Question

The cost to produce bottled spring water is given
by Cost = 16x - 63 − where x is the number of
thousands of bottles. The total income (revenue)
from the sale of these bottles is given by the function
R(x) = -x2 + 326 - 7463
How many bottles sold will produce the maximum
profit?
(a) 125 (b) 155
(c) 175 (d) 185
Can someone show the working also

Answer 1

155  bottles sold will produce the maximum profit

The profit made from selling x thousands of bottles can be found by subtracting the cost of production from the total income.

So, the profit function can be written as:

P(x) = R(x) - Cost

= $-x^2$+ 326x - 7463 - 16x + 63

= $-x^2$ + 310x - 7426

To find the maximum profit, we need to find the value of x that maximizes the profit function P(x).

To do this, we can find the derivative of the profit function and set it equal to zero.

Then, we can solve for x using the first derivative test.

$dP/dx$ = -2x + 310 = 0

x = 155

So, the maximum profit is obtained when 155 thousands of bottles are sold.

Hence, the answer is (b) 155

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