In a certain region, the number of bushels of corn per acre, B, is given by the function B(n)=-0.1n^2 + 10n, where n represents the number of seeds, in thousands, planted per acre.

If corn sells for $3/bushel and cost $2 for 1000 seeds, find the optimal number of seeds to be planted per acre.

I need help please!!!

Oh, don't worry, I'm here to help (while cracking a few jokes along the way)! So, let's figure out the optimal number of seeds to be planted per acre.

First, we need to find the revenue function, R(n), which is the product of the number of bushels and the selling price per bushel. In this case, R(n) = 3B(n), since corn sells for $3/bushel.

Now, we also need to find the cost function, C(n), which is the cost of seeds per acre. The cost per seed is $2/1000 seeds. So, C(n) = (2/1000)n, my friend.

The profit function, P(n), can then be calculated by subtracting the cost function from the revenue function: P(n) = R(n) - C(n).

To find the optimal number of seeds, we need to find the maximum of the profit function. And you know what they say, "Maximizing profit is like telling a joke – it's all about finding the punchline!"

So, let's derive the profit function and set it equal to zero to find the critical points. Then we can determine if it's a maximum or minimum.

P'(n) = (3B'(n)) - (C'(n))

To find the critical points, set P'(n) = 0 and solve for n. We can also check the second derivative to confirm if it's indeed a maximum.

Once we have the critical points, we can evaluate the profit at each point to determine the optimal number of seeds to be planted per acre. And voila! You'll have the answer you're looking for!

I hope that brought a smile to your face while tackling the problem. Just remember, even math problems can use a little humor! Good luck!

To find the optimal number of seeds to be planted per acre, we need to maximize the function B(n) which represents the number of bushels of corn per acre.

The cost of planting n seeds can be calculated as:
Cost(n) = 2 * (n/1000) = 0.002n

The revenue from selling the corn can be calculated as:
Revenue(n) = 3 * B(n)

The profit is the difference between the revenue and the cost:
Profit(n) = Revenue(n) - Cost(n)

We need to find the value of n that maximizes the profit function Profit(n).

To find the optimal value, we can take the derivative of the profit function with respect to n and set it equal to 0, then solve for n.

Profit'(n) = 0 when d(Profit(n))/dn = 0

Differentiating the profit function:
Profit'(n) = d(Revenue(n))/dn - d(Cost(n))/dn
= 3 * d(B(n))/dn - d(0.002n)/dn
= 3 * B'(n) - 0.002

Setting Profit'(n) = 0 and solving for n:
3 * B'(n) - 0.002 = 0
3 * d(B(n))/dn = 0.002
d(B(n))/dn = 0.002/3
B'(n) = 0.002/3

Now, let's find the derivative of the function B(n):
B(n) = -0.1n^2 + 10n

B'(n) = d(-0.1n^2 + 10n)/dn
= -0.2n + 10

Setting B'(n) equal to 0.002/3:
-0.2n + 10 = 0.002/3

Solving for n:
-0.2n = 0.002/3 - 10
-0.2n = 0.002/3 - 30/3
-0.2n = (0.002 - 30)/3
-0.2n = -29.998/3
n = (-29.998/3) / -0.2
n ≈ 49.997

Therefore, the optimal number of seeds to be planted per acre is approximately 49,997 seeds.

To find the optimal number of seeds to be planted per acre, we need to determine the value of n for which the function B(n) is maximized.

The first step is to write the equation for the profit, P, as a function of n. Profit is calculated by subtracting the cost from the revenue, which can be written as:
P(n) = (3 * B(n)) - (2 * n)

Next, we can substitute the given expression for B(n) into the profit equation:
P(n) = (3 * (-0.1n^2 + 10n)) - (2 * n)

Now, simplify the equation:
P(n) = -0.3n^2 + 30n - 2n
P(n) = -0.3n^2 + 28n

To find the optimal number of seeds, we need to find the maximum value of the profit function. One way to do this is by using calculus. The maximum value occurs when the derivative of the profit function, P'(n), is equal to zero.

Taking the derivative of P(n) with respect to n:
P'(n) = -0.6n + 28

Setting P'(n) equal to zero and solving for n:
-0.6n + 28 = 0
-0.6n = -28
n = -(-28) / 0.6
n ≈ 46.67

Since n represents the number of seeds planted per acre, it cannot be a decimal value. Therefore, we round it to the nearest whole number:
The optimal number of seeds to be planted per acre is approximately 47 thousand.