Long Test 3 1. Compute the directional derivative and gradient of at, yz) = 41' 9-263\" at [2. 1.0) in the direction of the unit vector {If parallel to {1.43}. (8 points) 2. Suppose you are running a factoryr producing some sort of widget that requires steel as a raw material. Your costs are predominantly human labor. which is $20 per hour for your workers. and the steel itself. which runs for $2. DOD per ton; Suppose your revenue R is loosely modelled by the following equation: BIZ-'15] = lilil'frg'mriil'll3 where it represents hour of labor and 3 represents tons of steel, If your budget is $20,000, what is the maximum possible revenue? (10 points) 3. Assuming that z is a differentiable function of .1: and y. use implicit differentiation to find 3; from the equation lnilrgy) - e!\" = tan\" 2. (5 points) 4. A rectangular box with square base is 10 cm tall and ,1 cm wide. If its height is increased by 0.5 cm. estimate the change in its width so that its volume stays the same. (8 points) 5. A steel plate is in the shape of a right triangle. Its height is decreasing at the rate of 0.5 emf min and its base is increasing at the rate of LI] cm ;' min. Find the rate of change of the area of the plate at an instant when its height is 10 cm and its base is 5 cm. (8 points)