The volume of two similar solids are125 in ^3 and 216 in ^3. The surface area of the larger solid is 288 in^2. What is the surface area of the smaller solid?
The volumes of similar solids are proportional to the cube of their
corresponding sides
125 : 216 = s1^3 : s2^3
5^3 : 6^3 = s1^3 : s2^2
s1 : s2 = 5 : 6
the surface area of similar solids is proportional to the square of their
corresponding sides
a1 : a2 = 5^2 : 6^2 = 25 : 36
if a2 is the larger,
a1 : 288 = 25:36
a1 = 288(25)/36 = 200 <----- surface area of the smaller solid
To find the surface area of the smaller solid, we can use the concept of similarity between the two solids. Since the solids are similar, their corresponding sides are in the same ratio.
Let's say the ratio of the lengths of corresponding sides of the two solids is k. Then, the ratio of their volumes would be k^3 (since volume is a three-dimensional measurement), and the ratio of their surface areas would be k^2 (since surface area is a two-dimensional measurement).
Given that the larger solid has a volume of 216 in^3, and the smaller solid has a volume of 125 in^3, we can set up the following equation:
k^3 = 216/125
Taking the cube root of both sides, we find:
k = (216/125)^(1/3) = 1.2
Now that we know the ratio of the lengths of corresponding sides, we can find the surface area of the smaller solid.
Let's say the surface area of the smaller solid is S. Since the ratio of the surface areas of the two solids is k^2, we have:
S/288 = 1.2^2
S/288 = 1.44
Multiplying both sides by 288, we get:
S = 414.72 in^2
Therefore, the surface area of the smaller solid is approximately 414.72 in^2.
To find the surface area of the smaller solid, we need to use the concept of similar figures.
The volume of two similar solids is related by the ratio of their side lengths cubed. So,
(Volume of smaller solid) / (Volume of larger solid) = (Side length of smaller solid)³ / (Side length of larger solid)³
Given that the volumes of the smaller and larger solids are 125 in³ and 216 in³ respectively, we can set up the equation:
125 / 216 = (Side length of smaller solid)³ / (Side length of larger solid)³
To find the ratio of their side lengths, we take the cube root of both sides of the equation:
(∛125 / ∛216) = (Side length of smaller solid) / (Side length of larger solid)
Simplifying this ratio, we get:
(5 / 6) = (Side length of smaller solid) / (Side length of larger solid)
Now we can find the side length of the smaller solid:
(Side length of smaller solid) = (5 / 6) * (Side length of larger solid)
Given that the surface area of the larger solid is 288 in², we can find the surface area of the smaller solid using the concept that the surface area of similar figures is related by the square of the ratio of their side lengths. So,
(Surface area of smaller solid) / (Surface area of larger solid) = (Side length of smaller solid)² / (Side length of larger solid)²
Plugging in the values, we get:
(Surface area of smaller solid) / 288 = [(5 / 6) * (Side length of larger solid)]² / (Side length of larger solid)²
Simplifying this equation, we get:
(Surface area of smaller solid) / 288 = (25 / 36)
Now we can find the surface area of the smaller solid:
(Surface area of smaller solid) = (25 / 36) * 288
Solving this equation, we find:
(Surface area of smaller solid) ≈ 200 in²
Therefore, the surface area of the smaller solid is approximately 200 in².