SUGGESTED PROBLEMS
Exercises 12-14 refer to the map in Figure 12.1.
13.
Sketch a possible graph of the predicted high temperature T on a line north-south through
Topeka.
19.
The balance, B , in dollars, in a bank account depends on the amount deposited, A dollars, the annual interest rate, r %, and the time, t , in months since the deposit, so
B = f ( A, r, t ) .
(a) Is f an increasing or decreasing function of A ? Of r ? Of t ?
(b) Interpret the statement f (1250 , 1 , 25)
≈
1276 . Give units.
(a)
•
As the amount deposited increases, so too does the bank balance
⇒ f increases with increases to A .
•
As the interest rate increases, so does the amount earned and therefore the final bank balance
⇒ f increases with increases to A .
•
As the length of time increases, more interest is earned
⇒ f increases with increases to A .
(b) f (1250 , 1 , 25)
≈
1276 means that an initial deposit put in an account at 1% interest for 25 months will grow to approximately 1276 dollars..
22.
A car rental company charges $ 40 a day and 15 cents a mile for its cars.
(a) Write a formula for the cost, C , of renting a car as a function, f , of the number of days, d , and the number of miles driven, m .
(b) If C = f ( d, m ) , find f (5 , 300) and interpret it.
(a) C = f ( d, m ) = 40 d + 0 .
15 m

MATH 122 - Section H-H 12.1 Solutions 2
(b) f (5 , 300) = 40(5) + 0 .
15(300) = $245. It costs $245 to a rent a car for five days and drive it for 300 miles.
26.
A cube is located such that its top four corners have the coordinates (-1, -2, 2), (-1, 3,
2), (4, -2, 2) and (4, 3, 2). Give the coordinates of the center of the cube.
By drawing the top four corners, we find that the length of the edge of the cube is 5. We also notice that the edges of the cube are parallel to the coordinate axis.
The x -coordinate of the the center is halfway between two x coordinates
−
1 + 4
= 1 .
5
2
Similarly, the y -coordinate of the the center equals
−
2 + 3
= 0 .
5
2
The z -coordinate of the the center are 2.5 below the top side, at
2
−
2 .
5 =
−
0 .
5
29.
Find a formula for the shortest distance between a point ( a, b, c ) and the y -axis.
The coordinates of the point closest to ( a, b, c ) on the y axis will be (0 , b, 0). Therefore, the distance to the y -axis can be determined using our 3D distance formula between two points:
Distance = p
( a
−
0) 2 + ( b
− b ) 2 + ( c
−
0) 2
= p a 2 + c 2
TEST PREPARATION PROBLEMS
3.
You are at the point (-1, -3, -3), standing upright and facing the yz -plane. You walk 2 units forward, turn left, and walk for another 2 units. What is your final position? From the point of view of an observer looking at the coordinate system in Figure 12.2, are you

MATH 122 - Section H-H 12.1 Solutions 3 in front of or behind the yz -plane? To the left or to the right of the xz -plane? Above or below the xy -plane?
You are walking in the positive x direction 2 units. When you take your left turn, you are pointing in the positive y direction, and move 2 more units. Your final position is therefore (
−
1 + 2 ,
−
1 + 2 ,
−
3) = (1 ,
−
1 ,
−
3). Therefore, you are
• in front of the yz -plane ( x > 0),
• to the left of the xz -plane ( y < 0),
• and below the xy -plane ( z < 0).
Sketch graphs of the equations in Exercises 9-11 in 3-space.
9.
x =
−
3
The graph is a plane parallel to the yz-plane, and passing through the point (-3, 0, 0).
23.
The temperature adjusted for wind-chill is a temperature which tells you how cold it feels, as a result of the combination of wind and temperature.2 See Table 12.2.
(a) If the temperature is 0 o F and the wind speed is 15 mph, how cold does it feel?
(b) If the temperature is 35 o F, what wind speed makes it feel like 24 o F?
(c) If the temperature is 25 o F, what wind speed makes it feel like 12 o F?
(d) If the wind is blowing at 20 mph, what temperature feels like 0 o F?
Table 12.2
Temperature adjusted for wind-chill ( o F) as a function of wind speed and temperature
Wind Speed (mph)
Temperature ( o F)
35 30 25 20 15 10 5 0
5 31 25 19 13 7 1 -5 -11
10 27 21 15 9 3 -4 -10 -16
15 25 19 13 6 0 -7 -13 -19
20 24 17 11 4 -2 -9 -15 -22
25 23 16 9 3 -4 -11 -17 -24
(a) According to Table 12.2 of the problem, it feels like -19 o F.
(b) A wind of 20 mph, according to Table 12.2.

MATH 122 - Section H-H 12.1 Solutions 4
(c) About 17.5 mph, halfway between the table entries 15 mph (13 o F) and 20 mph (11 o F).
(d) About 16.6
o F. One third of the way from the table entries 15 o F (feels like
−
2 o F ) and 20 o F (feels like 4 o F).
30.
Find the equations of planes that just touch the sphere ( x
−
2) 2 and are parallel to
+ ( y
−
3) 2 + ( z
−
3) 2 = 16
(a) The xy -plane
(b) The yz -plane
(c) The xz -plane
This is a sphere of radius 4, and with a center at (2 , 3 , 3).
(a) Planes at z = 3 + 4 = 7 and z = 3
−
4 =
−
1 will just touch the sphere.
(b) Planes at x = 2 + 4 = 6 and x = 2
−
4 =
−
2 will just touch the sphere.
(c) Planes at y = 3 + 4 = 7 and y = 3
−
4 =
−
1 will just touch the sphere.
31.
Find an equation of the largest sphere contained in the cube determined by the planes x = 2 , x = 6 , y = 5 , y = 9 and z =
−
1 , z = 3 .
This is a cube with sides of length 6
−
2 = 4. Therefore the largest sphere we can fit inside will have diameter 4, or radius 2.
The center of the sphere must be at the center of the cube. This is halfway along each of the edges, at x = 4, y = 7 and z = 1. Therefore, the equation for the largest sphere inside this cube is
( x
−
4) 2 + ( y
−
7) 2 + ( z
−
1) 2 = 2 2