Find the point p on the line yequals4x that is closest to the point (51 comma 0 ). what is the least distance between p and (51 comma 0 )? let d be the distance between the two points. what is the objective function in terms of one number, x?

Step-by-step explanation:

Use the distance formula D = √[ (x - x0)² + (y - y0)² ], using (x0, y0):  (51, 0).  Replace 'y' with '4x':

D = √[ (x - 51)² + (4x - 0)² ], or (after simplification)

D = √[ x² - 102x + 2601) + 16x² ], or  D = √[ 17x² - 102x + 2601 ].

This is the objective function in terms of one variable (x).

You don't specify the method to be used.  The main method choices you have are (1) calculus and (2) algebra.  

We want the x value at which this distance D is a minimum.  We can find that x value by finding the minimum of 17x² - 102x + 2601), whose graph is that of a parabola with minimum at x = -b/(2a).  Here, that x is x = 102/(34), or x = 3.  

We conclude that the distance between the given point and the given line is a minimum when x = 3.

That distance is D = √[17x² - 102x + 2601], evaluated at x = 3:

√[ 17(9) - 102(3) + 2601 ] = 49.5 approximately

i leave the calculator work to you.

7. diameter 4 miles, circumference 4π miles

8. radius 0.2 miles, circumference 0.4π miles

[this is a bad problem because the actual radius of a car on the track is less than 0.2 miles so the answer is less than 0.4π miles]

9. radius 6 feet, circumference 12π feet

10. diameter 14 cm, circumference 14π cm

11. diameter 4.8 feet, circumference 4.8π feet

4.6875 pounds

step-by-step explanation: