Lois C. answered • 04/01/20
BA in secondary math ed with 20+ years of classroom experience
This is a related rate problem in which the key equation is the Pythagorean Theorem and where x is fixed at 3 feet ( 96 inches) for the distance from the bottom of the ladder to the base of the wall, y represents the height on the wall at the point where the ladder makes contact with the wall, and h is the length of the ladder.
We must first identify the known and unknown rates for all quantities involved. Measurements must be consistent in the unit of measure used, so we should convert everything either to feet or to inches. I prefer inches to avoid working with fractions in the key equation. For x, there is no rate of change as the distance from the bottom of the ladder to the base of the wall does not change throughout the problem. Regarding y, dy/dt is the rate we seek since we want to know how fast the height along the wall where the ladder makes contact with the wall is changing as the ladder is extended. Regarding h, dh/dt is given to us as 1 in/second.
We now set up the key equation based on the Pythagorean Theorem, were x 2 + y2 = h2. We then take the derivative of each side of the equation. Since the x quantity is fixed, its derivative is 0. So the new equation becomes 0 + 2y dy/dt = 2h dh/dt ( we must use the Chain Rule to take each derivative with respect to time). At the moment in question, we need the y value. By the Pythagorean Theorem, x2 + y2 = h2 becomes 362 + y2 = 962. Solving for y, we see that y is approximately equal to 88.9944. Inserting this into the derivative version of the equation, we have 2(88.9944) dy/dt = 2(96)(1). Solving this equation for dy/dt, we see that the rate of change of y at the moment in question is approximately 1.08 in/sec.
