ohaganbooks is offering a wide range of online books, including current best-sellers. a colleague has determined that the demand for the latest best selling book is given by q=(-p^2)+33p+9 (18<p<28) copies sold per week when the price is p dollars. can you help me determine what price the company should charge to obtain the largest revenue? the cost function is C=9q+100.

To determine the price at which the company should charge to obtain the largest revenue, we need to find the maximum point of the revenue function.

The revenue function can be obtained by multiplying the demand function (q) by the price (p). So, the revenue function (R) is given by:

R = p * q = p * [(-p^2) + 33p + 9]

To find the maximum point of the revenue function, we need to find the value of p that maximizes R. This can be done by finding the critical points of the revenue function.

1. Find the derivative of R with respect to p:
R' = d(R)/d(p) = d(p * [(-p^2) + 33p + 9])/d(p) = (-p^2) + 33p + 9 - 2p

2. Set the derivative equal to zero and solve for p:
0 = (-p^2) + 33p + 9 - 2p
0 = -p^2 + 31p + 9

This equation can be solved using the quadratic formula:
p = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -1, b = 31, and c = 9. Plugging these values into the quadratic formula, we get:
p = (-(31) ± √((31)^2 - 4(-1)(9))) / (2(-1))

After simplifying, we find two possible values for p:
p ≈ 0.33 or p ≈ 30.67

Since the price must be between 18 and 28, we can discard the value p ≈ 30.67. Therefore, the critical point for p is approximately p ≈ 0.33.

3. To confirm that this point is a maximum, we can analyze the second derivative of R.
R'' = d^2(R)/d(p)^2 = d(-p^2 + 31p + 9 - 2p)/d(p) = -2 - 2 = -4

Since the second derivative is negative, p ≈ 0.33 corresponds to a maximum point. This means that the company should charge approximately $0.33 to obtain the largest revenue.

Finally, to find the maximum revenue, substitute the value of p ≈ 0.33 into the revenue function:
R = p * q = 0.33 * [(-0.33^2) + 33(0.33) + 9]
R ≈ $9.40 (rounded to two decimal places)

Therefore, by charging approximately $0.33 per book, the company will obtain the largest revenue of approximately $9.40.