The height of one right circular cylinder is 7 centimeters and its radius is 2 centimeters. The height of the second right circular cylinder is 28 centimeters and its radius is also 2 centimeters. What is the ratio of the volume of the larger cylinder to the volume of the smaller cylinder? A. 4:1 B. 5:1 C. 10:1 D. 25:1

Explanation:

The volume of a cylinder formula is

V = pi*r^2*h

where r is the radius and h is the height.

For the first cylinder, we have r = 2 and h = 7. This leads us to...

V = pi*r^2*h

V = pi*2^2*7

V = pi*4*7

V = pi*28

V = 28pi

This is the exact volume in terms of pi.

Repeat those steps for the other cylinder (r = 2 and h = 28)

V = pi*r^2*h

V = pi*2^2*28

V = pi*4*28

V = 112pi

To recap so far, we have these two cylinder volumes:

We see that cylinder B has the larger volume (since 112 > 28).

The ratio of the volumes from larger to smaller is 112pi : 28pi

Divide both parts by pi to have those terms cancel and we get the ratio 112:28

Lastly, we divide each part by the GCF 28

112/28 = 4

28/28 = 1

Therefore, the ratio 112:28 fully reduces to 4:1 which makes the final answer choice A.

The larger volume is 4 times that of the smaller volume.

A fairly quick way to see why we get this ratio is to notice the smaller height h = 7 jumps to the larger height h = 28 by a factor of 4.

In other words,

larger height = 4*(smaller height)

28 = 4*7

The fact that the radius is kept the same at r = 2 won't have an effect on the volume ratio. These parts effectively cancel out.

So this jump of "times 4" directly connects to the ratio 4:1

We multiplied the height by 4, so the volume is also multiplied by 4 (when going from the smaller cylinder to the larger one).

the value of x is 9 ⇒ answer b

step-by-step explanation:

* we can solve this problem using cosine and sine rule

- in δ abc

∵ ab = 36 , bc = 28 , ac = 16

- lets find the measure of angle b using the cosine rule

∵ cos b = [ab² + bc² - ac²]/2(ab)(bc)

∴ cos b = [36² + 28² - 16²]/2(36)(28)

∴ cos b = 19/21

∴ m∠b = cos^-1(19/21) ≅ 25°

- lets find the measure of angle a using the sine rule

∵ sina/bc = sinb/ac = sinc/ab

∴ sina/28 = sin25°/16 ⇒ by using cross multiplication

∴ sina = 28 × sin25°/16

∴ sina = 0.7396

∴ m∠a = sin^-1(0.7396) ≅ 48°

- from the figure m∠abd = m∠dbc

∵ m∠abc = 25°

∴ m∠abd = 25°/2 = 12.5°

- in δabd

∵ m∠a = 48° , m∠abd = 12.5°

∵ the sum of the measures of the interior angles of any δ is 180°

∴ m∠adb = 180° - (48° + 12.5°) = 119.5°

∵ ab = 36

∵ ad = x

- use the sine rule to find x

∵ sin∠adb/ab = sin∠abd/x

∴ sin119.5°/36 = sin12.5°/x ⇒ by using cross multiplication

∴ x = 36 × sin12.5°/sin119.5° = 8.95 ≅ 9

* the value of x is 9