# Math

1. Air is being pumped into a spherical balloon at the rate of 1.754 cubic centimeters per second. The balloon maintains a spherical shape throughout. How fast is the radius of the balloon changing when the diameter is 2.736 centimeters in length?

2. A wall stands upright and perpendicular from the flat ground. A 25-footlong ladder leans against the wall. The bottom of the ladder is moved away from the building horizontally (along a line perpendicular to the wall) at a rate of 30 inches per minutes (until the ladder lies flat on the ground). How fast is the top of the ladder sliding down the building when the top of the ladder is 10 feet above the ground?

3. A ladybug crawls out of a small hole in a large wall and crawls to the right at a rate of 3 inches per minute. Forty-five seconds later, a tick crawls out of the hole and crawls up at a rate of 2 inches per minute. how fast is the distance between the ladybug and the tick changing four minutes after the ladybug crawls out of the hole?

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1. #1
I think you have a typo here. Since the r = d/2, the radius always grows half as fast as the diameter.

#2
If the base is x from the wall, and the top is y in height, then you have

x^2+y^2 = 25^2
so,
2x dx/dt + 2y dy/dt = 0
or, getting rid of the 2's,

x dx/dt + y dy/dt = 0
when y=10, x=√525 = 5√21
So, now you know x,y, and dx/dt.

#3
At time t minutes,
the ladybug has gone 3t inches
the tick has gone 2(t-3/4)

So, the distance z between them is

z^2 = (3t)^2 + (2(t-3/4))^2
z^2 = 1/4 (52t^2 - 24t + 9)
so, as in the problem above,

2z dz/dt = 1/4 (104t-24) = 26t-6
at t=4, z = √(12^2 + 6.5^2) = 13.65

So, now you know z, so figure dz/dt at t=4

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2. i still don't understand how you did all of them. can you please elaborate? the answers for #2 is supposed to be 5.728 ft/min and #3 is 3.59in/min

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