Assignment Paper

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]