the radius of a sphere is 7 cm,if the radius is increased by 5o%by how much percent is the volume increased?

Question

please mention all steps

Answer 1

1.5^3 = 3.375

volume increases as the cube of the linear dimension

(3.375 -1) * 100 = 237.5 percent increase in volume

Answer 2

The volume of any sphere is given by the formula V(s) = (4/3)pi r³.

Let the radius of the larger sphere be r' and its volume be V'. Let the radius of the smaller sphere be r, and its volume be V. Then clearly:

V' / V = [(4/3)pi (r')³] / [(4/3)pi (r)³].

Everything cancels out but the radii. So obviously the ratio of the two volumes is dependent on the cube of the ratio of the two radii. It doesn't matter what the actual numbers are. All that matters in this case is their relationship to each other. One could actually substitute the values 10.5 cm. and 7.0 cm into the following equation, but since they divide out to 1.5, that is the figure which is important here. Mathematically it can be stated like this:

V' / V = (r' / r)³.

Substituting in 1.5 for r' and 1.0 for r, we get this equation:

V' / V = (1.5 / 1.0)³ = 3.375 ----> (V' / V) = 3.375 or V' = 3.375 V

But this doesn't give us the percentage increase. It only tells us the relative volumes of the two spheres.

To calculate the percentage increase, we first calculate how much the volume has changed. Then we divide that by the original volume, because we are comparing the change to the original volume. Then we multiply the decimal result obtained from that calculation by 100 to convert it to a percentage. The final result gives us the percentage increase in volume.

[(V' - V) / V] (100) ----> (V'/ V - V/ V) (100) ----> [(V' / V) - 1] (100).

Substituting the appropriate values into the final form, we obtain:

(3.375 - 1) (100) = (2.375) (100) = 237.5 %.

So, the sphere has increased in volume by 237.5% when its radius is increased by 50%.

Answer 3

The volume of a sphere is (4πr^3)/3 so the volume is proportional to the cube of the radius. Thus, if the radius increases by 1.5 times (which is a 50% increase), the volume will increase by 1.5^3 = 3.375 times.

Hope that helps ☺


Doug

Answer 4

7+ 1/2*7=10.5.
To work out volume of sphere 4/3*pi*radius^3
original volume = 4/3*pi=4.188790205 then 4.188790205* 7^3 = 1436.75504
THEN 4/3*pi*10.5 cubed=4849.048261
To work out percentage of increase
1436.75504 divided by 48.49048261=29.629629...
Answer = 29.6% increase in volume

Answer 5

Assume you mean 50%.

Volume of a sphere = 4/3*(pi)*r*r*r
If r is increased by 50%, then substitute r=1.5*r.
So the volume of the sphere = 4/3*(pi)*1.5*r*1.5*r*1.5*r
= 4/3*(pi)*r*r*r*3.375 = 4/3*(pi)*r*r*r*27/8 = 9/2*(pi)*r*r*r
where r = the original radius of the sphere

Answer 6

V1 = (4/3) π 7³ cm³

V2 = (4/3) π (1.5 x 7)³ cm³

V2/V1 = 1.5³ = 3.375

V2 = 3.375 V1

V2 = 337.5% x V1

Increase in volume from 100% to 337.5% = 237.5%

Answer 7

Volume = 4/3 pi r^3
If new radius is 1.5 r,
then new volume is 1.5 * 1.5 * 1.5 = 3.375

This is an increase of 237.5%