Rational Expressions and Functions
Unit Portfolio
Directions: Complete each of the tasks outlined below.
Task 1
Volume and surface area are often compared by manufacturers in order to
maximize how much of something can go inside of a package (volume) while
keeping how much material is required to create the package (surface area) low.
Pick a product that might be packaged in the shape of a rectangular prism. A
rectangular prism has three dimensions: length, width, and height. The surface
area of a rectangular prism can be found using the formula SA = 2lw + 2wh + 2lh.
The volume of a rectangular prism can be found using the formula V = lwh. Write
an expression for the ratio of surface area to volume for the figure.
Choose an appropriate length, width, and height for your package so that it can fit
the product you are shipping. Using these dimensions, what is the ratio of surface
area to volume?
Task 2
John, Rick, and Molli paint a room together.
a. Pick a reasonable amount of time in which the three friends can paint the
room together. Also pick a reasonable amount of time in which John can
paint the room alone and a reasonable amount of time in which Rick can
paint the room alone.
b. What is the hourly rate for John, Rick, and Molli (when working
together)? Use rooms per hour as the unit for your rates.
c. What is the hourly rate for John? What is the hourly rate for Rick? Refer
to the amount of time you determined in which John and Rick can paint
the room alone. Use rooms per hour as the unit for your rates.
d. Write an equation comparing the group rate to the sum of the individual
rates. How should the group rate and the sum of the individual parts
compare? Use parts (b) and (c) to help you write the equation.
e. What is the least common denominator for the equation you found in part (c)?
f. Solve the equation and determine how long it will take Molli to paint the
room alone.
Task 3
Suppose you are having a birthday party at the local bowling alley. You are trying
to figure out how many people you can afford to invite.
a. The number of guests you can invite to your party varies inversely with the
price per bowler at the alley. Explain what this means.
b. How much money are you willing to spend to host this bowling party?
c. Set up an equation that shows the inverse relationship between the number
of guests at your party and the price per bowler. Your answer to part (b)
should be part of this equation.
d. Research two local bowling alleys. Record the price per bowler at each of
these two alleys.
Calculate how many guests you will be able to invite to your party at each of the
bowling alleys. Which alley would you choose for your party? Why?
Task 1:
Product: A box of cereal
Dimensions: Length = 10 inches, Width = 4 inches, Height = 12 inches
Surface Area: SA = 2lw + 2wh + 2lh = 2(10)(4) + 2(4)(12) + 2(10)(12) = 328 square inches
Volume: V = lwh = (10)(4)(12) = 480 cubic inches
Ratio of Surface Area to Volume: SA/V = 328/480 = 0.6833
Task 2:
a. Reasonable amount of time: 4 hours
John's time alone: 6 hours
Rick's time alone: 8 hours
b. Group rate: 1/4 room per hour
c. John's rate: 1/6 room per hour
Rick's rate: 1/8 room per hour
d. Group rate = John's rate + Rick's rate + Molli's rate
1/4 = 1/6 + 1/8 + Molli's rate
e. Least common denominator: 24
f. Solving the equation:
1/4 - 1/6 - 1/8 = Molli's rate
1/24 = Molli's rate
Molli can paint the room alone in 24 hours.
Task 3:
a. Inverse relationship means that as one variable increases, the other variable decreases. In this case, as the price per bowler at the alley increases, the number of guests that can be invited decreases.
b. $200
c. Let x be the number of guests and y be the price per bowler. Then, we have:
xy = 200
d.
Bowling Alley 1: $5 per bowler
Number of guests: x = 200/5 = 40 guests
Bowling Alley 2: $6 per bowler
Number of guests: x = 200/6 = 33.33 guests (rounded down to 33)
I would choose Bowling Alley 1 because it can accommodate more guests within my budget.
Which of the following describes a claim?(1 point)
Responses
a statement that connects evidence to a reason
a statement that connects evidence to a reason
a statement that presents the speaker’s position or opinion
a statement that presents the speaker’s position or opinion
a statement that will grab the audience's attention
a statement that will grab the audience's attention
a statement that everyone in the audience will agree with
a statement that presents the speaker’s position or opinion
Task 1:
To find the expression for the ratio of surface area to volume for the rectangular prism, we can divide the surface area formula (SA = 2lw + 2wh + 2lh) by the volume formula (V = lwh).
Expression for the ratio of surface area to volume (SA/V):
SA/V = (2lw + 2wh + 2lh)/(lwh)
For the specific dimensions of the rectangular prism, you need to choose appropriate values for length, width, and height based on the product you are shipping in the package. Once you have these values, plug them into the expression. For example, let's say the length is 10 cm, width is 5 cm, and height is 7 cm:
SA/V = (2 * 10 * 5 + 2 * 5 * 7 + 2 * 10 * 7)/(10 * 5 * 7)
Simplifying the expression:
SA/V = (100 + 70 + 140)/(350)
SA/V = 310/350
The ratio of surface area to volume for a rectangular prism with length 10 cm, width 5 cm, and height 7 cm is 310/350, which can be further simplified if needed.
Task 2:
a. Pick reasonable amounts of time for the entire group, John alone, and Rick alone to paint the room. Let's say the group can paint the room in 4 hours, John alone can paint it in 6 hours, and Rick alone can paint it in 8 hours.
b. To find the hourly rate for the group, divide the number of rooms painted by the total time taken:
Group hourly rate = 1 room / 4 hours = 1/4 rooms per hour
c. To find the hourly rate for John, divide the number of rooms painted by the time taken for John alone:
John's hourly rate = 1 room / 6 hours = 1/6 rooms per hour
To find the hourly rate for Rick, divide the number of rooms painted by the time taken for Rick alone:
Rick's hourly rate = 1 room / 8 hours = 1/8 rooms per hour
d. The equation comparing the group rate to the sum of the individual rates can be written as:
Group rate = John's rate + Rick's rate
1/4 = 1/6 + 1/8
e. The least common denominator for 6 and 8 is 24, so we need to multiply the equation by 24 to clear the fractions:
24 * (1/4) = 24 * (1/6) + 24 * (1/8)
6 = 4 + 3
f. Solving the equation:
6 = 7
Since the equation 6 = 7 is not true, there is no solution. Therefore, it is not possible to determine how long it will take Molli to paint the room alone based on the given information.
Task 3:
a. "The number of guests you can invite to your party varies inversely with the price per bowler at the alley." This means that as the price per bowler increases, the number of guests you can invite decreases, and vice versa.
b. Determine how much money you are willing to spend to host the bowling party. Let's say you are willing to spend $200.
c. Set up an equation to represent the inverse relationship between the number of guests (g) and the price per bowler (p):
g = k/p
Where k is a constant. Substituting the value for money spent:
g = k/200
d. Research the price per bowler at two local bowling alleys. Let's say Alley 1 charges $10 per bowler and Alley 2 charges $8 per bowler.
To calculate the number of guests at each alley, substitute the respective values into the equation:
Number of guests at Alley 1 = k/10
Number of guests at Alley 2 = k/8
To compare the number of guests you can invite at each alley, substitute the value of k from part (b):
Number of guests at Alley 1 = 200/10 = 20 guests
Number of guests at Alley 2 = 200/8 = 25 guests
Based on this calculation, you would choose Alley 2 for your party because you can afford to invite more guests (25) compared to Alley 1 (20) within your budget.