Question

two cylinders are given. the larger cylinder is a dilation of the smaller cylinder. part a. determine the scale factor. show or explain all the necessary steps (2 points) part b. if the volume of the smaller cylinder is 24π m³, show how the scale factor can be used to determine the volume of the larger cylinder. (2 points)

Explanation:

Step1: Calculate scale - factor

The scale factor $k$ of two similar solids is the ratio of corresponding linear dimensions. For two cylinders, if the radii are $r_1$ and $r_2$, then $k=\frac{r_2}{r_1}$. Here, $r_1 = 2m$ and $r_2=7m$, so $k=\frac{7}{2}$.

Step2: Determine volume - relationship

For two similar solids, if the scale - factor is $k$, the ratio of their volumes $V_1$ and $V_2$ is $k^{3}$. We know $V_1 = 24\pi m^{3}$ and $k=\frac{7}{2}$. Then $V_2=V_1\times k^{3}$. Substitute the values: $V_2 = 24\pi\times(\frac{7}{2})^{3}$.

$$
LATEXBLOCK0
$$

Answer:

Part A: The scale factor is $\frac{7}{2}$.
Part B: The volume of the larger cylinder is $1029\pi m^{3}$.