Answer:
3.45 inches
Step-by-step explanation:
To determine the height of the mega dish, calculate the volume of the small dish and use the given relationship between the volumes of the small and mega dishes.
The formula for the volume of a cylinder is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Cylinder}}\\\\V=\pi r^2 h\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}[/tex]
In the case of the small dish:
Substitute the values into the volume formula to calculate the volume of the small dish:
[tex]V_{\text{small}}=\pi \cdot 1.57^2 \cdot 2\\\\V_{\text{small}}=\pi \cdot 2.4649 \cdot 2\\\\V_{\text{small}}=4.9298\pi \; \rm in^3[/tex]
Given that the mega dish holds three times the volume of the small dish, the volume of the mega dish is:
[tex]V_{\text{mega}}=3 \cdot 4.9298\pi\\\\V_{\text{mega}}=14.7894\pi \; \rm in^3[/tex]
The diameter (d) of a circle is twice the length of its radius. So, the diameter of the small dish is:
[tex]d_{\text{small}}=2 \cdot 1.57\\\\d_{\text{small}}=3.14\; \rm in[/tex]
Given that the diameter of the mega dish is 1 inch longer than the diameter of the small dish, the diameter of the mega dish is:
[tex]d_{\text{mega}}=3.14+1\\\\d_{\text{mega}}=4.14\; \rm in[/tex]
Therefore, the radius of the mega dish is:
[tex]r_{\text{mega}}=\dfrac{4.14}{2}\\\\\\r_{\text{mega}}=2.07\; \rm in[/tex]
To find the height of the mega dish, substitute its radius (r = 2.07) and its volume (V = 14.7894π) into the volume formula and solve for h:
[tex]14.7894\pi = \pi \cdot 2.07^2 \cdot h\\\\\\14.7894\pi = 4.2849\pi \cdot h\\\\\\h=\dfrac{14.7894\pi}{4.2849\pi}\\\\\\h=\dfrac{14.7894}{4.2849}\\\\\\h=3.451515787...\\\\\\h=3.45\; \rm in\;(nearest\;hundredth)[/tex]
Therefore, the height of the mega dish rounded to the nearest hundredth is:
[tex]\Large\boxed{\boxed{3.45\; \rm inches}}[/tex]
