Question

Once​ Kate's kite reaches a height of 50 ft ​(above her​ hands), it rises no higher but drifts due east in a wind blowing 7 ft divided by s. How fast is the string running through​ Kate's hands at the moment that she has released 107 ft of​ string?

Answer 1

Answer:

dz  = 7.136 (answer)

Explanation:

given height  for kate's kite   = 50 ft  (say y)

due to drift it move towards east =  dx = 7 ft  

string maximum length          = 107 ft ( say z)

we have to find change in z  

that is  dz

it will form a right angle triangle for x , y and z  where x is base y is height and z is hypotenuse

so we get according to Pythagoras Theorem

$z^2 = x^2+y^2$...............(i)

by derivative both side  consider y as constant

$2zdz = 2xdx\\dz = {xdx }/z$

from (i)  equation

$x = \sqrt{z^2-y^2}$

$dz = \sqrt{z^2-y^2}dx/z$

now put the value and find dz

$dz = (\sqrt{120^2-50^2)}7/107\\ =((\sqrt{14400 -2500} )\times7)/107$

after solving these  we get

$dz =\frac{109.08\times7}{107}$  

dz  = 7.136 (answer)