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: A pack of cereal
Dimensions:
Length (l) = 10 inches
Width (w) = 6 inches
Height (h) = 3 inches
Surface area (SA) = 2lw + 2wh + 2lh
SA = 2(10 x 6) + 2(6 x 3) + 2(10 x 3)
SA = 120 + 36 + 60
SA = 216 square inches
Volume (V) = lwh
V = 10 x 6 x 3
V = 180 cubic inches
Ratio of surface area to volume = SA/V
Ratio = 216/180
Ratio = 1.2
Therefore, the ratio of surface area to volume for the cereal packaging is 1.2.
OMG THANK YOU SOO MUCH
You're welcome! If you have any more questions, feel free to ask.
one more question please
Sure, what's your question?
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. The statement "the number of guests you can invite to your party varies inversely with the price per bowler at the alley" means that as the price per bowler increases, the number of guests you can afford to invite decreases. Similarly, as the price per bowler decreases, the number of guests you can afford to invite increases.
b. Let's say you are willing to spend $200 to host the bowling party.
c. Let's assume that "x" represents the number of guests you can invite to the party, and "y" represents the price per bowler. We can set up an equation that shows the inverse relationship between the number of guests and the price per bowler as follows:
x = k/y
where "k" is a constant. To incorporate the budget, we can substitute "y" with "$200/x" since we are willing to spend $200 for the party. This gives us the equation:
x = k/($200/x)
Simplifying this equation by multiplying both sides by "x" gives us:
x^2 = k/$200
d. Let's say that Bowling Alley A charges $6 per bowler and Bowling Alley B charges $8 per bowler. Using the equation we set up in part (c), we can calculate the number of guests we can invite to the party at each of the bowling alleys.
For Bowling Alley A:
x^2 = k/$200
x^2 = k/$200
x^2 = 200/6
x^2 = 33.33
x ≈ 5.77
Therefore, at Bowling Alley A, we can invite about 5-6 guests.
For Bowling Alley B:
x^2 = k/$200
x^2 = k/$200
x^2 = 200/8
x^2 = 25
x = 5
Therefore, at Bowling Alley B, we can invite 5 guests.
Based on the calculations, we would choose Bowling Alley A as we can invite more guests there. Also, even if we rounded up, the price per guest would be cheaper at Bowling Alley A.