A spherical balloon is being inflated. Estimate the rate at which its surface area is changing with respect to the radius when the radius measures 20 cm.
a = 4 pi r^2
da/dr = 8 pi r
plug and chug
To estimate the rate at which the surface area of the spherical balloon is changing with respect to the radius, we need to use the derivative of the surface area formula with respect to the radius. The surface area of a sphere can be calculated using the formula:
A = 4πr^2
where A is the surface area and r is the radius.
To find the rate at which the surface area is changing with respect to the radius (dA/dr), we need to differentiate the surface area formula with respect to r:
dA/dr = d(4πr^2)/dr = 8πr
Substituting the given radius of 20 cm into the equation:
dA/dr = 8π(20) = 160π cm^2.
Therefore, when the radius measures 20 cm, the rate at which the surface area is changing with respect to the radius is estimated to be 160π cm^2.