Answer:
The answers are:
A. 7.5 m³
B. 90 m²
C. 180 m³
Step-by-step explanation:
Let us answer the questions about the volume. We know that the volume of a prism is V=l*w*h, where l stands for the length, w stands for the width and h stands for the height.
Volumes.
A prism with half the original sizes. From the original prism we know that V=l*w*h=60 m³. The new prism have dimension l'=l/2, h'=h/2 and w'=w/2. Then, its volume V' is
V' = l'*h'*w' = (l/2)*(h/2)*(w/2) = (l*w*h)/8=V/8 = 60/8 = 7.5 m³.
A prism with height tripled. In this case all the dimension are the same, except for the height: this l'=l, w'=w and h'=3*h. Then, the new volume V' is
V' = l'*h'*w' = l*(3*h)*w = 3*l*h*w=3*V = 3*60 = 180 m³.
Surface area.
A prism with half the original sizes.
The formula for the surface area of a prism is
[tex]A = 2A_b + P_bh[/tex]
where [tex]A_b[/tex] stands for the area of the base and [tex]P_b[/tex] stands for the perimeter of the base. As the base is rectangle, [tex]A_b=lw[/tex] and [tex]P_b = 2(l+w)[/tex].
Hence,
[tex]A = 2lw + 2(l+w)h.[/tex]
Then, the area of the new prism is A' ()recall that the dimension of the new prism are l'=l/2, h'=h/2 and w'=w/2).
[tex]A' = 2(l')(w') + 2(l'+w')h' = 2\frac{l}{2}\frac{w}{2} + 2(\frac{l}{2}+\frac{w}{2})\frac{h}{2} = \frac{2lw}{4} + 2\frac{l+w}{2}\frac{h}{2} = \frac{2lw}{4} + \frac{2(l+w)h}{4}.[/tex]
In this expression we can extract a common factor 1/4, thus
[tex]A' = \frac{1}{4}(2lw + 2(l+w)h) = \frac{1}{4}A = 360/4 = 90 m^2[/tex]
