Question

assume that the flask shown in the diagram can be modeled as a combination of a sphere and a cylinder. based on this assumption, the volume of the flask is cubic inches. if both the sphere and the cylinder are dilated by a scale factor of 2, the resulting volume would be times the original volume.

Explanation:

Step1: Calculate original sphere volume

The formula for the volume of a sphere is $V_s=\frac{4}{3}\pi r^3$. The radius of the sphere $r = \frac{4.5}{2}= 2.25$ inches. So the original sphere volume $V_{s1}=\frac{4}{3}\pi(2.25)^3$.

Step2: Calculate original cylinder volume

The formula for the volume of a cylinder is $V_c=\pi r^2h$. The radius of the cylinder $r=\frac{1}{2}$ inch and height $h = 3$ inches. So the original cylinder volume $V_{c1}=\pi(\frac{1}{2})^2\times3=\frac{3}{4}\pi$.

Step3: Calculate original total volume

The original volume of the flask $V_1=V_{s1}+V_{c1}=\frac{4}{3}\pi(2.25)^3+\frac{3}{4}\pi$.

Step4: Consider dilation

When a three - dimensional object is dilated by a scale factor $k$, the new volume $V_2$ is related to the original volume $V_1$ by the formula $V_2=k^3V_1$. Here $k = 2$. So the new volume is $2^3 = 8$ times the original volume.

Answer:

8