The volume V of a pyramid varies jointly as the area of it’s base B and its height h. If the volume of the pyramid is 24 cubic meters when the area of it’s base is 18 square meters and its height is 4 meters find the volume of a pyramid when the area of it’s base is 21 square meters and its height is 7 meters.
v = k B h
24 = k* 18 * 4
k = 24 / 72 = 1/3 If you did not already know volume of pointy object with straight sides = (1/3) * base area * height
so
V = (1/3) * 21 * 7 = 7 * 7 = 49
Volume = k(base)(height)
given: Vol = 24 m^3, base = 18 m^2 , height = 4 m
24 m^3 = k(18)(4) m^3
k = 24/(18*4) = 1/3
Vol = (1/3)(base)(height)
when base = 21, height = 7
Vol = (1/3)(21 m^2)(7 m) = 49 m^3
To solve this problem, we need to find the constant of variation relating the volume of the pyramid to the area of its base and height. Once we have the constant of variation, we can use it to find the volume of the pyramid in the given scenario.
Let's assign variables to the different quantities involved:
- Volume of the pyramid: V
- Area of the base: B
- Height of the pyramid: h
We are given that the volume V varies jointly as the area of the base B and the height h, which can be represented by the equation:
V = k * B * h
where k is the constant of variation.
Now, we can use the given information to find the value of k. Given that the volume of the pyramid is 24 cubic meters when the area of the base is 18 square meters and the height is 4 meters, we can substitute these values into the equation:
24 = k * 18 * 4
Simplifying this equation, we get:
24 = 72k
To solve for k, divide both sides of the equation by 72:
k = 24 / 72
Simplifying further, we find that k = 1/3.
Now that we have the value of k, we can use it to find the volume of the pyramid when the area of its base is 21 square meters and its height is 7 meters. Substituting these values into the equation:
V = (1/3) * 21 * 7
Simplifying this equation, we find:
V = 49 cubic meters
Therefore, the volume of the pyramid in the given scenario is 49 cubic meters.