Section 4.7: Do the following problems.

a) It is given that the sum of two numbers is 30. Find the maximum product that can be reached by these two numbers. Can there be a minimum product for the two numbers? Explain your answer.

b) Manisha has two positive numbers, and the product of them is 81. What is the minimum sum that she can obtain by these numbers? Explain why there can not be a largest sum.

c) Ryan has 1200 ft of fencing material to build a rectangular corral. One side of the corral will be formed by a barn and requires no fence. Three exterior fences and three interior fences partition the corral into four rectangular regions. What are the dimensions of the corral that maximize the enclosed area? What is the area of that corral?

d) Find the area of the largest rectangle that can be inscribed in an isosceles triangle with sides 18 cm, 18 cm and 10 cm.

e) An open-top box is made by cutting similar square shape pieces from each corner of a rectangular piece of paper of size 16 10 and folding the remain piece of paper. Find the side length of the square piece that can be cut-off to maximize the volume of the box.

f) An athlete has an option to either run along a beach with a speed of 120 m/min or swim with a speed of 30 m/min. If he sees an object 100 m off the beach and 200 m ahead to the nearest point of the beach, how far does he need to run before he swims to retrieve the object quickest?

g) A bob attached to a spring undergoes in a damp vibrating motion given by = 2 4 sin(4) + 1. Find the highest and lowest points it can reach with a justification. Also, provide the desmos graph of the position of the bob indicating the corresponding values of for the highest and the lowest point along with your answer.

h)* (Extra Credit) A frog is aiming at a bee that flies on a parabolic path given by = 2 + 1. Assuming the frog is at the (5, 0) position, find the shortest distance between them.