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?
Product: Cereal Box
Length: 10 inches
Width: 3 inches
Height: 12 inches
Surface Area: SA = 2lw + 2wh + 2lh = 2(10)(3) + 2(3)(12) + 2(10)(12) = 276 sq. in.
Volume: V = lwh = 10(3)(12) = 360 cubic inches
Ratio of Surface Area to Volume: SA/V = 276/360 ≈ 0.767
Therefore, the ratio of surface area to volume for the cereal box is approximately 0.767.
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.
a. Reasonable time for all three friends to paint the room together: 4 hours
Reasonable time for John to paint the room alone: 6 hours
Reasonable time for Rick to paint the room alone: 4 hours
b. Let x be the hourly rate for John, Rick, and Molli together.
1 Room painted by all 3 friends in 4 hours = 1/4 rooms per hour (rate)
Therefore, the hourly rate for all three together is x = 1/4 rooms per hour
c. Let y be the hourly rate for John and z be the hourly rate for Rick.
John paints 1 room in 6 hours = 1/6 rooms per hour (rate)
Therefore, the hourly rate for John alone is y = 1/6 rooms per hour
Rick paints 1 room in 4 hours = 1/4 rooms per hour (rate)
Therefore, the hourly rate for Rick alone is z = 1/4 rooms per hour
d. Equation: Group rate = Sum of individual rates
x = y + z + Molli's rate
e. Least common denominator for (c) is 12.
f. To solve for Molli's hourly rate, we can substitute given values into the equation we found in (d):
1/4 = 1/6 + 1/4 + Molli's rate
Solving for Molli's rate:
Molli's rate = 1/4 - 1/6 - 1/4 = 1/12
Therefore, Molli can paint one room alone in 12 hours at a rate of 1/12 rooms per hour.
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?
a. When two values vary inversely, it means that they change in opposite directions. In this case, as the price per bowler at the alley increases, the number of guests you can invite to your party decreases.
b. You are willing to spend $200 to host the bowling party.
c. Let n be the number of guests you can invite, and let p be the price per bowler at the alley. The equation showing the inverse relationship between the number of guests and the price per bowler is:
n = k/p
where k is a constant of proportionality. Since we know that you are willing to spend $200, we can use this information to solve for k:
n = k/p
200 = k/p
k = 200p
Substituting k back into the equation, we get:
n = 200/p
d.
Bowling Alley A: Price per bowler is $15.
Using the equation we just derived in part (c), we can calculate the number of guests you can invite:
n = 200/15
n = 13.3
Since you cannot invite a fraction of a person, the maximum number of guests you can invite to Bowling Alley A is 13.
Bowling Alley B: Price per bowler is $12.
Using the equation, we can calculate the number of guests you can invite:
n = 200/12
n = 16.7
Again, you cannot invite a fraction of a person, so the maximum number of guests you can invite to Bowling Alley B is 16.
Based on these calculations, you would choose Bowling Alley B for your party, since you can invite more guests (16 guests) as compared to Bowling Alley A (13 guests) given your budget and the inverse relationship between the number of guests and price.
To find the expression for the ratio of surface area to volume for a rectangular prism, we first need to determine the length, width, and height of the package. Let's say the length is L, the width is W, and the height is H.
The surface area of the rectangular prism can be calculated using the formula:
SA = 2lw + 2wh + 2lh
The volume of the rectangular prism can be calculated using the formula:
V = lwh
Now, let's substitute the given values into the formulas:
SA = 2lw + 2wh + 2lh
V = lwh
Assuming the length (L) is 10 units, the width (W) is 5 units, and the height (H) is 3 units, we can calculate the surface area and volume:
SA = 2(10)(5) + 2(5)(3) + 2(10)(3) = 100 + 30 + 60 = 190 square units
V = (10)(5)(3) = 150 cubic units
The ratio of surface area to volume is given by:
SA/V = 190/150 = 1.26
So, the ratio of surface area to volume for this rectangular prism is 1.26.
To find the ratio of surface area to volume for a rectangular prism, we need to first calculate the surface area and volume of the prism using the given formulas. Let's choose specific values for the dimensions of our rectangular prism to find the ratio.
Let's say the length (l) of our package is 10 units, the width (w) is 5 units, and the height (h) is 3 units. Plugging these values into the formulas gives us:
Surface area (SA) = 2lw + 2wh + 2lh
= 2(10)(5) + 2(5)(3) + 2(10)(3)
= 100 + 30 + 60
= 190 square units
Volume (V) = lwh
= 10(5)(3)
= 150 cubic units
Now, we can calculate the ratio of surface area to volume by dividing the surface area by the volume:
Ratio = Surface area / Volume
= 190 / 150
= 1.27
Therefore, the ratio of surface area to volume for our rectangular prism package is approximately 1.27.