Review Exercise 10

Review Exercise 10 Solutions

Syllabus checkpoints, self-grading quizzes, decay/growth models, cubic curves, supply/demand, and break-even functions.

Question 1 MCQ Quiz
Choose the correct option. Test your conceptual understanding using the interactive self-grading quiz below.
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Question 2 Exponential & Reciprocal
Plot the graph of the following functions:
(i) y = 3-x for x from -2 to 4     (ii) y = 2/x for x ≠ 0
Solution Step-by-Step
(i) y = 3-x (Exponential Decay)
Substitute values:
x-2-101234
y9310.330.110.040.01
Plot these coordinates. This curve decays towards 0 as x increases. y-intercept is (0, 1).
(ii) y = 2/x (Reciprocal)
Substitute values for x ≠ 0:
x-8-4-2-11248
y-0.25-0.5-1-2210.50.25
The y-axis (x = 0) is a vertical asymptote, and the x-axis (y = 0) is a horizontal asymptote. The graph consists of two branches in Quadrants I and III.
Question 3 Magazine Sales
Sales for a new magazine are expected to grow according to the equation: S = 200000(1 - e-0.05t), where t is given in weeks.
(a) Plot the graph of sales for the first 50 weeks.
(b) Calculate the number of magazines sold when t = 5 and t = 35.
Solution Step-by-Step
(a) Calculate coordinates for first 50 weeks:
Weeks t01020304050
Sales S078,694126,424155,374172,932183,583
Plot points and join with a smooth growth curve leveling off towards the horizontal asymptote at S = 200,000.
(b) Calculate for t = 5 and t = 35:
• For t = 5 weeks:
S = 200000(1 - e-0.05 × 5) = 200000(1 - e-0.25)
= 200000(1 - 0.7788) = 200000(0.2212) = 44,240 magazines

• For t = 35 weeks:
S = 200000(1 - e-0.05 × 35) = 200000(1 - e-1.75)
= 200000(1 - 0.1738) = 200000(0.8262) = 165,245 magazines
Question 4 Parabolas
Plot the graph of the following for x from -5 to 5:
(i) y = x² - 3     (ii) y = 15 - x²
Solution Step-by-Step
(i) y = x² - 3
Calculate coordinates:
x-5-3-10135
y226-2-3-2622
This is a parabola opening upward (since a = 1 > 0). Vertex is at (0, -3).
(ii) y = 15 - x²
Calculate coordinates:
x-5-3-10135
y-1061415146-10
This is a parabola opening downward (since a = -1 < 0). Vertex is at (0, 15).
Question 5 Cubic Curve
Plot the graph of y = 0.5(x + 4)(x - 1)(x - 3) from -5 to 4.
Solution Step-by-Step
Calculate coordinates for the given range:
x-5-4-3-2-101234
y-24012151260-3012
Plot the points and connect them with a smooth, continuous cubic curve. The curve intercepts the x-axis at the roots of the equation: x = -4, x = 1, and x = 3. It cuts the y-axis at y = 6.
Question 6 Supply & Demand
The supply and demand functions for a particular market are given by the equations: Ps = Q² + 5 and Pd = Q² - 10Q. Sketch the graph of each function over the interval Q = -20 to Q = 20.
Solution Step-by-Step
Calculate coordinates for supply and demand curves:
Q-20-1001020
Ps4051055105405
Pd60020000200
Plot the two parabolas on the same Cartesian coordinate plane:
• Supply Ps = Q² + 5 is a standard parabola opening upward shifted 5 units up.
• Demand Pd = Q² - 10Q is a parabola opening upward with vertex at (5, -25) and roots at Q = 0 and Q = 10.
Question 7 LED TV breakeven
A television manufacturing company makes 40-inch LEDs. The cost of manufacturing x LEDs is C(x) = 60,000 + 250x and the revenue from selling x LEDs is R(x) = 1200x. Find the break-even point and find the profit or loss when 100 LEDs are sold. Identify the break-even point graphically.
Solution Step-by-Step
1. Find break-even point:
Revenue equals cost:
R(x) = C(x)
1200x = 60,000 + 250x
950x = 60,000
x = 60,000950 ≈ 63.16 ≈ 63 LEDs
2. Profit/loss for x = 100:
Profit P(x) = R(x) - C(x) = 950x - 60,000
P(100) = 950(100) - 60,000
= 95,000 - 60,000 = Rs. 35,000 profit
3. Graphical interpretation:
The break-even point occurs at the intersection of the two lines at approximately (63.16, 75789.50).
Interactive Sandbox Equation Grapher