7. Chloe and Ben are in motorboats at the center of a lake. At time t=0, Chloe b

Question

7. Chloe and Ben are in motorboats at the center of a lake. At time t=0, Chloe begins traveling south at a speed of 60km/h. One minute later, Ben takes off,heading east at a speed of 40km/h. At what rate is the distance between them increasing at t=12min? Draw a picture and label variables. Show the equations and the work used to find your answer.
8. For the function y=x^3-2x^2+3, find:
Local Max
Local Min
Increasing interval(s)
Decreasing interval(s)
Inflection point(s)
Concave up
Concave Down

SHOW ALL WORK

Explanation / Answer

7)

draw a picture and you get a right triangle. label sides x,y, z where:
x=chloes distance from origin
y=baos distance from origin
z=distance betw them

x^2 + y^2 = z^2

at t=12min we get:
x=60(12/60) = 12 km
y=40(11/60) = 22/3 km
z= sqrt(12^2+(22/3)^2) = sqrt(1780/9)

speeds are given by:
dx/dt = 60
dy/dt=40
dz/dt is what we need to find

x^2 + y^2 = z^2
differentiate wrt time:
2x(dx/dt)+2y(dy/dt)=2z(dz/dt)
x(dx/dt)+y(dy/dt)=z(dz/dt)
12*60 + (22/3)(40) = ((sqrt1780)/3)(dz/dt)
dz/dt = approx 72.05 km/hr

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