two similar solids have surface areas 48cm^2 and 108cm^2 respectively. find the volume of the smaller solid if the bigger one has a volume of 162cm^3.
the volumes scale as the 3/2 power of the areas
162 * (48/108)^(3/2) = 162 * (4/9)^(3/2) = 162 * 8/27 = 48 cm^3
Answer
To find the volume of the smaller solid, we can use the concept of ratios.
Let's assume that the smaller solid has a surface area of A cm² and a volume of V cm³.
According to the problem, the ratio of the surface areas of the two solids is 48 cm² to 108 cm². This can be written as:
48/108 = A₁/A₂
Simplifying this ratio gives:
2/3 = A₁/A₂
Next, let's consider the ratio of the volumes of the two solids. The problem states that the volume of the bigger solid is 162 cm³, which means:
V₁/V₂ = V/V₂ = 162 cm³/V₂
Therefore, the ratio of the volumes can be written as:
V/V₂ = 162/V₂
Now, we can set up an equation using the ratio of surface areas and the ratio of volumes:
A₁/A₂ = V/V₂
2/3 = 162/V₂
To find the volume of the smaller solid, we need to solve for V₂.
Multiplying both sides of the equation by V₂ gives:
2V₂/3 = 162
Now, solve for V₂:
2V₂ = 486
V₂ = 243 cm³
Finally, we can calculate the volume of the smaller solid (V₁) by using the ratio of volumes:
V₁/V₂ = V/V₂
V₁/243 = 162/243
V₁ = 162 * 243 / 243
V₁ = 162 cm³
Therefore, the volume of the smaller solid is 162 cm³.
To find the volume of the smaller solid, we need to use the concept of similarity between the two solids.
Similar solids have proportional dimensions, including their surface areas and volumes. The ratio of the surface areas of two similar solids is equal to the square of the ratio of their corresponding dimensions. Similarly, the ratio of their volumes is equal to the cube of the ratio of their corresponding dimensions.
Let's represent the dimensions of the smaller solid as "x" and the dimensions of the bigger solid as "y".
The surface area of the smaller solid is 48 cm^2, and the surface area of the bigger solid is 108 cm^2. Therefore, we have the equation:
(48/108)^2 = (x/y)^2
Simplifying the equation:
16/36 = x^2/y^2
1/9 = x^2/y^2
Now, we know that the volume of the bigger solid is 162 cm^3. Using the ratio of their volumes, we can write another equation:
(162/Volume of smaller solid) = (y/x)^3
Simplifying the equation:
162/Volume of smaller solid = y^3/x^3
Plugging in the value of x^2/y^2 from the first equation, we get:
162/Volume of smaller solid = 9/1
Cross-multiplying:
162 * 1 = 9 * Volume of smaller solid
Volume of smaller solid = 162 * 1 / 9
Therefore, the volume of the smaller solid is:
Volume of smaller solid = 18 cm^3