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The function s(x) = 3600/60+x=3600(60+x)^-1 gives a person's average speed in miles per hour if he or she travels on mile in x seconds more or less than 60 seconds. Use a linear approximation to s at 0 to find a person's approximate average speed if he or she travels one mile in 69 seconds. What is his or her exact speed? The person's approximate average speed is mi/hr.

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Raymond

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Step 1/2

To find the approximate average speed using a linear approximation, we can use the formula for linear approximation:

`L(x) = f(a) + f'(a)(x - a)`

where:

- `f(x)` is the original function,- `a` is the point at which we are making the approximation,- ` f'(a)` is the derivative of the function at `a` , and- `L(x)` is the linear approximation.In this case, the function is `s(x) = \frac{3600}{60 + x} ` , and need to approximate the speed when `\( x = 9 )` (since the person travels one mile in `69` seconds, which is `9` seconds more than `60` seconds).The derivative `s'(x) ` :

`s(x) = \frac{3600}{60 + x}`

`s(x) = 3600(60+x)^-1`

`s'(x) = 3600d/dx(60+x)^-1`

`s'(x) = (-3600)/(60+x)^2`

Explanation:

The formula used in this step:

Explanation:

`d/dx(ky)=kdy/dx`

Explanation:

`d/dx(x^n)=nx^(n-1)`

Explanation:

`(dy)/dx=dy/(du)(du)/dx`
So, `s'(x)=(-3600)/(60+x)^2`

Step 2/2

`s'(x) ` at `x = 0` (the point of approximation):

` s'(0) = -\frac{3600}{60^2}`

` s'(0) = -1`

The linear approximation formula:

`L(x) = s(0) + s'(0)(x - 0)`

` = \frac{3600}{60 +0}+(-1)x`

`L(x) = 60-x`

The linear approximation at `x = 9` :

` L(9) \approx s(0) + s'(0)(9 - 0)`

` = s(0) - 9 `

` = 60-9`

` L(9) \approx 51`

So, the approximate average speed when a person travels one mile in `69` seconds is `51` mph.

Explanation:

To find the approximate average speed using a linear approximation, we can use the formula for linear approximation: `L(x) = f(a) + f'(a)(x - a)`
Now, to find the exact speed when `x = 9 ` :

`s(9) = \frac{3600}{60 + 9} `

` = \frac{3600}{69}`

` = 1200/23`

`s(9) \approx 52.17391`

The person's exact speed is approximately `( 52.174) ` mph.

Final Answer

The approximate average speed when a person travels one mile in `69` seconds is `51` mph.The person's exact speed is approximately `( 52.174) ` mph.

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