Mathematics: The radius of a circular oil spill after t minutes is given by r ( t ) = √ 25 t . Find the instantaneous rate at which the radius is growing after 31 minutes.

I’m pretty sure the unit rate is $1.85 per muffin....

The radius of a circular oil spill after t minutes

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20.01.2022

The instantaneous rate at which the radius is growing after 31 minutes is $0.449 \ units/min$

Explanation:

It is given that the radius of a circular oil spill after t minutes is given by

$r(t)=\sqrt{25t}$

Simplifying, we have,

$r(t)=5\sqrt{t}$

The instantaneous rate of change can be determine by differentiating with respect to t.

Thus, we have,

$\frac{dr}{dt} =5\left(\frac{1}{2}(t)^{\frac{-1}{2}}\right)$

$\frac{dr}{dt} =\left\frac{5}{2}(t)^{\frac{-1}{2}}\right$

$\frac{dr}{dt} =\left\frac{5}{2\sqrt{t} }$

We need to determine the instantaneous rate of the radius after 31 minutes.

Hence, substituting t= 31 in the above expression, we get,

$\frac{dr}{dt} =\left\frac{5}{2\sqrt{31} }$

Substituting the value of $\sqrt{31}$ , we get,

$\frac{dr}{dt} =\left\frac{5}{2(5.568)}$

Multiplying the denominator, we get,

$\frac{dr}{dt} =\left\frac{5}{11.136}$

Dividing, we have,

$\frac{dr}{dt} =0.449$

Thus, the instantaneous rate at which the radius is growing after 31 minutes is $0.449 \ units/min$

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