Name:

You may use your notes, R, or Wolfram Alpha for this quiz.

  1. Given the following Marginal Benefit (MB) and Marginal Cost (MC) functions, solve for the quantity (x) where MB = MC. Assume x is greater than or equal to 0.

\[ MB = \frac{360}{x} \]

\[ MC = 10 + 6x \] \[ 0 = 10x + 6x^{2} - 360 \]

QUAD <- function(a,b,c) {
    pos_sol <- (-b + sqrt(b^2-4*a*c))/(2*a)
    neg_sol <- (-b - sqrt(b^2-4*a*c))/(2*a)
    vertex <- -b/(2*a)
    
  sol <- c(pos_sol,neg_sol,vertex)
  
}

QUAD(6,10,-360)

soln <- QUAD(6,10,-360)[1]

soln
## [1] 6.957331

\[ x = 6.9573306\]

  1. Use R to answer the following questions about a continuously compounding future value function:

\[ FV = PVe^{rt} \]

  1. Plot the function starting at t = 0 to t = 30. Let PV = 1000 and r = 0.10.
# Plot of FV Function

t <- seq(from=0,to=30,by=1)

FV <- 1000*exp(0.10*t)

plot(t,FV,type="l")

FV_20 <- FV[21]
  1. Determine the value of the investment at t = 20.

7389.0560989

  1. Approximatley how long will it take for this investment to double?

\[ t = \frac{70}{10} \approx 7 yrs\]

  1. Use the following revenue and cost functions for the next set of questions.

\[R(Q) = 25Q \] \[C(Q) = 0.5Q^{2} \]

  1. Find an expression for profits (R(Q)- C(Q))

\[ \pi = -0.5Q^{2} + 25Q\]

  1. Use your QUAD function to find the 0’s for the profit function. Plot the profit function using these results to form your starting value and ending value.

Q = 0 Q = 50

# Profit Plot
QUAD <- function(a,b,c) {
    pos_sol <- (-b + sqrt(b^2-4*a*c))/(2*a)
    neg_sol <- (-b - sqrt(b^2-4*a*c))/(2*a)
    vertex <- -b/(2*a)
    
    sol <- c(pos_sol,neg_sol,vertex)
    print(sol)
}

QUAD(-0.5,25,0)
## [1]  0 50 25
pmax <- QUAD(-0.5,25,0)[3]
## [1]  0 50 25
Q <- seq(from=0,to=50,by=1)
profit <- -0.5*Q^2 + 25*Q

plot(Q,profit,type="l")

  1. At what quantity are profits maximized? (Hint: Use your results from the QUAD function)

\[ Q = 25 \]