Answer:
73.77 ft/s
Step-by-step explanation:
Let's imagine this situation as a right triangle where the distance between the two points is the hypotenuse.
-Applying Pythagoras theorem:
[tex]z^2=x^2+y^2\\\\z=\sqrt{x^2+y^2}\\\\x=dx*t+x_i, x_i=0\\\\y=dy*t+y_i\\\\y_i=160\ ft[/tex]
#Take the derivative of the first equation and solve for [tex]dz[/tex]:
[tex]d(z^2=x^2+y^2)\\\\2z*dz=2x*dx+2y*dy\\\\dz=\frac{x*dx+y*dy}{z}\\\\dz=\frac{x*dx+y*dy}{\sqrt{x^2+y^2}}\\\\\\dz=\frac{dx^2*t+dy(dy*t+160)}{\sqrt{(dx*t)^2+(dy*t)}}[/tex]
#We then substitute the values given in the question to solve for [tex]dz[/tex]:
[tex]dz=\frac{7*66^2+14(14*7+160)}{\sqrt{(66*7)^2+7*14+160}}\\\\=73.78[/tex]
Hence, the rate of change of the distance between the motorcycle and the balloon 7 seconds later is 73.77 ft/s
