1. Diameter and Area!

As you know, the formula for the area of a circle is a power function. If the diameter of a circle is 37 μm, then for this circle to increase its area 32-times, what diameter, to the nearest whole μm, does this larger circle need to have?

2. Hello, littlesohi!p

If the diameter of a circle is 37 mm,
then for this circle to increase its area 32 times, what diameter,
to the nearest whole mm, does this larger circle need to have?

Let the original diameter be $d_1$
. . The area of the original circle is: . $A_1 \:=\:\pi\left(\frac{d_1}{2}\right)^2 \:=\:\frac{\pi}{4}(d_1)^2$

Let the diameter of the larger circle be $d_2$
. . The area of the larger circle is: . $A_2 \:=\:\pi\left(\frac{d_2}{2}\right)^2 \:=\:\frac{\pi}{4}(d_2)^2$

Since $A_2 = 32\!\cdot\!A_1$ we have: . $\frac{\pi}{4}(d_2)^2 \;=\;32\cdot\frac{\pi}{4}(d_1)^2 \quad\Rightarrow\quad (d_2)^2 \:=\:32(d_1)^2$

. . Hence: . $d_2\;=\;4\sqrt{2}\,d_1$

If $d_1 = 37$, then: . $d_2 \:=\:4\sqrt{2}\,(37) \;=\;209.3036072$

Therefore: . $d_2 \;\approx\;209\text{ mm}$