A souvenir shop sells 300 key chains each month for $5 each.the shop owner estimates that for each 50 cents increase in price he will sell 15 fewer key chains per month.write an equation to model this. How many increases should make to maximize revenue

r = (300 - 15 n) (5 + .50 n) = 1500 + 75 n - 7.5 n^2

the max is on the axis of symmetry ... n = -b / (2 a)

To model the relationship between the price of the keychains and the number of keychains sold, we can use a linear equation. Let's define the variables:

Let x be the price increase in dollars.
Let y be the decrease in the number of keychains sold.

From the given information, we know that for each 50 cents increase in price, the shop will sell 15 fewer keychains. Therefore, the relationship can be expressed as:

y = (x / 0.5) * 15

We divide x by 0.5 since every 50 cents is equivalent to 0.5 dollars, and then multiply by 15 to account for the 15 fewer keychains.

Now, to find the price that maximizes revenue, we need to consider the revenue equation. Revenue is calculated by multiplying the price by the number of keychains sold. Let's denote R as the revenue:

R = (300 - y) * (5 + x)

Since the number of keychains sold decreases by y, the remaining quantity will be 300 - y. The price also increases by x, so the new price will be $5 + x.

To find the value of x that maximizes revenue, we need to find the value that yields the highest R. We can do this by computing the derivative of R with respect to x, setting it equal to zero, and solving for x. However, since the relationship between the number of keychains sold and the price increase is linear, we can see that the revenue will be maximized when the price increase is zero (x = 0).

Therefore, the shop owner should not make any price increases to maximize revenue.