You are designing a high-rise apartment building. Your architects and engineers

Question

You are designing a high-rise apartment building. Your architects and engineers have already designed three possible plans for each floor. Plan A has 6 one-bedroom apartments, 3 two- bedroom apartments, and 4 three-bedroom apartments on a single floor. Plan B has 4 one-bedroom, 5 two-bedroom, and 3 three-bedroom apartments on a floor. Plan C has 8 one-bedroom, 3 two-bedroom, and 3 three-bedroom apartments on a floor. (a) Is it possible to design the building with exactly 80 one-bedroom, 40 two-bedroom, and 30 three-bedroom apartments? If so, is there more than one way to do this? Explain your answer. (b) Is it possible to design the building with exactly 80 one-bedroom, 40 two-bedroom, and 40 three-bedroom apartments? If so, is there more than one way to do this? Explain your answer. You are designing a high-rise apartment building. Your architects and engineers have already designed three possible plans for each floor. Plan A has 6 one-bedroom apartments, 3 two- bedroom apartments, and 4 three-bedroom apartments on a single floor. Plan B has 4 one-bedroom, 5 two-bedroom, and 3 three-bedroom apartments on a floor. Plan C has 8 one-bedroom, 3 two-bedroom, and 3 three-bedroom apartments on a floor. (a) Is it possible to design the building with exactly 80 one-bedroom, 40 two-bedroom, and 30 three-bedroom apartments? If so, is there more than one way to do this? Explain your answer. (b) Is it possible to design the building with exactly 80 one-bedroom, 40 two-bedroom, and 40 three-bedroom apartments? If so, is there more than one way to do this? Explain your answer. You are designing a high-rise apartment building. Your architects and engineers have already designed three possible plans for each floor. Plan A has 6 one-bedroom apartments, 3 two- bedroom apartments, and 4 three-bedroom apartments on a single floor. Plan B has 4 one-bedroom, 5 two-bedroom, and 3 three-bedroom apartments on a floor. Plan C has 8 one-bedroom, 3 two-bedroom, and 3 three-bedroom apartments on a floor. (a) Is it possible to design the building with exactly 80 one-bedroom, 40 two-bedroom, and 30 three-bedroom apartments? If so, is there more than one way to do this? Explain your answer. (b) Is it possible to design the building with exactly 80 one-bedroom, 40 two-bedroom, and 40 three-bedroom apartments? If so, is there more than one way to do this? Explain your answer. You are designing a high-rise apartment building. Your architects and engineers have already designed three possible plans for each floor. Plan A has 6 one-bedroom apartments, 3 two- bedroom apartments, and 4 three-bedroom apartments on a single floor. Plan B has 4 one-bedroom, 5 two-bedroom, and 3 three-bedroom apartments on a floor. Plan C has 8 one-bedroom, 3 two-bedroom, and 3 three-bedroom apartments on a floor. (a) Is it possible to design the building with exactly 80 one-bedroom, 40 two-bedroom, and 30 three-bedroom apartments? If so, is there more than one way to do this? Explain your answer. (b) Is it possible to design the building with exactly 80 one-bedroom, 40 two-bedroom, and 40 three-bedroom apartments? If so, is there more than one way to do this? Explain your answer.

Explanation / Answer

Possible Designs:

A: 6,3,4

B: 4,5,3

C: 8,3,3

a) 80, 40 , 30

Let there be A number of A floors , B number of B floors and C number of C floors. Then

6A+4B+8C = 80

3A+5B+3C = 40

4A+ 3B + 3C = 30

Solve these 3 equations in 3 variables:

Subtract equation 3 from equation 2 to get:

2B-A = 10

A = 2B- 10

Now subtract 2 times equation 2 from equation 1 to get:

-6B+2C =0

Therefore, C = 3B

Put value of A and C in terms of B in equation 2 to get:

3( 2B-10) + 5B + 3( 3B) = 40

6B - 30 + 5B + 9B = 40

20B = 70

B = 7/2 = 3.5

A = 2B- 10 = -3

C = 3B = 10.5

Which is not possible as A,B,C should be integers

Therefore such a building is not possible

b)Now Lets do this similar to Part a)

Here the plan required for the building is: 80, 40 , 40

Let there be A number of A floors , B number of B floors and C number of C floors. Then

6A+4B+8C = 80

3A+5B+3C = 40

4A+ 3B + 3C = 40

Solve these 3 equations in 3 variables:

Subtract equation 3 from equation 2 to get:

2B-A = 0

A = 2B

Now subtract 2 times equation 2 from equation 1 to get:

-6B+2C =0

Therefore, C = 3B

Put value of A and C in terms of B in equation 2 to get:

3( 2B) + 5B + 3( 3B) = 40

6B + 5B + 9B = 40

20B = 40

B = 40/20 = 2

A = 2B= 4

C = 3B = 6

Therefore such a building is possible with 2 floors with plan B , 4 floors with plan A and 6 floors with plan C. Only one possible combination is there.

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