and is being filled at a rate of 12 gallons per second. The second vat has 23 gallons and is bein10 gallons per second. After how many seconds will both vats have the same amount of liquid?
Let x be the number of seconds it takes for both vats to have the same amount of liquid.
The amount of liquid in the first vat after x seconds is 4 + 12x.
The amount of liquid in the second vat after x seconds is 23 + 10x.
Setting these two equations equal to each other, we get:
4 + 12x = 23 + 10x
2x = 19
x = 9.5
Therefore, after 9.5 seconds, both vats will have the same amount of liquid.
David wants to find the height of a building. He places his 36 feet ladder against the very top of the building. David then measures the base of the building to his ladder which is 7 feet. How tall is the building? Round your answer to the nearest tenth.
We can use the Pythagorean Theorem to solve this problem. The ladder, the height of the building, and the base of the building form a right triangle. Let the height of the building be h.
According to the Pythagorean Theorem: (height of the building)^2 + (base of the building)^2 = (length of the ladder)^2
h^2 + 7^2 = 36^2
h^2 + 49 = 1296
h^2 = 1296 - 49
h^2 = 1247
h = √1247
h ≈ 35.3
Therefore, the height of the building is approximately 35.3 feet.
A ball is placed into a cubed box as shown below. Find the difference in space from the sphere and the box. Use 3.14 for
.
The space between the sphere and the box is about cubic inches.
To find the space between the sphere and the box, we first need to calculate the volume of both the sphere and the box.
Volume of a sphere: V = (4/3) * π * r^3
Given that the radius of the sphere is half the length of the side of the cube, the radius of the sphere is 3 inches (half of 6).
V_sphere = (4/3) * 3.14 * 3^3
V_sphere = (4/3) * 3.14 * 27
V_sphere = 113.04 cubic inches
Volume of a cube: V = s^3 (where s is the length of a side of the cube)
V_cube = 6^3
V_cube = 216 cubic inches
The difference in space between the sphere and the box is:
Difference = V_cube - V_sphere
Difference = 216 - 113.04
Difference ≈ 102.96 cubic inches
Therefore, the space between the sphere and the box is about 102.96 cubic inches.
Find the volume of a pyramid that has a square base where the length of the base is 43 feet and the height of the pyramid is 51 feet.
The volume of a pyramid can be calculated using the formula:
V = (1/3) * base area * height
Since the base of the pyramid is a square, the base area is given by the formula:
Base Area = side * side
Given that the length of the base is 43 feet, the base area is:
Base Area = 43 * 43
Base Area = 1849 square feet
Now, we can find the volume of the pyramid using the base area and the height:
V = (1/3) * 1849 * 51
V = (1/3) * 94399
V ≈ 31466.33 cubic feet
Therefore, the volume of the pyramid is approximately 31466.33 cubic feet.