Question 595197: I have a room: L:12 feet 4 inches, W: 10 feet 6 inches, Height: 8 feet 3 inchesConvert to inches and find the surface area.. How many gallons of paint if a gallon covers 350 Sq Feet? Cost of a gallon is $22.95 plus 8% tax, what is the total cost?, Convert to centimeters. Find the volume in cubic centimeters. If the dimensions are doubled, what happens to the volume of the room?
This is like a terrifying never-ending word problem! I am overwhelmed!
Answer by math-vortex(648)   (Show Source):
You can put this solution on YOUR website! Hi, there—
I can see how this problem might seem overwhelming because you actually have at least five problems smashed into this paragraph. Let’s take them one by one…
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[I] Convert (dimensions) to inches and find the surface area.
We use the unit conversion that there are 12 inches in each foot, so we multiply the number of feet by 12 and add the left over inches.
LENGTH: 12 feet 4 inches = 12 x 12 + 4 = 148 in.
WIDTH: 10 feet 6 inches = 10 x 12 + 6 = 126 in.
HEIGHT: 8 feet 3 inches = 8 x 12 + 3 = 99 in.
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Once we have the dimensions in inches, we can figure out the surface area. It might help to stand in a rectangular room to figure this part out. Notice that the surface area of a room includes the area of the ceiling plus each wall. Notice also that the walls opposite each other have the same dimensions (and the same area!) The problem doesn’t mention painting the floor, or anything about windows or doors, so we don’t need to worry about those (o:
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We will use the formula for the area of a rectangle to find the dimensions of each surface and then add them up. The units of our answer will be in square inches (sq. in.)
Area of the Ceiling:
length x width = 148 in. x 126 in. = 18,648 sq. in.
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Area of Wall 1 and of the wall opposite it:
length x height = 148 in. x 99 in. = 14,652 sq. in.
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Height of Wall 2 and the one opposite it:
width x height = 126 in. x 99 in. = 12,474 sq in.
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Now add up the areas of the ceiling and the five walls to get the total surface area:
18,648 + 14,652 + 14,652 + 12,474 + 12,474 = 72,900 sq. in.
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[II] How many gallons of paint if a gallon covers 350 square feet (sq. ft.)?
Since we know the paint coverage in square FEET, we need to convert the surface area of the room to square feet. Use the unit conversion that 1 square foot is equivalent to 144 square inches. We want to know how many square feet are in 72,900 sq. in., so we divide: 72,900 / 144 = 506.25 sq. ft.
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One gallon of paint covers 350 sq. ft. We want to know how made 350 sq. ft. sections there are in 506.25, so we divide: 506.25 / 350 = 1.45 gallons.
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This makes sense when you think about it because 1 gallon of paint covers 350 sq. ft., so 2 gallons would cover twice that area, or 700 sq. ft. Since 506.25 sq. ft. is halfway between 350 and 700 sq. ft., it takes about 1.5 gallons of paint to do the job.
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[III] Find the cost of the paint.
You will need to decide how you want to interpret this problem. Normally, you cannot purchase 1.45 gallons of paint. You would need to purchase two 1-gallon cans and have some left over. (On the other hand, this problem assumes that you are painting a room with no windows or doors.) Let’s go with the two cans of paint approach for now. One can of paint costs $22.95, so two cans cost twice that much: $22.95 x 2 = $44.90
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To calculate the tax, we must first convert 8% to a decimal; that’s 0.08 (8 cents for each dollar you spent.) We want to know the tax on $44.90, so we multiply: 44.90 x 0.08 = $3.59. Now add the tax to the cost of the paint, $44.90 + 3.59 = $48.49.
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[IV] Convert the dimensions to centimeters (cm), and find the volume of the room in cubic centimeters.
We use the unit conversion that there are 2.54 cm in one inch. Go back to part [I] and convert inches to centimeters for the length, width, and height. For the length, every inch is equivalent 2.54 cm. We want to know how many cm in 148 in. so we multiply:
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LENGTH: 148 in. x 2.54 = 375.9 cm
WIDTH: 126 in. x 2.54 = 320.0 cm
HEIGHT: 99 in. x 2.54 = 251.5 cm
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We calculate the volume of the room by using the formula, volume equals length times width times height.
V = (375.9)(320.0)(251.5) = 30,252,432 cubic cm
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[V] If the dimensions are doubled, what happens to the volume of the room?
This might be the most interesting question of the whole paragraph. Let’s think about this for a minute…It turns out that the answer is the same no matter what size room we have! Check this out…
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We found the volume of the room by multiplying length times width times height. Write that as an equation:
V = (L)( W)( H)
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If we double the length of the room, then whatever the length was, now it's 2 times bigger. In algebra, L is now 2 times L, or 2L; same for the width and height: W becomes 2W, and H becomes 2H. The formula for the room with dimensions doubled is
V = (2L)( 2W)( 2H)
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We can simplify by multiplying the twos together to give
V = (8)(L)(W)(H)
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So…the original volume and the new volume differ by a factor of 8 because (8)(L)(W)(H) means 8 times (L)(W)(H). Since we figured this out using the variables, L, W, and H, the dimensions of the room could be any positive number, so we know this is true for any room.
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I hope this helps! Please email me if any of this doesn’t make sense. I’ll be happy to go over it in more detail.
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Mrs.Figgy
math.in.the.vortex@gmail.com
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