Thread: differential calculus rate of change problems

1.) Johann is 6 ft. tall and he walks at a rate of 5 ft/sec toward a street light that is 16 ft above the ground. At what rate is the tip of his shadow moving? At what rate is the length of his shadow changing when he is 10 ft from the base of the light?

2.) A light is at the top of a pole 50 ft high. A ball is dropped from the same height from a point 30 ft away from the light. How fast is the shadow of the ball moving along the ground 0.5 second later? (assume the ball falls a distance s = 16t^2 ft in t seconds).

Originally Posted by fishlord40

1.) Johann is 6 ft. tall and he walks at a rate of 5 ft/sec toward a street light that is 16 ft above the ground. At what rate is the tip of his shadow moving? At what rate is the length of his shadow changing when he is 10 ft from the base of the light?

2.) A light is at the top of a pole 50 ft high. A ball is dropped from the same height from a point 30 ft away from the light. How fast is the shadow of the ball moving along the ground 0.5 second later? (assume the ball falls a distance s = 16t^2 ft in t seconds).

Have you tried anything at all? It is far better to show what you have done rather than just post the question.

For the first one, draw a picture. You should see that the light pole, the line from the base of the light pole to the end of Johann's shadow, the line from the light to the end of Johann's shadow, and the light pole make a right triangle. Further, Johann himself is a vertical line inside that triangle and so forms another right triangle that is similar to the first. You can use the basic property of similar triangles, that ratios of lengths of corresponding sides are equal, to set up an equation that gives length of the shadow as a function of his distance from the shadow. Then differentiate both sides with respect to time to get an equation connecting their rates of change.

For the second, much the same thing except that now it is the vertical side of the right triangle you are given a rate of change for, rather than the horizontal side.