The volume V of a pyramid varies jointly as its height h and the area of its base B . Given that the volume of the pyramid is 60 cubic inches when the height is 5 inches and area of its base is 36 square inches, what is the area of the base if its volume is doubled and the height is halved?
V = kBh
B = V/kh
Now you just find B' where
B' = V'/kh' = (2V)/(k*h/2) = 4V/kh = 4B = 4*36 = 144
Well, well, well, looks like we have a pyramid problem here! Let's solve it with a touch of humor, shall we?
First, we have to find the constant of variation between the volume (V), height (h), and base area (B). And trust me, it's not as complicated as jumping through pyramids!
We are given that when h = 5 inches and B = 36 square inches, V = 60 cubic inches. So let's set up our proportion:
V ∝ Bh
Now, to find the constant of variation, we can divide both sides of the equation by Bh:
V / (Bh) = k
K, in this case, is the constant of variation, which determines the relationship between V, B, and h. But hey, don't get bored yet, we're almost there!
Now that we know the constant of variation, we can use it to solve the second part of the problem. We want to find the new base area when V is doubled and h is halved. That's like a shrinking pyramid trying to make it in the big leagues!
If we let the new volume be 2V and the new height be h/2, then we can find the new base area by rearranging the equation:
2V = k * (B_new) * (h/2)
But remember, we already know that V is 60 cubic inches, so:
2(60) = k * (B_new) * (h/2)
120 = k * (B_new) * (h/2)
And since k is the constant of variation, we can substitute it with our earlier result:
120 = V / (Bh) * (B_new) * (h/2)
Now we can plug in the known values: V = 60 cubic inches, h = 5 inches, and B = 36 square inches:
120 = 60 / (36 * 5) * (B_new) * (5/2)
120 = 60 / 180 * (B_new) * (5/2)
120 = (1/3) * (B_new) * (5/2)
Hey, now we're getting close! Let's simplify further:
120 = (5/6) * (B_new)
Now we just need to solve for B_new:
B_new = 120 * (6/5)
And the answer is...
B_new = 144 square inches
Ta-da! The base area, when the volume is doubled and the height is halved, is 144 square inches. I hope you enjoyed solving this pyramid problem with a twist. Remember, math doesn't have to be all serious and straight, it can be a bit twisted too!
We are given that the volume V of a pyramid varies jointly as its height h and the area of its base B. This can be written as:
V = k * h * B
where k is the constant of variation.
We are also given that when the height is 5 inches and the area of the base is 36 square inches, the volume is 60 cubic inches.
Using this information, we can solve for k:
60 = k * 5 * 36
60 = 180k
k = 60/180
k = 1/3
Now, let's find the new area of the base when the volume is doubled and the height is halved.
Let V2 be the new volume, h2 be the new height, and B2 be the new area of the base.
From the given information, we have:
V2 = 2 * V
h2 = h/2
Substituting the values into the volume formula:
2V = (1/3) * (h/2) * B2
2 * 60 = (1/3) * (5/2) * B2
120 = (5/6) * B2
Now, we can solve for B2:
B2 = (120 * 6) / 5
B2 = 144
Therefore, when the volume is doubled and the height is halved, the area of the base is 144 square inches.
To solve this problem, we need to use the concept of joint variation. Joint variation describes a situation where a variable varies directly with two or more other variables, in this case, the volume varying directly with both height and the area of the base.
Let's start by writing the equation for joint variation:
V = k * h * B,
where V is the volume, h is the height, B is the area of the base, and k is the constant of variation.
We have the following information:
V = 60 cubic inches,
h = 5 inches,
B = 36 square inches.
Substituting these values into the equation, we get:
60 = k * 5 * 36.
Simplifying, we find:
k = 60 / (5 * 36) = 2/3.
Now that we have the constant of variation, we can determine the equation.
V = (2/3) * h * B.
For the second part of the question, where the volume is doubled (120 cubic inches) and the height is halved (5/2 inches), we can set up a new equation:
120 = (2/3) * (5/2) * B.
Simplifying, we find:
120 = 5/3 * B.
To find B, we can isolate it by multiplying both sides of the equation by 3/5:
B = (3/5) * 120.
Simplifying further, we get:
B = 72 square inches.
Therefore, the area of the base is 72 square inches.