The height of a cylinder with constant volume is inversely proportional to the square of its radius. If h = 8 cm when r = 4 cm, what is r when h = 2 cm?

Question

Eli Assouline · Accepted Answer

undefined To find the value of r when h = 2 cm, we can use the inverse proportionality relationship between the height and the square of the radius.

Let's denote the height as h and the radius as r. According to the given information, we know that the volume of the cylinder remains constant.

Using the inverse proportionality, we can write the equation as:

h ∝ 1/r^2

To solve for the constant of proportionality, we can substitute the given values into the equation:

8 ∝ 1/4^2

Simplifying this equation, we get:

8 ∝ 1/16

To find the constant of proportionality, we can multiply both sides of the equation by 16:

8 * 16 ∝ 1

128 ∝ 1

Now, we can use this constant of proportionality to find the value of r when h = 2 cm:

2 ∝ 1/r^2

Substituting the constant of proportionality, we have:

128 ∝ 1/r^2

To solve for r, we can rearrange the equation:

r^2 ∝ 1/128

Taking the square root of both sides, we get:

r ∝ 1/√128

Simplifying this expression, we have:

r ∝ 1/8√2

Therefore, when h = 2 cm, r is approximately equal to 1/8√2 cm.

Hazel Anglin · Answer

see the explanation..

#Height prop 1/(radius^2)# This is what the above statement says about the inverse relationship between HEIGHT and SQUARE OF RADIUS. Now in next step when removing the proportional sign #(prop)# we use an equal to sign and multiply #color(RED)"k"# on either of the sides like this; #Height = k*1/(Radius^2)# {where k is constant (of volume)} Putting the values of height and radius^2 we get; #8 = k*1/4^2 # #8 * 4^2= k# #8 * 16= k# #k= 128# Now we have calculated our constant value #color(red)"k"# which is #color(red)"128"#. Moving towards your question where radius is to be calculated. Plugging the values into the equation: #Height = k*1/(Radius^2)# #2 = 128*1/r^2# {r is for radius} #r^2=128/2# #r^2=64# #sqrt(r^2) =sqrt 64# #r = 8# Hence, for height of 2 cm with a constant of 128 we get the #color(blue)(radius)# of #color(blue)(2 cm)#