the lateral area of two similar solids are 16pie cmsquared and 9pie cmsquared. The volume of the larger solid is 128pie cm cubed. find the volume of the smaller solid please explain
area of similar objects is proportional to the square of their sides
16/9 = 4^2/3^2
so the ratio of their sides is 4:3
the volume of similar objects is proportional to the cube of their sides
4^3/3^3 = 128/x
64/27 = 128/x
64x = 27(128)
x = 27(128)/64 = 54
7.5 ft 37ft
To find the volume of the smaller solid, we can use the concept of scaling ratios.
Let's assume that the scaling ratio between the larger solid and the smaller solid is "x". This means that the corresponding lengths, widths, and heights of the smaller solid would be x times smaller compared to those of the larger solid.
Given that the lateral area of the larger solid is 16π cm² and the lateral area of the smaller solid is 9π cm², we can set up the following proportion:
Lateral area of larger solid / Lateral area of smaller solid = (length of larger solid / length of smaller solid)²
Using the given values, we have:
16π / 9π = (length of larger solid / length of smaller solid)²
Simplifying, we get:
16 / 9 = (length of larger solid / length of smaller solid)²
Taking the square root of both sides, we find:
√(16 / 9) = length of larger solid / length of smaller solid
Simplifying further:
4/3 = (length of larger solid / length of smaller solid)
Since the volumes of two similar solids are proportional to the cubes of their corresponding lengths, we can write:
Volume of larger solid / Volume of smaller solid = (length of larger solid / length of smaller solid)³
Plugging in the values, we have:
128π / V(smaller solid) = (4/3)³
Simplifying further, we get:
128π / V(smaller solid) = 64 / 27
Cross-multiplying, we find:
V(smaller solid) = (128π * 27) / 64
Calculating this expression, we get:
V(smaller solid) = 54π cm³
Therefore, the volume of the smaller solid is 54π cm³.