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.
this sucks i really need help and wish someone could answer rip
Still nothing :(
Task 1:
Sure, let's talk about packaging shapes! How about we package a clown nose in the shape of a rectangular prism? A clown's nose can be quite circular, so we'll need a box that can fit it nicely.
Let's say the length of the clown nose box is 4 inches, the width is 2 inches, and the height is 2 inches. Now, let's find the surface area and volume of this box.
The surface area (SA) of a rectangular prism is given by the formula SA = 2lw + 2wh + 2lh. Plugging in our values, we get SA = 2(4)(2) + 2(2)(2) + 2(4)(2) = 8 + 8 + 16 = 32 square inches.
The volume (V) of a rectangular prism is given by the formula V = lwh. Plugging in our values, we get V = (4)(2)(2) = 16 cubic inches.
Now, let's find the ratio of surface area to volume (SA/V). Plugging in our values, we get SA/V = 32/16 = 2. So, the ratio of surface area to volume for this clown nose box is 2.
Task 2:
a. Let's say John, Rick, and Molli can paint the room together in 4 hours. John can paint the room alone in 6 hours, and Rick can paint the room alone in 8 hours.
b. To find the hourly rate when working together, we can simply take the reciprocal of the time it takes for them to finish the job together. So, the hourly rate for John, Rick, and Molli is 1/4 rooms per hour.
c. For John's hourly rate, we take the reciprocal of the time it takes for him to paint the room alone, which is 1/6 rooms per hour. Similarly, for Rick's hourly rate, we take the reciprocal of the time it takes for him to paint the room alone, which is 1/8 rooms per hour.
d. To compare the group rate to the sum of the individual rates, we can write the equation 1/4 = 1/6 + 1/8. The group rate should be equal to the sum of the rates of John and Rick.
e. The least common denominator for 1/6 and 1/8 is 24. So, let's rewrite the equation with the common denominator: 1/4 = 4/24 + 3/24.
f. Solving the equation, we get 1/4 = 7/24. Multiply both sides by 24 to isolate Molli's rate, and we get Molli's hourly rate as 6/24 or 1/4 rooms per hour. So, it will also take Molli 4 hours to paint the room alone.
Task 1:
To find the ratio of surface area to volume for a rectangular prism, we need to write an expression for it.
The surface area of a rectangular prism is given by the formula SA = 2lw + 2wh + 2lh, and the volume is given by the formula V = lwh.
So, the ratio of surface area to volume can be expressed as SA/V = (2lw + 2wh + 2lh)/(lwh).
Now, choose appropriate values for the length (l), width (w), and height (h) of your package. Let's say l = 10 cm, w = 5 cm, and h = 3 cm.
Substituting these values into the expression for the ratio, we get SA/V = (2 * 10 * 5 + 2 * 5 * 3 + 2 * 10 * 3)/(10 * 5 * 3).
Simplifying the expression gives us SA/V = 130/150.
So, the ratio of surface area to volume for the rectangular prism with the given dimensions is 130/150.
Task 2:
a. Pick a reasonable amount of time for the three friends to paint the room together. Let's say it takes them 2 hours to complete the job. Also, pick reasonable amounts of time for John and Rick to paint the room alone. Let's say John takes 4 hours and Rick takes 3 hours.
b. To find the hourly rate for John, Rick, and Molli when they work together, we divide the number of rooms painted by the total time taken. Let's say they paint 2 rooms in 2 hours, so their hourly rate is 2/2 = 1 room per hour.
c. The hourly rate for John can be calculated by dividing the number of rooms he can paint alone by the time it takes him. Since John takes 4 hours to complete the job and he can paint 1 room, his hourly rate is 1/4 rooms per hour. Similarly, if Rick takes 3 hours to paint 1 room, his hourly rate is 1/3 rooms per hour.
d. To compare the group rate to the sum of the individual rates, we can write an equation. Let's call the group rate GR, John's rate JR, and Rick's rate RR. The equation is: GR = JR + RR.
e. The least common denominator for the equation in part (d) is the product of the denominators of the individual rates. In this case, it is 4 * 3 = 12.
f. To solve the equation, we substitute the individual rates into the equation: 1/2 = 1/4 + 1/3. Multiplying both sides of the equation by 12, we get 6 = 3 + 4. Therefore, Molli's hourly rate is 1/6 rooms per hour.
So, it will take Molli 6 hours to paint the room alone.
my lord nothing-
2020 sucks BrUggggGGGggg