Question

which statement is not true? doubling the length of a prism will double its surface area. multiplying the length of a prism by a scale factor of 4 will multiply the volume by 4. there is a direct relationship between a change in the length and a change in the volume. doubling the length of a prism will double its volume.

Explanation:

1. Analyze "Doubling the length of a prism will double its surface area": The surface area of a prism depends on the lateral surface area (perimeter of the base × length) and the area of the two bases. If only the length is doubled, the lateral surface area (perimeter × length) will double, but the area of the two bases remains the same. So the total surface area will not simply double (since the base areas don't change proportionally to the length). This statement is not true.
2. Analyze "Multiplying the length of a prism by a scale factor of 4 will multiply the volume by 4": The volume of a prism is \( V = B\times l \) (where \( B \) is the area of the base and \( l \) is the length). If \( l \) is multiplied by 4, and \( B \) remains constant, then \( V' = B\times(4l)=4\times(B\times l) = 4V \). This statement is true.
3. Analyze "There is a direct relationship between a change in the length and a change in the volume": Since \( V = B\times l \) (and \( B \) is constant for a given prism base), \( V \) is directly proportional to \( l \) ( \( V\propto l \) ). So a change in length will cause a proportional change in volume. This statement is true.
4. Analyze "Doubling the length of a prism will double its volume": Using \( V = B\times l \), if \( l \) becomes \( 2l \), then \( V' = B\times(2l)=2\times(B\times l) = 2V \). This statement is true.

Answer:

Doubling the length of a prism will double its surface area.