Derivative word problems

[shahroze]

Can anyone help me to solve these derivative word problems pls
A sum of $100 is invested in an enterprise with an unknown risk. However after t months the value of the investment, in dollars, is
I (t) = 100 + 100t2 / t2 + 1
a) What is the Rate at which the investment is changing?
b) Simplify the expression and evaluate after 1 month?
An apartment complex has 250 apartments to rent. If they rent x apartments then their monthly profit, in dollars, is given by,
P(x) = -8x2 + 3200x – 80,000
a) How many apartments should they rent in order to maximize their profit?
b) What will be the Maximum Profit, if they rent all those apartments?
A production facility is capable of producing 60,000 widgets in a day and the total daily cost of producing x widgets in a day is given by,
C(x) = 250,000 + 0.08x +200,000,000 / x
a) How many widgets per day should they produce in order to minimize production costs?
b) What will be the minimum cost per day for producing those widgets?
The production costs per week for producing x widgets is given by,
C (x) = 500 +350x – 0.09x2
a) What is the cost to produce the 301st widget?
b) What is the rate of change of the cost at x = 300?
The total revenue function for a kind of t-shirt is R(x) = 16x – 0.01x2, where R is in dollars and x is the number of t-shirts sold. Find the following:
a) Find the revenue when 40 units are sold.
b) Find the marginal revenue function.
c) Find the marginal revenue at x = 40. What do you predict about the sale of the next unit?

 

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[stapel]

A sum of $100 is invested in an enterprise with an unknown risk. However after t months the value of the investment, in dollars, is
I (t) = 100 + 100t2 / t2 + 1

As posted, the above means \(\displaystyle I(t)\, =\, 100\, +\, \dfrac{100t_2}{t_2}\, +\, 1\). Was this what you meant?

a) What is the Rate at which the investment is changing?

What information were you given, relating the given equation to the specified item?

b) Simplify the expression and evaluate after 1 month?

If "\(\displaystyle t_2\)" stands for the time in months, then plug the given value in for the given variable in the given equation, and simplify to find the answer. (This works just like back in algebra.)

An apartment complex has 250 apartments to rent. If they rent x apartments then their monthly profit, in dollars, is given by,
P(x) = -8x2 + 3200x – 80,000
a) How many apartments should they rent in order to maximize their profit?

Like back in algebra, find the vertex of this upside-down parabola. Then interpret the coordinates in terms of the question.

b) What will be the Maximum Profit, if they rent all those apartments?

Apply the vertex information from part (a).

A production facility is capable of producing 60,000 widgets in a day and the total daily cost of producing x widgets in a day is given by,
C(x) = 250,000 + 0.08x +200,000,000 / x
a) How many widgets per day should they produce in order to minimize production costs?

Take the derivative. Set it equal to zero. Find the minimizing value.

b) What will be the minimum cost per day for producing those widgets?

Plug the minimizing value into the "cost" equation. Simplify for the answer.

The production costs per week for producing x widgets is given by,
C (x) = 500 +350x – 0.09x2
a) What is the cost to produce the 301st widget?

Apply the methodology they gave you for this. What result do you get?

The total revenue function for a kind of t-shirt is R(x) = 16x – 0.01x2, where R is in dollars and x is the number of t-shirts sold. Find the following:
a) Find the revenue when 40 units are sold.

Plug the given value in for the specified variable in the given equation (just like back in algebra). Simplify to obtain the numerical answer.

b) Find the marginal revenue function.
c) Find the marginal revenue at x = 40. What do you predict about the sale of the next unit?

How did your book / class relate "marginal revenue" to "total revenue"? Apply the methodology they gave you.

If you get stuck, please reply showing your work, especially on the only-algebra parts, so we can help you get un-stuck on the calculus parts. Thank you!