Question

Anouk is an engineer planning sound and lighting for a free concert in the park. The concert was advertised with a promise to use no more than108kilowatts(kW) of power. It was determined that the main contributors to power usage, speakers and floodlights, use1.8 kW and2.2kW, respectively. Anouk also must keep within her budget of$3,300. The rental company is charging$75for each speaker and$42for each floodlight. Which of the following combinations meets Anouk's requirements?
A.40 speakers and 30 floodlights
B.12 speakers and 54 floodlights
C.26 speakers and 13 floodlights
D. 38 speakers and 22 floodlights


SHOW WORK!

Answer 1

We start by choosing variables for unknown

So let number of speakers = x

Number of flood lights = y

power used by 1 speaker = 1.8KW

So power used by x speakers will be = 1.8x----------(1)

Power used by 1 flood light = 2.2KW

So power used by y floodlights will be = 2.2y-------------------------(2)

Now expression for total power used will be total of (1) and (2)

so 1.8x + 2.2y which should be "no more than" 108KW

so inequality will be 1.8x + 2.2y ≤ 108 -----------------------------------(3)

Now charge for 1 speaker = $75

so charge for x speakers = 75x-----------------------------(4)

Charge for 1 flood light = $42

So charge for y flood lights = 42y----------------------(5)

So total chrage of speakers and flood lights will be total of (4) and (5) which will be 75x + 42y which should be "within budget of $3,300"

so inequality for this will be

75x + 42y ≤ 3300-----------------------------------------(6)

Now we have to see which choice will satisfy inequalities (3) and (6) made

option (A) 40 speakers and 30 flood lights

So x= 40 and y =30

Plug these value in inequality (3)

1.8x + 2.2y ≤ 108

1.8(40) + 2.2(30) ≤ 108

72 + 66 ≤ 108

138 ≤ 108

which is false so option (A) is not correct

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Option (B) 12 speakers and 54 floodlights

so x = 12 and y = 54

Plug these value in inequality (3)

1.8x + 2.2y ≤ 108

1.8(12) + 2.2(54) ≤ 108

21.6 + 118.8 ≤ 108

140.4 ≤ 108

which is false so option (B) is not correct

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Option (C) 26 speakers and 13 floodlights

x = 26 and y = 13

Plug these value in inequality (3)

1.8x + 2.2y ≤ 108

1.8(26) + 2.2(13) ≤ 108

46.8 + 28.6 ≤ 108

75.4 ≤ 108

which is TRUE

So now lets plug x and y values in inequality (6) also

75x + 42y ≤ 3300

75(26) + 42(13) ≤ 3300

1950 + 546 ≤ 3300

2496 ≤ 3300

which is also TRUE

so option (C) is correct and the right answer

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Option (D) 38 speakers and 22 floodlights

so x = 38 and y = 22

Plug these value in inequality (3)

1.8x + 2.2y ≤ 108

1.8(38) + 2.2(22) ≤ 108

68.4 + 48.4 ≤ 108

116.8 ≤ 108

which is false so option (D) is not correct

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