Topic 4 - Final

Question 1

4.7
Ivan has a 200 metre roll of fencing. He wants to cut it into two pieces, use one piece to enclose a square area
and the other to enclose a circular area.
Where should he make the cut in order to minimize the total area enclosed by both?

Answer 1

In order to setup this problem, the fence for the square is x and the fence for the circle is 200-x

This would be the perimeter

so 4s = x
s = x/4
A = s^2
A = (x/4)^2
A = x^2/16

so 2𝜋r = 200-x
r = 200-x/2𝜋
A = 𝜋r^2
= 𝜋(200-x/2𝜋)^2
= (200-𝜋)^2/4𝜋

Add both functions Now take the derivative and set equal to 0 and you should get a root of

x = 1600/2𝜋+8 which is a minimum

​

Question 2

4.7

A fence 8 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length
of the shortest ladder that will reach from the ground over the fence to the wall of the building?

Answer 2

You want to draw the diagram in such a way that you form similar triangles nested in each other.

One with 8/x = y/4
and you can make an equation with y such that y =32/x

Now you can make another equation with the hypotenuses of the ladder which is the combination of the hypotensues of both similar triangles

L = L1 + L2
= sqrt(x^2 + 64) + sqrt(16 + y)

sub in the y value that you need from above, differentiate and solve for the minimum

Question 3

4.1.

A light is fastened to a rolling cart 1 m above the floor. The light moves at 2 m/s towards a 2 m tall
man standing 4 m from a wall. How fast is the tip of the man’s shadow on the wall moving when
the light is 7 m from the wall?

What is an important caveat you have to know?

Answer 3

The direction is important. The light moves towards the wall but the shadow away so the speeds of both will have the same magnitude but different directions (one positive and one negative

Question 4

4.4

Two cars are drag racing down a straight track. They start the same instant from the same place, and they cross the
nish line at the same instant as well. Show that at some instant during the race, the two cars were moving at exactly
the same speed

Answer 4

If you call them s1 and s2 then the difference in displacements is equal to 0 because theyt ravelled the same distance. They start at time T so you get

f’(c) = f(s1) - f(s2) / T - 0
0/ T- 0
f’(c) = 0
change in displacement is 0. change in displacement equals change in velocity

0 = v1 - v2
which means they’re equla

Question 5

What is the trick with this question?

Your alien friend Estraven has an unusual ability: they can walk along walls just as easily as they can walk on level ground.
Estra starts walking straight down from the top of the 10 m high wall of the Maanjiwe Nendamowinan (MN) building at a constant
speed of 2 m/s. Just as they start walking down the wall, a MAT135 student standing directly under Estra notices them and starts
running directly away from the wall at a non-constant speed.
2 seconds after the student starts running, they are 6 m from the wall and their speed is 3 m/s. At what rate is the distance between
Estra and the student changing at that time (2 seconds after the student starts running)?

Answer 5

Estra’s position is not a constant! Don’t treat it as such!

Also have to consider the direction. Estra is moving in the negative direction wihle the student is running in the positive direction, so have to factor this in

Question 6

4.7

Ezra is planning for a swimming race on a 1 km wide river (with very slow water). The finish line is 5 km downstream from the starting point, on the other side of the river. Ezra can swim at 3 km/h and run at 10 km/h. What is his quickest route?

Answer 6

Consider this as a total time problem, and that the time is distance/speed of swim time and run time

Swim distance is x and running distance is flat and is 5 -x

You know the swim distance based on a triangle of (√1+x^2) divided by the speed of 3 and 5-x divided by the speed of 10

Then you take the derivative and find the minimum