A function is a relationship that maps each member of a set of input values into only one member of the set of output values
The correct values for the costs per passenger are as follows:
(a) At 470 miles per hour, the cost is approximately <u>$189.48 per passenger</u>
At 590 miles per hour, the cost is approximately <u>$191.9 per passenger</u>
(b) The <em>domain </em>of the function <em>C </em> is <u>0 < x ≤ ∞</u>
(c) Please find attached the required graph
(d) The table is included in the solution
(e) The <em>ground speed</em> that minimizes the cost per passenger is <u>500 miles per hour</u>
The reason the above values are correct is as follows:
The known parameters are;
The <em>length </em>of the Atlantic ocean the airplane crosses<em> = 3,000 miles</em>
The airspeed with which the airplane crosses the Atlantic ocean <em>= 500 mi/hr</em>
The given function that gives the cost per passenger is presented as follows;
![C(x) = \mathbf{50 + \dfrac{x}{7} + \dfrac{34,000}{x}}](https://tex.z-dn.net/?f=C%28x%29%20%3D%20%5Cmathbf%7B50%20%2B%20%5Cdfrac%7Bx%7D%7B7%7D%20%2B%20%5Cdfrac%7B34%2C000%7D%7Bx%7D%7D)
Where x is the ground speed of the airplane = airspeed ± windspeed
(a) Required:
(i) The cost when the ground speed is <em>470 miles per hour</em>
Solution:
![The \ cost \ \mathbf{ C(470)} = 50 + \dfrac{470}{7} + \dfrac{34,000}{470} \approx \mathbf{ 189.48}](https://tex.z-dn.net/?f=The%20%5C%20cost%20%5C%20%5Cmathbf%7B%20C%28470%29%7D%20%3D%2050%20%2B%20%5Cdfrac%7B470%7D%7B7%7D%20%2B%20%5Cdfrac%7B34%2C000%7D%7B470%7D%20%5Capprox%20%5Cmathbf%7B%20189.48%7D)
The cost C, when the ground speed is <em>470 miles per hour</em> is approximately <u>$189.48 per passenger</u>
(ii) The cost when the ground speed is <em>590 miles per hour</em>
Solution:
![The \ cost \ \mathbf{ C(590)} = 50 + \dfrac{590}{7} + \dfrac{34,000}{590} \approx \mathbf{ 191.9}](https://tex.z-dn.net/?f=The%20%5C%20cost%20%5C%20%5Cmathbf%7B%20C%28590%29%7D%20%3D%2050%20%2B%20%5Cdfrac%7B590%7D%7B7%7D%20%2B%20%5Cdfrac%7B34%2C000%7D%7B590%7D%20%5Capprox%20%5Cmathbf%7B%20191.9%7D)
The cost C, when the ground speed is <em>590 miles per hour</em> is approximately <u>$191.9 per passenger</u>
(b) Required:
To find the domain of <em>C</em>
The domain of a function is given by the values of the function for which the function is defined, or possible, or for which there is an output
Given that the independent variable, <em>x</em>, is a denominator, we have that the function is not defined (<em>Does not exist</em>) at <em>x = 0</em>
<em />
The domain of the function <em>C </em> is 0 < x ≤ ∞
c) Required:
Graph the function using a graphing calculator
Please find attached the required graph of the function created with MS Excel
(d) Required:
(i) To create a table of values for the groundspeed
Please find the required TABLE as follows
![\begin{array}{|c|cc|}Airspeed&&Cost \ C\\0&&Does \ Not \ Exist\\50&&737.14\\100&&404.29\\150&&298.1\\200&&248.57\\250&&221.71\\300&&206.19\\350&&197.14\\400&&192.14\\450&&189.84\\500&&189.43\\550&&190.39\\600&&192.38\\650&&195.16\end{array}\right]](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7B%7Cc%7Ccc%7C%7DAirspeed%26%26Cost%20%5C%20C%5C%5C0%26%26Does%20%5C%20Not%20%5C%20Exist%5C%5C50%26%26737.14%5C%5C100%26%26404.29%5C%5C150%26%26298.1%5C%5C200%26%26248.57%5C%5C250%26%26221.71%5C%5C300%26%26206.19%5C%5C350%26%26197.14%5C%5C400%26%26192.14%5C%5C450%26%26189.84%5C%5C500%26%26189.43%5C%5C550%26%26190.39%5C%5C600%26%26192.38%5C%5C650%26%26195.16%5Cend%7Barray%7D%5Cright%5D)
(e) Required:
<em>To find the required ground speed</em> that gives the minimum cost per passenger
Solution:
By differentiation, we get;
![\dfrac{d\left( 50 + \dfrac{x}{7} + \dfrac{34,000}{x}\right)}{dx} = \dfrac{7 \cdot x(2 \cdot x+350)-7 \cdot \left(x^2+350 \cdot x +238000\right)}{(7 \cdot x)^2} = 0](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%5Cleft%28%2050%20%2B%20%5Cdfrac%7Bx%7D%7B7%7D%20%2B%20%5Cdfrac%7B34%2C000%7D%7Bx%7D%5Cright%29%7D%7Bdx%7D%20%20%3D%20%5Cdfrac%7B7%20%5Ccdot%20x%282%20%5Ccdot%20x%2B350%29-7%20%5Ccdot%20%5Cleft%28x%5E2%2B350%20%5Ccdot%20x%20%2B238000%5Cright%29%7D%7B%287%20%5Ccdot%20x%29%5E2%7D%20%20%3D%200)
Which gives;
7·x² - 1666000 = 0
7·x² = 1666000
x = √(1666000/7) ≈ 487.85
Therefore, to the nearest 50 miles per hour, the ground speed that minimizes the cost per passenger is <u>500 miles per hour</u>
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