Use differentials to estimate the amount of paint needed to apply a coat of paint 0.02 cm thick to a sphere with diameter 40 meters. (Recall that the volume of a sphere of radius r is V=(4/3)πr^3. Notice that you are given that dr=0.02.)
my dv is comming as 0.24m^3 ,fir finding volume you can add this dv to actual volume considering radius as 20m
Well, you wouldn't want to make the sphere look like a clown by applying too much or too little paint, right?
To estimate the amount of paint needed, we can use differentials. First, let's convert the diameter to a radius: r = 40 / 2 = 20 meters.
The volume of a sphere is given by V = (4/3)πr^3. Taking the derivative of V with respect to r, we get dV = 4πr^2 dr.
Since we're given that dr = 0.02 cm, which is equivalent to 0.0002 meters, we can substitute this value into the equation above: dV = 4π(20)^2 (0.0002) = 0.064π cubic meters.
So, the estimated amount of paint needed to apply a coat 0.02 cm thick to the sphere is roughly 0.064π cubic meters.
Just make sure you don't accidentally paint the clown's red nose on the sphere, as that might result in a rather funny-looking planet!
To estimate the amount of paint needed to apply a coat of paint 0.02 cm thick to a sphere with a diameter of 40 meters, we can use differentials.
First, let's find the radius of the sphere. The diameter is given as 40 meters, so the radius is half of the diameter, which is 20 meters.
We have the formula for the volume of a sphere, which is V = (4/3) * π * r^3. To find the change in volume, we differentiate the volume formula with respect to r:
dV = (4/3) * π * 3 * r^2 * dr
Since dr is given as 0.02 cm, we need to convert it to meters to match the units of the radius. 1 cm is equal to 0.01 meters, so:
dr = 0.02 cm * 0.01 m/cm = 0.0002 m
Now we can substitute the values into the formula:
dV = (4/3) * π * 3 * (20)^2 * 0.0002
= (4/3) * π * 3 * 400 * 0.0002
= 0.160 π
This means that the change in volume of the sphere when the thickness of the paint is 0.02 cm is approximately 0.160 times the value of π.
To estimate the amount of paint needed, we can assume that the paint spreads uniformly on the surface of the sphere. The thickness of the paint is 0.02 cm, so the volume of the paint can be calculated by multiplying the change in volume by the thickness:
Paint volume = dV * 0.02 cm
= 0.160 π * 0.02 cm
= 0.0032 π cm^3
Therefore, approximately 0.0032 times the value of π cubic centimeters of paint are needed to apply a coat of paint 0.02 cm thick to the given sphere.
v = 4/3 πr^3
dv = 4πr^2 dr
so, plug in your numbers...
i did.
Dv=128pi. But the answer is 8pi/25
Check your units. Their answer is in m^3, since it is
4π(20^2)(.02/100)