Question

100 Points and I will give brainliest! PLEASE HELP
Prism X is a dilation of Prism Y. The height of Prism X is 10 ⅓ ft, and the volume of Prism X is 74 ⅖ ft³. The height of Prism Y is 5 ⅙ ft.

What is the volume of Prism Y?

Enter your answer as a mixed number in simplest form by filling in the boxes.

Answer 1

9.3 ft³

• height of Prism X is 10 ⅓ ft,

• the volume of Prism X is 74 ⅖ ft³

• height of Prism Y is 5 ⅙ ft.

use the rule:

$\rightarrow \sf (\dfrac{V_1}{V_2} )= (\dfrac{h_1}{h_2} )^3$

solve:

$\rightarrow \sf (\dfrac{74\frac{2}{5} }{V_2} )= (\dfrac{10\frac{1}{3} }{5\frac{1}{6} } )^3$

$\rightarrow \sf (\dfrac{74\frac{2}{5} }{V_2} )= (2 )^3$

$\rightarrow \sf \dfrac{74\frac{2}{5} }{V_2} =8$

$\rightarrow \sf V_2 = \dfrac{74\frac{2}{5} }{8}$

$\rightarrow \sf V_2 = 9.3 \ ft^3$

Answer 2

Answer:

Given:

$\sf Height \ of \ Prism \ X = 10\frac13=\dfrac{(10 \times 3)+1}{3}=\dfrac{31}{3}$

$\sf Height \ of \ Prism \ Y = 5\frac16=\dfrac{(5 \times 6)+1}{6}=\dfrac{31}{6}$

$\sf Volume\ of \ Prism \ X = 74\frac25=\dfrac{(74 \times 5)+2}{5}=\dfrac{372}{5}$

Prism X is a dilation of Prism Y.

⇒ Scale factor of dilation = height of Prism X ÷ height of Prism Y

                                          $\sf =\dfrac{31}{3} \div \dfrac{31}{6}$

                                          $\sf =\dfrac{31}{3} \times \dfrac{6}{31}$

                                          $\sf =\dfrac63$

                                          $\sf =2$

If Prism X is a dilation of Prism Y by scale factor 2, then Prism Y is a dilation of Prism X by scale factor $\frac12$.

To find the volume of Prism Y, we simply need to multiply the volume of Prism X by the cube of scale factor $\frac12$, since volume is measured in cubic units.

$\sf \implies volume \ of \ Prism \ Y = volume \ of \ Prism \ X \times (scale \ factor)^3$

                                    $\sf = \dfrac{372}{5} \times \left(\dfrac12\right)^3$

                                    $\sf = \dfrac{372}{5} \times \dfrac18$

                                    $\sf =\dfrac{372}{40}$

                                    $\sf =\dfrac{93}{10}$

                                    $\sf = 9 \frac{3}{10} \ ft^3$