Which dimensions of two rectangular prisms have volumes of 100 m3 but different surface areas
2x5x10
4x5x5
To find two rectangular prisms with the same volume but different surface areas, we need to consider the relationship between volume and surface area.
The volume of a rectangular prism is given by the formula:
Volume = length × width × height
And the surface area of a rectangular prism is given by the formula:
Surface Area = 2 × (length × width + width × height + height × length)
Since we want the volume to be the same, we can set up the equation:
100 = length₁ × width₁ × height₁ = length₂ × width₂ × height₂
Now let's consider the surface area. We need to find two sets of dimensions that satisfy the equation above but result in different surface areas.
One way to do this is to choose dimensions where the length, width, and height are not in the same proportion. For example, let's consider the following dimensions:
For the first rectangular prism:
length₁ = 10 m
width₁ = 10 m
height₁ = 1 m
For the second rectangular prism:
length₂ = 20 m
width₂ = 5 m
height₂ = 1 m
Both of these sets of dimensions have a volume of 100 m³ since 10 × 10 × 1 = 100 and 20 × 5 × 1 = 100. However, their surface areas will be different.
Calculating the surface area for the first rectangular prism:
Surface Area₁ = 2 × (10 × 10 + 10 × 1 + 1 × 10) = 2 × (100 + 10 + 10) = 240 m²
Calculating the surface area for the second rectangular prism:
Surface Area₂ = 2 × (20 × 5 + 5 × 1 + 1 × 20) = 2 × (100 + 5 + 20) = 250 m²
Therefore, by choosing these dimensions, the two rectangular prisms have the same volume of 100 m³ but different surface areas (240 m² and 250 m²).