A squash patch currently has 120 lbs of butternut squash. For each

A squash patch currently has 120 lbs of butternut squash. For each additional day in the patch, the amount of butternut squash increases by 6 lbs. If the price of butternut squash is currently 80 cents per pound, but it decreases by 2 cents each day, how long should the butternut squash grow to maximize profit?

after x days, we have

lbs: 120 + 6x
price: .80-.02x

No idea about profit, since cost was not mentioned, but the income is certainly

(120+6x)(.80-.02x) = -0.12x^2 + 2.4x + 96

That is just a parabola with vertex at (10,108)

10 days, because the x of the vertex is the maximum profit

@Steve, where does the -0.12x^2 + 2.4x + 96 come from? I don't understand, please answer

thanks

it is from (120+6x)

Well, I must say, this is one mighty squash predicament! Let's crunch some numbers, shall we?

To find the optimal growth period, we need to calculate the profit for each additional day. The profit is defined as the difference between the revenue and the cost. Since revenue is selling price multiplied by the weight, and the selling price decreases each day, let's start with that.

On the first day, the selling price is 80 cents per pound, so the revenue is 80 cents times 120 pounds, equal to 96 dollars. However, remember that the selling price decreases by 2 cents each day, so we need to account for that.

The formula for the selling price is: Selling Price = Initial Price - (Number of Days x Decrease Rate)

Now, let's calculate the costs. The cost is fixed since it doesn't change with the number of days.

The overall formula for profit is: Profit = (Selling Price x Weight) - Cost

Alright, seize the butternut! We're diving into the abyss of calculations!

To maximize the profit, we need to find the number of days when the profit is the highest. So, we must calculate the profit for each day and compare the results until we find the peak.

Are you ready to wrangle some numbers and find the optimal day for maximum profit?

To find the length of time the butternut squash should grow to maximize profit, we need to consider the relationship between the amount of butternut squash, the price per pound, and the number of days.

Let's denote the number of days the squash grows by "n". The amount of butternut squash, "A", after "n" days can be calculated using the following formula:

A = 120 + 6n

The price per pound of butternut squash, "P", is given by:

P = 80 - 2n (since the price decreases by 2 cents each day)

We can express the profit, "Profit", as the product of the amount of butternut squash and the price per pound:

Profit = A * P = (120 + 6n) * (80 - 2n)

To maximize the profit, we need to find the value of "n" that maximizes this profit function.

First, let's expand the profit function:

Profit = (9600 + 240n - 240n - 12n^2) = 9600 - 12n^2

This is now a quadratic equation, where the coefficient of the quadratic term is negative. The maximum point of a quadratic function is given by -b/2a, where "a" is the coefficient of the quadratic term and "b" is the coefficient of the linear term.

In our case, "a" is -12 and "b" is 0. Substituting these values, we get:

n = -0/2(-12) = 0/24 = 0

Therefore, the maximum profit occurs at n = 0.

However, it is not reasonable to have 0 days since the squash needs time to grow. So, the squash should be grown for the maximum possible time, which is the time it takes for the price per pound to reach 0 or become negative.

Setting the price per pound to 0 and solving for "n":

0 = 80 - 2n

2n = 80

n = 40

Therefore, the butternut squash should grow for 40 days to maximize profit.