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 response values are as follows:
(a) At 470 miles per hour, the cost is approximately $189.48 per passenger
At 590 miles per hour, the cost is approximately $191.9 per passenger
(b) The domain of the function C is 0 < x ≤ ∞
(c) Please find attached the required graph of the function C = C(x)
(d) The table with TblStsart = 0 and Tbl = 50 is included in the following solution
(e) The ground speed that minimizes the cost per passenger is 500 miles per hour
The reason the above values are correct is as follows:
The known parameters are:
The length of the Atlantic ocean the airplane crosses = 3,000 miles
The airspeed with which the airplane crosses the Atlantic ocean = 500 mi/hr
The given function that gives the cost per passenger is presented as follows;
[tex]C(x) = \mathbf{50 + \dfrac{x}{7} + \dfrac{34,000}{x}}[/tex]
Where x is the ground speed of the airplane = airspeed ± windspeed
(a) Required:
(i) The cost when the ground speed is 470 miles per hour
Solution:
[tex]The \ cost \ \mathbf{ C(470)} = 50 + \dfrac{470}{7} + \dfrac{34,000}{470} \approx \mathbf{ 189.48}[/tex]
The cost C, when the ground speed is 470 miles per hour is approximately $189.48 per passenger
(ii) The cost when the ground speed is 590 miles per hour
Solution:
[tex]The \ cost \ \mathbf{ C(590)} = 50 + \dfrac{590}{7} + \dfrac{34,000}{590} \approx \mathbf{ 191.9}[/tex]
The cost C, when the ground speed is 590 miles per hour is approximately $191.9 per passenger
(b) Required:
To find the domain of C
Solution:
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, x, is a denominator, we have that the function is not defined (Does not exist) at x = 0
The domain of the function C is 0 < x ≤ ∞
c) Required:
Graph the function using a graphing calculator
Solution:
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 with TblStsart = 0 and Tbl = 50
Solution:
Please find the required TABLE as follows
[tex]\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][/tex]
(e) Required:
To find the required ground speed that gives the minimum cost per passenger
Solution:
By differentiation, we get;
[tex]\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[/tex]
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 500 miles per hour
Please find response summary at the top
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