Partial Fractions
Word Problems 7.1
Word Problems 7.2
Linear Programming #1
Linear Programming #2
100
Set up the partial Fraction Decomposition (DO NOT SOLVE):
(5x-7)/(x^2+4x+3)
A/(x+3)+B/(x+1)
100
Aiden runs a farm stand that sells apples and strawberries. Each pound of apples sells for $2 and each pound of strawberries sells for $3. Aiden made $80 from selling a total of 35 pounds of apples and strawberries. Write a system of equations that could be used to determine the number of pounds of apples sold and the number of pounds of strawberries sold. Define the variables.
a=number of pounds of apples sold
w=number of pounds of strawberries sold
2a+3w=80
a+w=35
100
At a school store, a notebook, a pen, and a pencil together cost $6. Two notebooks, one pen, and one pencil cost $9. One notebook, three pens, and two pencils cost $11. Write a system of equations to find the price of each item. Define your variables.
n=price of a notebook, p=price of a pen, c=price of a pencil
n+p+c=6
2n+p+c=9
n+3p+2c=11
100
A student runs a lemonade stand selling small and large cups of lemonade. Each small cup earns a profit of $2, and each large cup earns a profit of $3. She can sell at most 40 cups total per day. She also only has enough lemonade mix to make at most 25 large cups. How many of each cup of lemonade should she sell to maximize her profit? Define your variables, list the objective function AND write your constraints.
x=number of small cups, y=number of large cups
objective function:
z=2x+3y
constraints:
x+y<=40
y<=25
x>=0 y>=0
100
A toy company makes trucks and dolls. The company needs to make at least 20 cases of trucks to fulfill the orders already placed for Christmas. They also need to make at least 30 cases of dolls. All together the toy company cannot produce more than 80 cases of toys because of labor and material limits. The company makes $200 profit off a case of trucks and $180 profit off a case of dolls. Which combination of cases of trucks and cases of dolls will give the company the most profit? Define the variables, list the objective function, AND list the constraints.
x=# of cases of trucks, y= number of cases of dolls
objective function:
z=200x+180y
constraints:
x>=20
y>=30
x+y<=80
200
Set up the partial Fraction Decomposition (DO NOT SOLVE):
(4x)/((x-5)(x^2+3x-1)
A/(x+5)+(Bx+C)/(x^2+3x-1)
200
Adrian has $1 worth of nickels and dimes. He has twice as many nickels as dimes. Write a system of equations that could be used to determine the number of nickels and the number of dimes that Adrian has. Define the variables.
n=number of nickels
d=number of dimes
n=2d
0.05n+0.1d=1
200
At a school play, the total cost of two adult tickets, one student ticket, and one child ticket is $37. The total cost of one adult ticket, two student tickets, and one child ticket is $32. The total cost of one adult ticket, one student ticket, and two child tickets is $27. Find the price of each type of tickt.
a=adult ticket price, s=student ticket price, c=child ticket price
2a+s+c=37
a+2s+c=32
a+s+2c=27
200
A company prints basic and premium T-shirts. Each basic shirt earns a profit of $8, and each premium shirt earns a profit of $12. A basic shirt requires 2 hours of labor, and a premium shirt requires 3 hours of labor. The company has 120 labor hours available and must produce at least 10 premium shirts. How many shirts should they produce to maximize profit? Define the variables, list the objective function, AND list the constraints.
x=number of basic shirts, y=number of premium shirts
Objective function:
z=8x+12y
Constraints:
2x+3y<=120
y>=10
x>=0
200
Aubrey makes two types of jewelry; necklaces and earrings. She is trying to maximize her profit from selling her jewelry. She has a limited number of boxes to package her jewelry, so at most she can make 12 pairs of earrings and 20 necklaces. Additionally, she needs 8 beads for each pair of earrings and 6 beads for each necklace, but only has 144 beads in total. Aubrey profits $21 for each pair of earrings and $15 for each necklace. How many pairs of earrings and necklaces can Aubrey make and sell to maximize her profit? Define the variables, list the objective function, AND list the constraints.
x=#of necklaces, y=# of earrings
objective function:
z=21y+15x
constraints:
y<=12
x<=20
8y+6x<=144
x>=0 y>=0
300
Set up the partial Fraction Decomposition (DO NOT SOLVE):
(3x+9)/(2x^2+9x+10)
A/(x+2)+B/(2x+5)
300
Jeremy and Livi have a combined age of 22. Jeremy is 4 years older than twice Livi's age. Write a system of equations to determine the age of Jeremy and Livi. Define the variables.
J=Jeremy's age
L=Livi's age
J+L=22
J=2L+4
300
On a small farm, there are chickens, goats and cows. There are 20 animals in total. Altogether, the animals have 56 legs. There are 8 more chickens than goats. How many of each animal are there? Define your variables.
c=number of chickens, g=number of goats, w=number of cows
c+g+w=20
2c+4g+4w=56
c=g+8
300
A bakery makes cakes and batches of cookies. Each cake earns $15 in profit and requires 4 cups of flour and 2 hours of labor. Each batch of cookies earns $10 and requires 3 cups of flour and 1 hour of labor. The bakery has 60 cups of flour and 30 hours of labor available. How many cakes and batches of cookies should the bakery make to maximize profit? Define the variables, list the objective function, AND list the constraints.
x=number of cakes, y=number of batches of cookies
Objective function:
z=15x+10y
constraints:
4x+3y<=60
2x+y<=30
x>=0 y>=0
300
A farmer has 600 acres of land to grow corn or soybeans. Because of a government subsidy he has to grow corn and he needs to grow at least 250 acres of soybeans. The farmer makes a profit of $80 for each acre of corn and$75 for each acre of soybeans. Find the number of acres of corn and soybeans the farmer should plant to earn the maximum profit. Define the variables, list the objective function, AND list the constraints.
x=# of acres of corn, y=# of acres of soybeans
Objective function:
z=80x+75y
constraints:
x+y<=600
x>0
y>=250
400
Set up the partial Fraction Decomposition (DO NOT SOLVE):
9/(x^2(x^2+4)^2
A/x+B/x^2+(Cx+D)/(x^2+4)+(Ex+F)/(x^2+4)^2
400
Heather works at an electronic store as a salesperson. She earns a 4% commission on the total dollar amount of all phone sales she makes, and earns a 5% commission on all computer sales. Heather had $600 more in computer sales than in phone sales and earned a total of $129 in commission. Write a system of equations that could be used to determine the dollar amount of phone sales she made and the dollar amount of computer sales she made. Define the variables.
p=number of dollars in phone sales
c=number of dollars in computer sales
0.04p+0.05c=129
c=p+600
400
A woman invests $10,000 into three accounts earning 3%, 5%, and 8% annual interest. She invests twice as much in the 3% account as in the 5% account. If her total annual interest is $480, how much is invested in each account? Define your variables.
x=amount invested at 3%, y=amount invested at 5%, z=amount invested at 8%
x+y+z=10,000
0.03x+0.05y+0.08z=480
x=2y
400
A furniture factory produces tables and chairs. Each table earns $40 profit and requires 5 hours of carpentry and 2 hours of finishing work. Each chair earns $25 profit and requires 3 hours of carpentry and 1 hour of finishing work. The factory has 150 carpentry hours and 60 finishing hours available. The factory policy requires at least twice as many chairs as tables. How many tables and chairs must the furniture factory produce to maximize profit? Define the variables, list the objective function, AND list the constraints.
x=number of tables, y=number of chairs
Objective function:
z=40x+25y
Constraints:
5x+3y<=150
2x+y<=60
y>=2x
x>=0 y>=0
400
Bob builds tool sheds. He used 10 sheets of dry wall and 15 studs for a small shed and 15 sheets of dry wall and 45 studs for a large shed. He has available 60 sheets of dry wall and 135 studs. If Bob makes $390 profit on a small shed and $520 on a large shed, how many of each type of sheds should Bob build to maximize his profit? Define the variables, list the objective function, AND list the constraints.
x=# of small sheds, y=# of large sheds
Objective function:
z=390x+520y
constraints:
10x+15y<=60
15x+45y<=135
x>=0 y>=0
500
Set up the partial Fraction Decomposition (DO NOT SOLVE):
(7x)/(6x^3+9x^2+4x+6)
(Ax+B)/(3x^2+2)+C/(2x+3)
500
A chemist wants to make 40 liters of a 35% acid solution by mixing a 25% acid solution with a 50% acid solution. How many liters of each solution should be used? Write a system of equations to solve the problem and define the variables.
x=number of liters of 25% solution
y=number of liters of 50% solution
x+y=40
0.25x+0.5y=14
500
A candy shop sells small, medium, and large boxes of candy. On Monday, 50 boxes were sold in total. Small boxes cost $4 each, medium boxes cost $6 each, and large boxes cost $10 each. The total revenue for the day was $310. The number of small boxes sold was equal to three times the number of large boxes plus the number of medium boxes. How many of each size were sold? Set up a system to solve this and define your variables.
s=number of small boxes, m=number of medium boxes, g=number of large boxes
s+m+g=50
4s+6m+10g=310
s=3g+m
500
A personal trainer is designing a daily meal plan using protein bars and smoothies. Each protein bar contains 10 grams of protein, 5 grams of fat, and costs $2. Each smoothie contains 15 grams of protein, 2 grams of fat, and costs $3. The meal plan must provide at least 120 grams of protein, no more than 40 grams of fat, include at least 3 protein bars, and include no more than 5 smoothies. How many protein bars and smoothies should the trainer use to minimize the total cost? Define the variables, list the objective function, AND list the constraints.
x=number of protein bars, y=number of smoothies
Objective function:
z=2x+3y
constraints:
10x+15y>=120
5x+2y<=40
x>=3
y<=5 y<=0
500
A commercial gardener wants to feed plants a very specific mix of nitrates and phosphates. Two kinds of fertilizer, Brand A and Brand B, are available, each sold in 50 pound bags, with the following quantities of each mineral per bag:
The gardener wants to put at least 30 lbs of nitrates and 15 lbs of phosphates on the gardens and not more than 250 lbs of fertilizer altogether. If Brand A costs $8.50 a bag and Brand B costs $3.50 a bag, how many bags of each would minimize fertilizer costs? Define the variables, list the objective function, AND list the constraints.
x=# of bags of Brand A, y=# of bags of Brand B
Objective function:
z=8.5x+3.5y
Constraints:
10x+5y>=30
2.5x+5y>=15
50x+50y<=250
x>=0 y>=0