Question

8
a rectangular wall is \\(\sqrt{240}\\) ft by \\(\sqrt{50}\\) ft. you need to paint the wall twice to cover the area with two coats of paint. if each can of paint can cover 60 square feet, how many cans of paint will you need?
\boxed{} cans

Explanation:

Step1: Calculate the area of the wall

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \). Here, length \( l=\sqrt{240} \) ft and width \( w = \sqrt{50} \) ft. So, \( A=\sqrt{240}\times\sqrt{50} \). Using the property \( \sqrt{a}\times\sqrt{b}=\sqrt{ab} \), we get \( A = \sqrt{240\times50}=\sqrt{12000} \). Simplify \( \sqrt{12000} \): \( \sqrt{12000}=\sqrt{100\times120} = 10\sqrt{120}=10\sqrt{4\times30}=10\times2\sqrt{30}=20\sqrt{30}\approx20\times5.477 = 109.54 \) square feet. But we need to paint it twice, so total area to paint is \( 2\times109.54 = 219.08 \) square feet. (Alternatively, we can do it symbolically first: twice the area is \( 2\times\sqrt{240}\times\sqrt{50}=2\times\sqrt{12000}=2\times\sqrt{100\times120}=2\times10\times\sqrt{120}=20\times\sqrt{4\times30}=20\times2\sqrt{30}=40\sqrt{30}\approx40\times5.477 = 219.08 \) square feet.)

Step2: Calculate the number of paint cans

Each can covers 60 square feet. Let \( n \) be the number of cans. Then \( n=\frac{\text{Total Area}}{\text{Area per can}}=\frac{40\sqrt{30}}{60}=\frac{2\sqrt{30}}{3}\approx\frac{2\times5.477}{3}=\frac{10.954}{3}\approx3.65 \). Since we can't buy a fraction of a can, we need to round up to the next whole number, so \( n = 4 \). Wait, wait, maybe I made a mistake in calculation. Wait, let's recalculate the area correctly. Wait, \( \sqrt{240}\times\sqrt{50}=\sqrt{240\times50}=\sqrt{12000} \). \( 12000 = 100\times120 = 100\times4\times30 = 400\times30 \), so \( \sqrt{12000}=\sqrt{400\times30}=20\sqrt{30}\approx20\times5.477225575 = 109.5445115 \). Then twice that is \( 2\times109.5445115 = 219.089023 \). Now divide by 60: \( 219.089023\div60\approx3.651 \). Since we can't have a fraction of a can, we need to round up to 4? Wait, but maybe my initial simplification was wrong. Wait, let's do it again: \( \sqrt{240} = \sqrt{16\times15}=4\sqrt{15} \), \( \sqrt{50}=\sqrt{25\times2}=5\sqrt{2} \). Then area is \( 4\sqrt{15}\times5\sqrt{2}=20\sqrt{30} \), same as before. Twice the area is \( 40\sqrt{30}\approx40\times5.477 = 219.08 \). Divide by 60: \( 219.08\div60\approx3.65 \). So we need 4 cans? Wait, but maybe I miscalculated the area. Wait, let's check \( 240\times50 = 12000 \), correct. \( \sqrt{12000} = \sqrt{100\times120} = 10\sqrt{120} \), \( \sqrt{120}=2\sqrt{30} \), so \( 10\times2\sqrt{30}=20\sqrt{30} \), correct. Twice is \( 40\sqrt{30}\approx219.08 \). 219.08 divided by 60 is approximately 3.65, so we need 4 cans because 3 cans would cover \( 3\times60 = 180 \) which is less than 219.08, and 4 cans cover \( 4\times60 = 240 \) which is more than 219.08.

Wait, but maybe there's a simpler way. Let's compute \( \sqrt{240}\times\sqrt{50} \) first: \( \sqrt{240} = \sqrt{4\times60}=2\sqrt{60} \), \( \sqrt{50}=\sqrt{25\times2}=5\sqrt{2} \), no, better to use \( \sqrt{a}\times\sqrt{b}=\sqrt{ab} \). So \( \sqrt{240}\times\sqrt{50}=\sqrt{240\times50}=\sqrt{12000} \). \( 12000 = 100\times120 = 100\times4\times30 = 400\times30 \), so \( \sqrt{12000}=20\sqrt{30}\approx109.54 \). Twice that is \( 219.08 \). Divide by 60: \( 219.08\div60\approx3.65 \), so we need 4 cans.

Wait, but maybe I made a mistake in the problem interpretation. Let's re-read: "A rectangular wall is \( \sqrt{240} \) ft by \( \sqrt{50} \) ft. You need to paint the wall twice to cover the area with two coats of paint. If each can of paint can cover 60 square feet, how many cans of paint will you need?"

So area of wall: \( \sqrt{240} \times \sqrt{50} = \sqrt{240 \times 50} = \sqrt{12000} \). Simplify \( \sqrt{12000} \): \( 12000 = 100 \times 120 = 100 \times 4 \times 30 = 400 \times 30 \), so \( \sqrt{12000} = 20\sqrt{30} \approx 20 \times 5.4772 = 109.544 \) sq ft. Two coats: \( 2 \times 109.544 = 219.088 \) sq ft. Number of cans: \( 219.088 \div 60 \approx 3.65 \). Since we can't buy a fraction of a can, we round up to 4. So the number of cans needed is 4.

Answer:

4