# Calculus

The profit (in millions of dollars) from the sale of x million units of Blue Glue is given by p= .7x-25.5. The cost is given by c= .9x +25.5

(a) Find the revenue equation.
(b) What is the revenue from selling 10 million units?
(c)What is the break-even point?

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Suppose you are the manager of a firm. The accounting department has provided cost estimates and the sales department sales estimates on a new product. You must analyze the data they give you, determine what it will take to break even, and decide whether or not to go ahead with the production of the new product.

(a) Cost is c=140x +3000 and revenue is 125x.

(b) Cost is c=1750x + 95,000 and revenue is r=1975x;no more than 600 units can be sold.

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Greg Tobin sells bottled water. He has found that the average amount of time he spends with each customer is related to his weekly sales volume by he function
f(x)=x(60-x),
where x is the number of minutes per customer and f(x) is the number of cases sold per week.
(a) How many cases does he sell if he spends 10 minutes with each customer?
20 minutes? 45 minutes?
(b)Choose and appropriate scale for the axes and sketch the graph f(x). Mark the points on the graph corresponding to 10, 20 and 45 minutes.
(c) Explain what the vertex of the graph represents.
(d)How long should Greg spend with each customer in order to sell as many cases per week as possible? If he does, how many cases will he sell?
----------------------------------------------------------------------------If an object thrown upward with an initial velocity of 32 ft per second, then it's height, in feet, above the ground after t seconds is given by

h=32t-16t^2

Find the maximum height attained by the object. Find the number of seconds it takes for the object to hit the ground.
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A rectangular garden bounded on one side by a river is the be fenced on the other three sides. Fencing material for the side parallel to the river costs \$30.00 per foot and material for the other two sides costs \$10.00 per foot.What are the dimensions of the garden of largest possible area, if \$1200.00 is to be spent for fencing material?
----------------------------------------------------------------------------
The function

f(x)= Kx/A+x
is used in biology to give the growth rate of a population in the presence of a quantity x of food. This is called Michaelis-Menten kinetics.
(a) What is a reasonable domain for this function, considering what x represents?
(b)Graph this function for K= 5, A=2 and x is greater than or equal to 0.
(c) Show that y=K is a horozontal asymptote.
(d) What do you think K represents?
(e) Show that A represents the quantity of food for which the growth rate is half of its maximum.
----------------------------------------------------------------------------
The manager of peach orchard is trying to decide when to arrange for picking the peaches. If they are picked now, the average yield for each tree will be 100lbs, which can be sold for .40 cents a pound. Past experience shows that the yield per tree will increase about 5 pounds per week , while the price will decrease about 2 cents per pound per week.
(a) Let x represent the number of weeks that the manager should wait. Find the income per pound.
(b) Find the number of pounds per tree.
(c) Find the total revenue fron a tree.
(d) When should the peaches be picked in order to produce the maximum revenue?
(e) What is the maximum revenue?

Welcome to Jiskha. I answered #1 for you earlier in an EMail and was hoping you would post the other questions separately. It maked it easier for others to join in answering.
================================
Regarding the second question,
<<Suppose you are the manager of a firm. The accounting department has provided cost estimates and the sales department sales estimates on a new product. You must analyze the data they give you, determine what it will take to break even, and decide whether or not to go ahead with the production of the new product.

(a) Cost is c =140x +3000 and revenue is 125x.

(b) Cost is c=1750x + 95,000 and revenue is r=1975x; no more than 600 units can be sold>>

The breakeven point occurs at the value of x when Cost = Revenue.
For case (a), that is when
140 x + 3000 = 125 x
15 x = 3000
x = 200
-----------------------------------
For case (b), breakeven is when
1750 x + 95,000 = 1975 x
225 x = 95,000
x = 422.
Since up to 600 can be sold, a profit is possible, but it will not exceed
(1975 - 1750)* 600 - 95,000 = 40,000.

Cases b and a must represent different marketing situations. Case b seems less appealing becasue of the high brfeakeven point and lmited profit potential.

This question requires algebra but no calculus

For a) I get not breakeven point is possible. For any positive value of x, costs are greater than revenue.

(DrWLS: step 2: 15x=-3000, x=-200)

Correct...the rational decision would be to not go into production.

I agree. My mistake.

Hi,
I wanted to know how wanted to know if i was doing this step right for my homework. the question is Given rt=<sin(t)-tcos(t), cos(t)+ tsin(t),t^2 sqrt 3 and the whole thing is divided by two.
The question is find the velocity vector and find V(pi)
for the first one i got cos(t)- 1(cos(t)-tsin(t) so that would give -tsint
the second integral would be sint+1*sin(t)+tcost so here it would be tcost. i don't know what the integral is for t^2sqrt 3/2 i want to say that its 2 t^2.
The second part asks to find the speed which is calculated by taking magntiude of the velocity vector.
which happents to be sqrt of t^2cos^2t+t^2sin^2t+4t^4you get that by using 2t^2.
so that becomes sqrt of t^2+ 4t^4 is there any way to simplify it further.
Thank you

<<If an object thrown upward with an initial velocity of 32 ft per second, then it's height, in feet, above the ground after t seconds is given by

h=32t-16t^2

Find the maximum height attained by the object. Find the number of seconds it takes for the object to hit the ground. >>

Using calculus, the maximum height occurs at time t when the first derivative of the function h(t) = 0
dh/dt = 32 - 32 t
t = 1 second
At that time, h(t) = 32 - 16 = 16 ft.

It hits the ground when h(t) = 0
32 t - 16 t^2 = 0
15 (2 - t) = 0
t = 0 or 2 seconds
(t=0 is the launch time, the answer they want is the t= 2s one)

<<A rectangular garden bounded on one side by a river is the be fenced on the other three sides. Fencing material for the side parallel to the river costs \$30.00 per foot and material for the other two sides costs \$10.00 per foot.What are the dimensions of the garden of largest possible area, if \$1200.00 is to be spent for fencing material? >>
Let x be the chosen length along the river and y the length perpendcular to the river.
The area is A = x y
The Cost is
C(x,y) = 30 x + 10 x + 20 y
= 30 (x + y)
Whatever x/y ratio you choose, you get the maximum area when you spend the full \$1200, so
1200 = 30 (x + y)
x + y = 40
We now have y as a funtion of x and can solve for Area in terms of x
A = x y = x (40 - x)= 40 x - x^2
Maximum area occurs when
dA/dx = 0 = 40 - 2 x
x = 20 feet
y = 40 - x = 20 feet.
Largest area = 20x20 = 400 ft^2.

The profit (in millions of dollars) from the sale of x million units of Blue Glue is given by p= .7x-25.5. The cost is given by c= .9x +25.5

(a) Find the revenue equation.
(b) What is the revenue from selling 10 million units?
(c)What is the break-even point?

Revenue= profit - cost. For b put in x, and calculate revenue. Break even point is when revenue is zero, or profit is equal to cost.

Greg Tobin sells bottled water. He has found that the average amount of time he spends with each customer is related to his weekly sales volume by he function
f(x)=x(60-x),
where x is the number of minutes per customer and f(x) is the number of cases sold per week.
(a) How many cases does he sell if he spends 10 minutes with each customer?
20 minutes? 45 minutes?
(b)Choose and appropriate scale for the axes and sketch the graph f(x). Mark the points on the graph corresponding to 10, 20 and 45 minutes.
(c) Explain what the vertex of the graph represents.
(d)How long should Greg spend with each customer in order to sell as many cases per week as possible? If he does, how many cases will he sell?

The vertex will represent the maximum number of sales per week. To calculate the maximum, take the derivitive of f(x), set to zero, solve for the x when f'(x)=0, then calculate f(x) at that x. Or, you can graph the f(x) and visually determine the maximum f(x) and read off x.

Income = revenue in this case. It is a function of x, the waiting time in weeks.

(a) Income per lb (\$)= P(x)
= 0.40 - 0.02 x
(b) Lbs per tree = Y(x) = 100 + 5 x
(c) Revenue per tree = P Y
= (0.40 - 0.02 x)(100 + 5 x)
= 40 - 2x + 2 x - 0.1 x^2
= 40 - 0.1 x^2
(d) Best time to pick: x=0,
(You can see this without using calculus. x>0 decreases revenue because of the -x^2 term. Linear terms in x cancel out.
(e) Max revenue = \$40 per tree

The function

f(x)= Kx/A+x
is used in biology to give the growth rate of a population in the presence of a quantity x of food. This is called Michaelis-Menten kinetics.
(a) What is a reasonable domain for this function, considering what x represents?
(b)Graph this function for K= 5, A=2 and x is greater than or equal to 0.
(c) Show that y=K is a horozontal asymptote.
(d) What do you think K represents?
(e) Show that A represents the quantity of food for which the growth rate is half of its maximum.

if x is food, it has to be non negative. f(x) is the growth rate, that is population rate of growth. x is the substrate concentration of "food". You can read about K, and A here

http://www.le.ac.uk/by/teach/biochemweb/tutorials/michment1.html

notice your algebra is a bit faulty, f(x)= Kx/(A+x). (e) is shown in the above link.

haha

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