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.
V = 5 in3
V = 40 in3
S.A. = 4 in2
To find the surface area of the larger figure, we need to use the concept of similarity. Since the figures are similar, their corresponding sides are proportional.
Let's denote the surface area of the larger figure as SA₁ and the surface area of the smaller figure as SA₂.
We know that the volumes of the two figures are in the ratio of the cubes of their corresponding sides:
(Volume of larger figure / Volume of smaller figure) = (Side of larger figure / Side of smaller figure)³
So, we can write:
(40 in³ / 5 in³) = (Side of larger figure / Side of smaller figure)³
Now let's solve for the side of the larger figure:
(40/5) = (Side of larger figure / Side of smaller figure)³
8 = (Side of larger figure / Side of smaller figure)³
Taking the cube root of both sides:
2 = (Side of larger figure / Side of smaller figure)
This means that the side of the larger figure is twice the side of the smaller figure.
Now, we know that the ratio of the areas of similar figures is the square of the ratio of their corresponding sides:
(SA₁ / SA₂) = (Side of larger figure / Side of smaller figure)²
Plugging in the values we know:
(SA₁ / 4 in²) = (2)²
(SA₁ / 4 in²) = 4
To find SA₁, we can solve for it:
SA₁ = 4 in² * 4
SA₁ = 16 in²
Therefore, the surface area of the larger figure is 16 square inches.
To find the surface area of the larger figure, we need to determine the scale factor between the two similar figures. The scale factor is the ratio of any corresponding lengths in the two figures (such as sides, heights, or radii).
Since the volumes of the two figures are given, we can use the formula for volume to find the scale factor. The formula for volume is V = l * w * h, where V represents the volume, l represents the length, w represents the width, and h represents the height.
Let's denote the length, width, and height of the smaller figure as l1, w1, and h1 respectively, and do the same for the larger figure as l2, w2, and h2.
Given:
V1 = 5 in^3 (volume of the smaller figure)
V2 = 40 in^3 (volume of the larger figure)
Using the volume formula, we have:
V1 = l1 * w1 * h1
V2 = l2 * w2 * h2
Dividing the two equations, we get:
V2 / V1 = (l2 * w2 * h2) / (l1 * w1 * h1)
Since we are only interested in the scale factor, we can cancel out the volume terms:
V2 / V1 = (l2 * w2 * h2) / (l1 * w1 * h1) = (l2 / l1) * (w2 / w1) * (h2 / h1)
Now, we can plug in the given values:
V2 / V1 = 40 / 5 = 8
So, the scale factor is 8. This means that corresponding lengths in the larger figure are 8 times the corresponding lengths in the smaller figure.
Now that we have the scale factor, we can find the surface area of the larger figure. The surface area of a figure is proportional to the square of its corresponding lengths. Therefore, we can find the surface area of the larger figure by multiplying the surface area of the smaller figure by the square of the scale factor.
Given:
S.A. = 4 in^2 (surface area of the smaller figure)
To find the surface area of the larger figure:
S.A. of larger figure = (S.A. of smaller figure) * (scale factor)^2
= 4 * (8)^2
= 4 * 64
= 256 in^2
Therefore, the surface area of the larger figure is 256 in^2.
since their volumes are in the ratio of 2^3:1, their areas are in the ratio of 2^2:1.
The larger figure has 4 times the area of the smaller one.
volume is proportional to the cube of a linear dimension
Area is proportional to the square of a linear dimension
The ratio of the volumes is 40/5 or 8
That means the linear dimensions have the ratio of ∛8 or 2
That means the areas have the ratio of 2² or 4
so the area of the larger s 4 x 4 in² = 16 in²