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lawyer [7]
3 years ago
13

A dog won a race at the local fair by running 3 and one fourth miles in exactly 2 hours. At this constant​ rate, how long does i

t take the same dog to run the 1 and three tenths ​-mile state fair​ race? Use ratio reasoning to solve.
Mathematics
1 answer:
Novosadov [1.4K]3 years ago
7 0

Answer:

\frac{4}{5} hour

Step-by-step explanation:

Let x represent time taken by dog to run the 1 and three tenths ​-mile state fair​ race.

We have been given that a dog won a race at the local fair by running 3 and one fourth miles in exactly 2 hours.

We will use proportions to solve our given problem as:

\text{Speed}=\frac{\text{Distance}}{\text{Time}}

We will equate both speeds as:

\frac{1\frac{3}{10}}{x}=\frac{3\frac{1}{4}}{2}

\frac{\frac{13}{10}}{x}=\frac{\frac{13}{4}}{2}

\frac{13}{10\cdot x}=\frac{13}{4\cdot 2}

Cross multiply:

13\cdot 10\cdot x=13\cdot 4\cdot 2

10\cdot x=4\cdot 2

\frac{10\cdot x}{10}=\frac{4\cdot 2}{10}

Therefore, it will take \frac{4}{5} hour to complete 1\frac{3}{10} mile state fair​ race.

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The function which is same as the function y = 3cos(2(x +π/2)) -2 is: Option A: y= 3sin(2(x + π/4)) - 2

<h3>How to convert sine of an angle to some angle of cosine?</h3>

We can use the fact that:

\sin(\theta) = \cos(\pi/2 - \theta)\\\sin(\theta + \pi/2) = -\cos(\theta)\\\cos(\theta + \pi/2) = \sin(\theta)

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<h3>Which trigonometric functions are positive in which quadrant?</h3>
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(this all positive negative refers to the fact that if you use given angle as input to these functions, then what sign will these functions will evaluate based on in which quadrant does the given angle lies.)

Here, the given function is:

y= 3\cos(2(x + \pi/2)) - 2

The options are:

  1. y= 3\sin(2(x + \pi/4)) - 2
  2. y= -3\sin(2(x + \pi/4)) - 2
  3. y= 3\cos(2(x + \pi/4)) - 2
  4. y= -3\cos(2(x + \pi/2)) - 2

Checking all the options one by one:

  • Option 1: y= 3\sin(2(x + \pi/4)) - 2

y= 3\sin(2(x + \pi/4)) - 2\\y= 3\sin (2x + \pi/2) -2\\y = -3\cos(2x) -2\\y = 3\cos(2x + \pi) -2\\y = 3\cos(2(x+ \pi/2)) -2

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Thus, this option is same as the given function.

  • Option 2: y= -3\sin(2(x + \pi/4)) - 2

This option if would be true, then from option 1 and this option, we'd get:
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Thus, this option is not same as the given function.

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This option's function simplifies as:

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The given function simplifies to:y= 3\cos(2(x + \pi/2)) - 2 = 3\cos(2x + \pi) -2 = -3\cos(2x) -2

The given option simplifies to:

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Thus, this function is not same as the given function.

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