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noname [10]
3 years ago
7

Gabe swam 2 hours each day, 5 days each week, for 5 weeks. In that time,gabe swam 1,500 laps. What was his average rate in laps

per hour?
Mathematics
1 answer:
Setler [38]3 years ago
7 0

Answer:

30 laps per hour

Step-by-step explanation:

The first thing you do is calculate the amount of hours he swam for. Since he swam 2 hours each day, 5 days each week, for 5 weeks we use the equation 2*5*5. The answer is 50. After that what we do is take the amount of laps and divide it by 50. We do 1500 divided by 50 and we get a total of 30 laps per hour.

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The area of a rectangle is 42 square millimeters. The length is 7 millimeters. What is the perimeter of the rectangle?
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3 years ago
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Use the change-of-basis identity,

\log_x(y) = \dfrac{\ln(y)}{\ln(x)}

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Use the product-to-sum identity,

\log_x(yz) = \log_x(y) + \log_x(z)

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xyz = \dfrac{(\ln(b) + \ln(c)) (\ln(a) + \ln(c)) (\ln(a) + \ln(b))}{\ln(a) \ln(b) \ln(c)}

Redistribute the factors on the left side as

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and simplify to

xyz = \left(1 + \dfrac{\ln(c)}{\ln(b)}\right) \left(1 + \dfrac{\ln(a)}{\ln(c)}\right) \left(1 + \dfrac{\ln(b)}{\ln(a)}\right)

Now expand the right side:

xyz = 1 + \dfrac{\ln(c)}{\ln(b)} + \dfrac{\ln(a)}{\ln(c)} + \dfrac{\ln(b)}{\ln(a)} \\\\ ~~~~~~~~~~~~+ \dfrac{\ln(c)\ln(a)}{\ln(b)\ln(c)} + \dfrac{\ln(c)\ln(b)}{\ln(b)\ln(a)} + \dfrac{\ln(a)\ln(b)}{\ln(c)\ln(a)} \\\\ ~~~~~~~~~~~~ + \dfrac{\ln(c)\ln(a)\ln(b)}{\ln(b)\ln(c)\ln(a)}

Simplify and rewrite using the logarithm properties mentioned earlier.

xyz = 1 + \dfrac{\ln(c)}{\ln(b)} + \dfrac{\ln(a)}{\ln(c)} + \dfrac{\ln(b)}{\ln(a)} + \dfrac{\ln(a)}{\ln(b)} + \dfrac{\ln(c)}{\ln(a)} + \dfrac{\ln(b)}{\ln(c)} + 1

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xyz = 2 + \dfrac{\ln(ac)}{\ln(b)} + \dfrac{\ln(ab)}{\ln(c)} + \dfrac{\ln(bc)}{\ln(a)}

xyz = 2 + \log_b(ac) + \log_c(ab) + \log_a(bc)

\implies \boxed{xyz = x + y + z + 2}

(C)

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