Multiple Choice Questions
Mathematical Modeling and Applications
Topic: Mathematical Modeling and Applications
Grade: 11
Question 1:
A company produces and sells widgets. The cost to produce x widgets is given by C(x) = 500 + 2x^2, and the revenue from selling x widgets is given by R(x) = 10x. What is the maximum profit the company can make?
Answer Choices:
A) $2,500
B) $5,000
C) $10,000
D) $20,000
Answer Explanation:
The profit function is given by P(x) = R(x) – C(x). To find the maximum profit, we need to find the critical points of P(x) by setting its derivative equal to zero. Taking the derivative of P(x), we get P\'(x) = 10 – 4x. Setting this equal to zero, we find x = 2. Plugging this value back into P(x), we get P(2) = 20 – (500 + 2(2)^2) = -$476. Since profit cannot be negative, the maximum profit the company can make is $0. Therefore, the correct answer is not listed.
Example:
Let\’s consider a simpler example where C(x) = 10x and R(x) = 5x. In this case, the profit function is given by P(x) = 5x – 10x = -5x. The critical point of P(x) is x = 0, and plugging this value back into P(x), we get P(0) = -5(0) = $0. Therefore, the maximum profit the company can make is $0.
Question 2:
A population of bacteria is modeled by the function P(t) = 1000e^(0.05t), where P(t) represents the population at time t. If the initial population is 1000, what is the population after 1 hour?
Answer Choices:
A) 1000
B) 1050
C) 1100
D) 1150
Answer Explanation:
To find the population after 1 hour, we substitute t = 1 into the given function. P(1) = 1000e^(0.05(1)) = 1000e^(0.05) ≈ 1051.27. Therefore, the correct answer is B) 1050.
Example:
Consider a simpler example where P(t) = 100e^(0.1t), with an initial population of 100. To find the population after 1 hour, we substitute t = 1 into the function. P(1) = 100e^(0.1(1)) = 100e^(0.1) ≈ 110.52. Therefore, the population after 1 hour is approximately 110.52.
Note: The exponential function e^x is commonly used to model population growth or decay, where x represents time. The constant in the exponent determines the rate of growth or decay.