HW13

1 Solve the Integral below:

\[ \int 4e^{-7x}dx \]

\[ \frac{4}{-7} e^{-7x} + C \]

2 Biologists are treating a pond contaminated with bacteria. Find a function N( t ) to estimate the level of contamination if the level after 1 day was 6530 bacteria per cubic centimeter.

$$

=-220 \ dN = (-220)dt \ N = dt-220dt \ N(1) = N_0 - - 220(1) \ N_0 = 6530 + 1050 + 220 \ N_0 = 7800 \ N = 7800 - - 220t

$$

3 Find the total area of the red rectangles in the figure below, where the equation of the line is f ( x ) = 2x  9.

func <- function(x) {
  2*x-9
}

integrate(func, 4.5, 8.5)$value

## [1] 16

4 Find the area of the region bounded by the graphs of the given equations.

func1 <- function(x) {
   x^2-2*x-2 
}
func2 <- function(x) {
x+2
}


integrate(func2, -1, 4)$value - integrate(func1, -1, 4)$value  

## [1] 20.83333

5 A beauty supply store expects to sell 110 flat irons during the next year. It costs $3.75 to store one flat iron for one year. There is a fixed cost of $8.25 for each order. Find the lot size and the number of orders per year that will minimize inventory costs.

x == size of order n == number of orders

\[ n*x = 110 \]

If we designate a storage spot for each iron in an order, but no additional spots (in other words, we wait till we are out to reorder), we can model our costs as follows:

\[ 3.75*x + 8.25*n \] Substituting …

\[ 3.75*110 / n + 8.25*n \\ 412.5/n + 8.25*n \\ \]