Few rate related problems

Question

Wanted to check my work. How does it look?

1. A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?

Can someone draw me a diagram? How does one do this on mathjax?

$$\theta = \arctan{\frac{h}{x}}$$

I think I have to use chain rule and quotient rule here right? $$\frac{d\theta}{dt} = \frac{1}{1 + \frac{h}{x}^2} \cdot \frac{-h}{x^2} \frac{dx}{dt}$$

I drew a triangle where one side is 100 and the hypotenuse is 200. $\theta$ is opposite the 100 length side and 173.20 is the missing side. Does that sound right?

$$\frac{d\theta}{dt} = \frac{1}{1+0.33} \cdot \frac{-100}{30000} \cdot 8ft/s = -.03556$$

Is this right? Did I do something wrong?

2.

If a snowball melts so that its surface area decreases at a rate of 1 cm^2/min, find the rate at which the diameter decreases when the diameter is 10 cm.

So area of a sphere is $A = 4 \pi r^2$

$$\frac{da}{dt} = 8 \pi r \frac{dr}{dt}$$

so when d = 10, r = 5, then:

$$-1 = 40 \pi \frac{dr}{dt}$$

$$\frac{dr}{dt} = \frac{-1}{40} \pi$$

Answer 1

Yes, these are right, and the diagram for the first you drew sounds right. Just make sure to include units. For the first problem, you're dealing with $radians/s$ because $\arctan x$ gives you radians. In the second it would be $cm/min$. Also, keep your variables consistent! So not $\frac{da}{dt}$ but $\frac{dA}{dt}$, etc.

Answer 2

$\frac{d\theta}{dt} = \frac{1}{1 + \left(\frac{h}{x}\right)^2} \cdot \frac{-h}{x^2} \frac{dx}{dt} = -\frac {h}{x^2 + h^2} \frac {dx}{dt}\\ x^2 + h^2 = 200^2\\ -\frac {100}{40,000} 8=0.02$

I think that the math that you show above is correct, but our answers look a little different. Not sure why.