(a) Express the perimeter in terms of r only. P (r ) = (b) Find P '(r) = (c) Determine the radius of the cross-section which minimizes the perimeter. ft 4. [-/3.5 Points] DETAILS Find the point on the parabola shown below that is closest to the point (2, 0). (a) Express the square of the distance between the parabola and the point (2, 0) in terms of x only. Note: Minimizing the square of the distance can be used in this case to avoid working with square roots. S (x) = (b) Find the coordinates of the point on the parabola to two decimal places. 5. [-/4 Points] DETAILS A wire of length 110 inches will be used to make the frame (edges) of a box with a square bottom. (a) Find a formula for the volume of the box in terms of the width only. Use lower case w. V(W) = cubic inches (b) Find V'(w) = (c) Find the non-zero critical point of V(w). W = inches (d) Does your critical point minimize or maximize the volume of the box? o minimize o maximize3. [-I4.5 Points] DETAILS The cross-section of a tunnel is a rectangle of height h surmounted by a semicircular roof section of radius r. (See figure below). Suppose the cross-sectional area is 1200 R2. (a) Express the perimeter in terms of r only. P(r) = (b) Find P '(r) = (C) Determine the radius of the cross-section which minimizes the perimeter. r = ft I. [-I3.5 Points] DETAILS Find the point on the parabola shown belnw that is closest to the point (z, o). (a) Express the square of the distance between the parabola and the point (2, 0) in terms ofx only. Note: Minimizing the square of the distance can be used in this case to avoid working with square roots. S(x) = (b) Find the coordinates of the point on the parabola to two decimal places. ( i ) A wire of length 110 inches will be used to make the frame (edges) of a box with a square bottom. (a) Find a formula for the volume of the box in terms of the width only. Use lower case w. V ( W ) = cubic inches (b) Find V '(w) = (c) Find the non-zero critical point of V(w). W = inches (d) Does your critical point minimize or maximize the volume of the box? minimize o maximize 6. [-/4 Points] DETAILS The hypotenuse of a right triangle has one end at the origin and one end on the curve y(x) = xsex, with x 2 0 . One of the other two sides is on the x-axis, the other side is parallel to the y-axis. (a) Express the area of the triangle in terms of x only. A (x) = (b) Find A '(x) = (c) Find the critical point for x > 0. X = (d) Find the maximum area of the triangle. Give an exact answer. Area =