Problem Set #2 Assignment

1. A packaging design team has determined that a cost-efficient way of manufacturing cylindrical

containers for their products is to have the volume, V, in cubic centimeters, modeled by: V(x) = 9πx3 +

51πx2 + 88πx + 48π. The height, h, in centimeters, of each cylinder is a linear function given by h(x) =

x + 3. [4]
a. Divide the polynomial 9πx3 + 51πx2 + 88πx + 48π by x + 3.

b. Interpret the meaning of the quotient in terms of this context.

c. Determine an expression for the radius in terms of x.

2. An engineer designs a rollercoaster so that a section of the ride can be modeled by

the function shown in the graph below (between the ■ marks). x is the horizontal

distance from the boarding platform, in metres, and h is the height, in metres, above or

below the boarding platform. Determine the equation of this polynomial. You can

leave the equation in factored form. Round where appropriate. [4]
3. Solve for x: Round to 2 decimal places if required.

a. log7(x + 4) + log7(x – 2) = 1 [5]
b.

2 2 1

2 3

+ +

=

x x

[4]
4. Estimate the instantaneous rate of change of

x

f x

−

=

2

4

( )

at point (0, 2) by copying and completing

the table. Do not use derivatives. [4]
Interval Δf(x) Δx Average Rate of Change

0 ≤ x ≤ 1

0 ≤ x ≤ 0.5

0 ≤ x ≤ 0.1

0 ≤ x ≤ 0.001

5. Find the instantaneous rate of change of the curve

x

x

f x

2 3

( )

−

=

at x = 2 using first principles. [4]
6. Sketch by hand: f(x) = -2×3 + 26x – 24, including the full algebraic calculations of the following. You

may use Calculus for this question. Round to 2 decimal places where required.

a) x & y-intercepts (Do NOT use guess & check for all x-intercepts) [3]
b) relative maxima and minima [5]
c) points of inflection [2]
7. There are two strings connected to the front/middle of a sled. Billy Bob

pulls on one of the strings on the right side of the sled with a force of 450-

N at an angle of 70 from the horizontal. Peggy Sue pulls on the other

string on the left side of the sled with a force of 670-N at an angle of 50

from the horizontal. Determine the net force (magnitude and direction) on

the sled. Round to the nearest whole number. Please note: You should

interpret this question as viewing it from a bird’s eye view. Thus,

‘horizontal’ is in relation to the picture, not the ground. [6]
8. Determine the parametric equations for the line that passes through the two points A = (5, 1, -3) and

B = (4, 5, -1). At what point does this line intersect the x-y plane? [4]
9. The following data is a sample of daily maximum

temperatures in Toronto in March (from 2006-2008).

a. Calculate the mean (1 decimal place) and

standard deviation (2 decimal places) of this

data. You may use technology to answer this

question. Only the final answer is required. [3]
b. Determine a reasonable interval size and number

of intervals. Produce a properly labeled

histogram for the grouped data using technology.

Paste this graph into your solutions. [4]
c. In March, the temperatures in Calgary are

normally distributed with a mean daily

maximum temperature of 4.5ºC and a standard

deviation of 6.25ºC. What percent of days

would you predict would be between 0ºC and

10ºC? This question must be answered algebraically. [3]
10. A paper bag contains a mixture of three types of candy. There are ten gum balls,

seven candy bars, and three packages of toffee. Five candies are randomly put

into a loot bag for a birthday party. What is the probability that the bag contains

at least 2 gum balls? Round to 3 decimal places. [5]
0.4 0.8 0.6 7.2 5.8 -1.5 12.2

0.8 0.6 0.9 -0.4 5.7 4.1 7.9

12.8 3.4 0.8 3.2 7 14.9 -1.7

-0.4 6 0 0.1 15.8 11.7 4.6

-0.1 4.8 3.7 -1.2 14.4 7.6 11.4

3.6 -0.5 5 -2.7 4.7 3 3.3

-0.7 -0.7 5.9 -12.3 -2.1 20.3 -1

-5.7 3.2 3.8 -6.1 0.5 18.9 4.5

-4.2 3.3 1.9 -3.4 2.4 9.6 15.6

-0.6 2.7 4.2 0.2 2.6 8.3 1.6

11. It has been estimated that about 30% of frozen chicken contain enough salmonella

bacteria to cause illness if improperly cooked. A consumer purchases 12 frozen chickens.

What is the probability that the consumer will have less than 2 contaminated chickens?

Round to 3 decimal places. [5]