differentiation question?

Question

Hi! I dont understand how you can tell if they want you to find dt/dA or dA/dt in this question...
1. A circular puddle is expanding. Its radius is increasing at the rate of 0.5 cm/s. The rate at which the area of the puddle is increasing when the radius is 10cm is...

so if a question says 'the rate' of (something), its always (something) in relation to time??

Answer 1

They asked you to find the "rate at which the AREA ... is increasing". So they are asking for the rate of change of the area with respect to time, which is dA/dt.

So find dA/dt and plug in the value of the radius (10cm) to get your answer.

Answer 2

dt/dA means the changing rate of area in relation to time while dA/dt means the other way...if you'd like to question why, it's the law of calculus

Answer 3

dA/dt because area is what's changing with rerspect to time. biggest indicator is the units. cm/s is saying that for every second that passes, the puddle expands .5 cm.

Answer 4

Area of puddle = pi * r square.
They have given the rate at which radius is expanding, i.e d(r)/dt
You have to find d(A)/dt

d(A)/dt
= d(pi * r^2)/dt
= pi * d(r^2)/dt
= pi * 2r * d(r)/dt
= pi * 2r * 0.5
= pi * r
= 3.14 * 10
= 31.4 cm^2/s

Answer 5

dr / dt = 0.5 cm/s
A = πr²
dA/dt = 2πr.dr/dt
dA / dt = πr
when r = 10 cm
dA/ dt = 10π cm²/s