A cylinder fits inside a square prism as shown. For every cross section, the ratio of the area of the circle to the area of the square is or .

Since the area of the circle is the area of the square, the volume of the cylinder equals

the volume of the prism or (2r)(h) or πrh.
the volume of the prism or (4r2)(h) or 2πrh.
the volume of the prism or (2r)(h) or r2h.
the volume of the prism or (4r2)(h) or r2h.

"For every cross section, the ratio of the area of the circle to the area of the square is or ."

To find this ratio we need to find areas of the circle and the square.
Circle:
[tex]Area= radius^{2} * \pi \\ A_{1} = r^{2} * \pi [/tex]
Square:
[tex]Area= side^{2} \\ A_{2} = (2r)^{2} \\ A_{2} 4r^{2}= [/tex]
Now we divide these two areas to find ratio:
[tex]ratio= \frac{ A_{1} }{ A_{2} } \\ ratio= \frac{r^{2} * \pi }{4r^{2}} \\ ratio= \frac{ \pi }{4} [/tex]

"Since the area of the circle is the area of the square,"
From the ratio above we can see that areas are not same.

"the volume of the cylinder equals"
Formula for volumes of cylinder and prism follow the formula:
[tex]Volume=base*height[/tex]
For cylinder:
[tex]V=r^{2} * \pi *h[/tex]
For prism:
[tex]V= (2r)^{2} \\ V=4 r^{2} [/tex]

astha8579 astha8579 · Answer 2 · 2022-05-05T07:14:33+02:00

The ratio of the area of the cross sectional circle and area of the cross sectional square is π : 4

What is the ratio of two quantities?

Suppose that we've got two quantities with measurements as 'a' and 'b'

Then, their ratio(ratio of a to b) a:b or [tex]\dfrac{a}{b}[/tex]

We usually cancel out the common factors from both the numerator and the denominator of the fraction we obtained. Numerator is the upper quantity in the fraction and denominator is the lower quantity in the fraction).

Suppose that we've got a = 6, and b= 4, then:

[tex]a:b = 6:2 = \dfrac{6}{2} = \dfrac{2 \times 3}{2 \times 1} = \dfrac{3}{1} = 3\\or\\a : b = 3 : 1 = 3/1 = 3[/tex]

Remember that the ratio should always be taken of quantities with same unit of measurement. Also, ratio is a unitless(no units) quantity.

For this case, we're specified that:

Radius of the circle of the cross section of the cylinder = r units
Side length of the square cross section of the square prism = 2r units

Then, the area of the circle is:

[tex]\pi r^2 \: \rm unit^2[/tex]

and the area of the square is: [tex]\rm side^2 = (2r)^2= 4r^2 \: \rm unit^2[/tex]

The ratio of the area of the circle to the area of the square is:

[tex]\dfrac{\pi r^2}{4r^2} = \dfrac{\pi}{4} = \pi : 4[/tex]

Thus, the ratio of the area of the cross sectional circle and area of the cross sectional square is π : 4

Learn more about ratio here:

brainly.com/question/186659

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