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Minimum Length A farmer plans to fence a rectangular pasture adjacent to a river (see figure). The pasture must contain 405,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. What dimensions will require the least amount of fencing?

Maximum Volume A rectangular solid (with a square base) has a surface area of 337.5 square centimeters. Find the dimensions that will result in a solid with maximum volume.

Maximum Volume A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)

Surface Area and Volume A shampoo bottle is a right circular cylinder. Because the surface area of the bottle does not change when it is squeezed, is it true that the volume remains the same? Explain.

Minimum Surface Area A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 14 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

Area The measurement of the side of a square floor tile is 10 inches, with a possible error of \(\frac{1}{32}\) inch. (a) Use differentials to approximate the possible propagated error in computing the area of the square. (b) Approximate the percent error in computing the area of the square.

Applying the First Derivative Test In Exercises 19- \(40,(a)\) find the critical numbers of \(f,\) if any, (b) find the open intervals on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results. \(f(x)=|x+3|-1\)

Minimum Area The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.

Volume and Surface Area The radius of a spherical balloon is measured as 8 inches, with a possible error of 0.02 inch. (a) Use differentials to approximate the possible propagated error in computing the volume of the sphere. (b) Use differentials to approximate the possible propagated error in computing the surface area of the sphere. (c) Approximate the percent errors in parts (a) and (b).

Writing In Exercises \(33-36\) , explain why the Mean Value Theorem does not apply to the function \(f\) on the interval \([0,6]\) . \(f(x)=|x-3|\)