In a right triangular prism, the legs of the base are each doubled, and the height is reduced by half. How does the new volume compare to the original volume of the same prism?
A.the new volume will be 1/2 as big as the original
B.the new volume will be two times bigger than original
C. the new volume will be 1/8 as big as original
D. the new volume will be 8 times bigger than original.
the volume of a triangular prism is given by
V = (1/2)*(B*H*L)
where
B = base
H = height
L = length
thus the new volume becomes,
V' = (1/2)*(2*B)(2*H)*L
V' = 2*(1/2)*(B*H*L) or
V' = 2*V
thus the new volume is twice the original.
note that first statement is tricky. i interpreted it as "the legs (base and height of triangle) of the base (meaning the base of the prism, which is a right triangle) are each doubled" thus saying that both base, B, and height, H is doubled, and not 1/2 of H. that's why in V', the H is multiplied by 2.
hope this helps~ :)
To find the new volume, we need to compare the dimensions of the original prism to the new prism.
Let's assume the original triangular prism has leg length 'a', leg length 'b', and height 'h'. The original volume would be given by V = (1/2) * a * b * h.
According to the problem, the legs of the base are each doubled and the height is reduced by half. Therefore, in the new prism, the leg length would be 2a, leg length 2b, and the height would be (1/2)h.
The new volume, Vnew, would be given by Vnew = (1/2) * (2a) * (2b) * (1/2h) = abh.
Comparing the new volume to the original volume: Vnew/V = abh / [(1/2) * a * b * h].
Simplifying the expression: Vnew / V = (ab * h) / [(1/2) * ab * h] = 2 / (1/2) = 4.
Therefore, the new volume is 4 times bigger than the original volume.
So, the correct answer is: D. the new volume will be 8 times bigger than the original.
To compare the new volume to the original volume of the right triangular prism, we need to consider the effect of doubling the legs of the base and reducing the height by half.
First, let's establish the formula for the volume of a right triangular prism. The volume (V) of a right triangular prism can be calculated by multiplying the area of the base (B) with the height (h), which can be expressed as:
V = B * h
When the legs of the base are doubled, the area of the base (B) will increase by a factor of 4 (2^2) because the base of a right triangular prism is a triangle, and the area of a triangle is calculated by multiplying the base by the height and dividing by 2.
Therefore, the new area of the base (B') can be written as:
B' = 4B
On the other hand, when the height (h) is reduced by half, the new height (h') can be written as:
h' = 1/2 h
Now, let's substitute the new values for B' and h' in the volume formula:
V' = B' * h'
= (4B) * (1/2 h)
= 2Bh
Comparing this to the original volume:
V' / V = (2Bh) / (Bh)
= 2
This means that the new volume (V') is twice the original volume (V).
Therefore, the answer is B. The new volume will be two times bigger than the original volume of the same prism.