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Hey guys,
Ive been working on my Calc hw for over 2 hrs now, and ive only solved problem #1...I was just wondering if you guys could possible help me out with any of these problems..
Thanks alot:
2. (1 pt) Suppose that the cost of operating a truck in Mexico
is 45+0.32v cents per mile when the truck runs at a steady
speed of v miles per hour. The top speed of the truck is 100
mph. Assume that the driver is paid 12 dollars per hour to drive
the truck, and he is to begin a 2600 mile trip.
Write the cost of operating the truck in dollars, as a function
of the speed v, for the planned trip .
Write the cost of driver’s wages in dollars, as a function of the
speed v, for the planned trip .
The total cost of the planned trip, as a function of the speed v, is
the sum of the first two costs.
Find the most economic speed for
the planned trip, i.e., the speed that minimize the total cost is v=
.
3. (1 pt) A company must manufacture a closed rectangular
box with a square base. The volume must be 980 cubic inches.
The top and the bottom squares are made of a material that costs
6 dollars per square inch. The vertical sides are made of a different
material that costs 4 dollars per sqare inch.
What is the minimal cost of a box of this type? .
4. (1 pt) Find the rectangle of largest area that can be inscribed
in a semicircle of diameter 39, assuming that one side of
the rectangle lies on the diameter of the semicircle.
The largest possible area is .
5. (1 pt) A poster is to contain 119 square cm of printed matter,
with margins of 4 cm each top and bottom and 3.5 on each
of the sides. Find the minimal cost of the poster if it is to be
made of material costing 17 cents per square cm.
The minimal cost (in dollars) is .
6. (1 pt) A rancher wants to fence in an area of 1,600,000
square feet in a rectangular field and then divide it in half with a
fence down the middle parallel to one side.
What is the shortest length of fence that the rancher can use?
7. (1 pt) It is estimated that t years from now the population
of a certain country will be
P(t) =
125
1+10e−0.03t .
million. When will the population be growing most rapidly?
In order to solve the problem, you have to maximize the function
R(t) that gives the growth rate of the population, i.e. R(t) =
P0(t). Check that
R(t) =
37.5e−0.03t
(1+10e−0.03t )2 .
Now you need the critical numbers of the rate function. Check
that
R0(t) =
1.125e−0.03t (10e−0.03t −1)
(1+10e−0.03t )3 .
and use it to find the critical number that corresponds to the
maximal rate
t = .
8. (1 pt) Suppose the total cost (in dollars) of manufacturing
x units of a certain commodity is
C(x) = 3x2+18x+147.
At what level of production is the average cost per unit the
smallest. .
At what level of production is the average cost per unit equal to
the marginal cost. .
9. (1 pt) During a cough, the diameter of the trachea decreases.
If the radius of the trachea of a person is 15 mm in a
relaxed state, the velocity v of the air in the trachea during a
cough may be modeled by the formula
v = Ar2(15−r)
where a is a constant, and r is the radius of the trachea during
the cough, 0 _ r _ 15.
Find the radius of the trachea when the velocity is greatest. r=
.
Find the maximal velocity of air, v-max= . Your answer
will be an expression that depends on A (use the capital
letter).
10. (1 pt) Suppose the total cost of producing x units of a
certain commodity is
C(x) =
1
4
x4+ −43
3
x3+
391
2
x2+1587x+100
Determine the largest and the smallest values of the MARGINAL
cost C0(x) for 0 _ x _ 16.
The largest value of the marginal cost is .
The smallest value of the marginal cost is .
Hint. The function f (x) that you have to MIN/MAX is the
derivative of cost, i.e. f (x) =C0(x).
Ive been working on my Calc hw for over 2 hrs now, and ive only solved problem #1...I was just wondering if you guys could possible help me out with any of these problems..
Thanks alot:
2. (1 pt) Suppose that the cost of operating a truck in Mexico
is 45+0.32v cents per mile when the truck runs at a steady
speed of v miles per hour. The top speed of the truck is 100
mph. Assume that the driver is paid 12 dollars per hour to drive
the truck, and he is to begin a 2600 mile trip.
Write the cost of operating the truck in dollars, as a function
of the speed v, for the planned trip .
Write the cost of driver’s wages in dollars, as a function of the
speed v, for the planned trip .
The total cost of the planned trip, as a function of the speed v, is
the sum of the first two costs.
Find the most economic speed for
the planned trip, i.e., the speed that minimize the total cost is v=
.
3. (1 pt) A company must manufacture a closed rectangular
box with a square base. The volume must be 980 cubic inches.
The top and the bottom squares are made of a material that costs
6 dollars per square inch. The vertical sides are made of a different
material that costs 4 dollars per sqare inch.
What is the minimal cost of a box of this type? .
4. (1 pt) Find the rectangle of largest area that can be inscribed
in a semicircle of diameter 39, assuming that one side of
the rectangle lies on the diameter of the semicircle.
The largest possible area is .
5. (1 pt) A poster is to contain 119 square cm of printed matter,
with margins of 4 cm each top and bottom and 3.5 on each
of the sides. Find the minimal cost of the poster if it is to be
made of material costing 17 cents per square cm.
The minimal cost (in dollars) is .
6. (1 pt) A rancher wants to fence in an area of 1,600,000
square feet in a rectangular field and then divide it in half with a
fence down the middle parallel to one side.
What is the shortest length of fence that the rancher can use?
7. (1 pt) It is estimated that t years from now the population
of a certain country will be
P(t) =
125
1+10e−0.03t .
million. When will the population be growing most rapidly?
In order to solve the problem, you have to maximize the function
R(t) that gives the growth rate of the population, i.e. R(t) =
P0(t). Check that
R(t) =
37.5e−0.03t
(1+10e−0.03t )2 .
Now you need the critical numbers of the rate function. Check
that
R0(t) =
1.125e−0.03t (10e−0.03t −1)
(1+10e−0.03t )3 .
and use it to find the critical number that corresponds to the
maximal rate
t = .
8. (1 pt) Suppose the total cost (in dollars) of manufacturing
x units of a certain commodity is
C(x) = 3x2+18x+147.
At what level of production is the average cost per unit the
smallest. .
At what level of production is the average cost per unit equal to
the marginal cost. .
9. (1 pt) During a cough, the diameter of the trachea decreases.
If the radius of the trachea of a person is 15 mm in a
relaxed state, the velocity v of the air in the trachea during a
cough may be modeled by the formula
v = Ar2(15−r)
where a is a constant, and r is the radius of the trachea during
the cough, 0 _ r _ 15.
Find the radius of the trachea when the velocity is greatest. r=
.
Find the maximal velocity of air, v-max= . Your answer
will be an expression that depends on A (use the capital
letter).
10. (1 pt) Suppose the total cost of producing x units of a
certain commodity is
C(x) =
1
4
x4+ −43
3
x3+
391
2
x2+1587x+100
Determine the largest and the smallest values of the MARGINAL
cost C0(x) for 0 _ x _ 16.
The largest value of the marginal cost is .
The smallest value of the marginal cost is .
Hint. The function f (x) that you have to MIN/MAX is the
derivative of cost, i.e. f (x) =C0(x).
