The volumes of two similar figures are given. The surface area of the smaller figure is given. Find the surface area of the larger figure.
Smaller figure:
V = 2 yd^3
SA = 13 yd^2
Larger figure:
V = 250 yd^3
SA = ???
Please help? Thanks
The surface areas of two similar figures are given. The volume of the larger figure is given. Find the volume of the smaller figure.
S.A.= 450 in^2
S.A.= 800 in^2
V= 1472 in^3
To find the surface area of the larger figure, you can use the concept of ratios between the volumes and surface areas of similar figures.
First, let's find the ratio of the volumes between the larger and smaller figures:
Ratio of volumes = Volume of larger figure / Volume of smaller figure
= 250 yd^3 / 2 yd^3
= 125
Next, since the two figures are similar, the ratio of their surface areas will be the square of the ratio of their volumes:
Ratio of surface areas = (Ratio of volumes)^2
= 125^2
= 15,625
Now, multiply the surface area of the smaller figure by the ratio of surface areas to find the surface area of the larger figure:
Surface area of larger figure = Surface area of smaller figure x Ratio of surface areas
= 13 yd^2 x 15,625
= 203,125 yd^2
Therefore, the surface area of the larger figure is 203,125 yd^2.
To find the surface area of the larger figure, we need to use the concept of similarity. Similar figures have the same shape but might have different sizes. The ratio of corresponding sides of similar figures is constant.
Let's denote the ratio of the corresponding sides of the figures as k. In this case, the ratio of their volumes is also k^3, as volume is a three-dimensional measurement. Therefore, we can set up the following equation:
(Volume of the larger figure) / (Volume of the smaller figure) = k^3
250 / 2 = k^3
Simplifying, we have:
125 = k^3
Taking the cube root of both sides, we find:
∛125 = k
k = 5
Now that we know the ratio of the sides is 5, we can find the surface area of the larger figure. Since the figures are similar, the ratio of their surface areas would be k^2, which is 5^2 = 25.
Finally, we can calculate the surface area of the larger figure by multiplying the surface area of the smaller figure by the square of the ratio:
Surface area of the larger figure = (Surface area of the smaller figure) × (k^2)
= 13 yd^2 × 25
= 325 yd^2
Therefore, the surface area of the larger figure is 325 yd^2.