Physical Applications of the Definite Integral

A rectangular plate 12 ft long and 8 ft wide is submerged vertically with the longer edge in the surface of the water. Find the force on one side of the plate.

• 384w (lb)

• 348w (lb)

• 834w (lb)

• 483w (lb)

A plate in the shape of a triangle is submerged vertically and the base which is 3 ft long is in the surface of the water. Find the altitude of the triangle if the force due to the water pressure against the face of the plate is 50w where w is the weight per cu.ft. of water.

• 14 ft

• 12 ft

• 10 ft

• 8 ft

An elliptical plate, major axis 6 ft, minor axis 4 ft, is submerged vertically until its center is 2 ft below the surface and the major axis is horizontal. Find the force on one side of the plate.

• $12\pi w\ \left(lb\right)$

• $10\pi w\ \left(lb\right)$

• $11\pi w\ \left(lb\right)$

• $14\pi w\ \left(lb\right)$

A trough of length 6 ft has for its vertical cross section an isosceles trapezoid. The upper and lower bases are 6 ft and 2 ft respectively and the altitude is 2 ft. If the trough is full of water, find the force on the slant side of the trough.

• $1214\sqrt{2}w\ \left(lb\right)$

• $1213w\ \left(lb\right)$

• $1212\sqrt{2}w\ \left(lb\right)$

• $1215\sqrt{2}w\ \left(lb\right)$

An unstretched spring is 10 ft long. A pull of 40 lb stretches the spring by ½ ft. Find the work done in stretching the spring from 10 ft to 14 ft.

• 640 ft-lb

• 645 ft-lb

• 604 ft-lb

• 600 ft-lb

If a force of 5 lb produces a stretch of one-tenth the natural length L ft of a spring, how much work will be done in stretching the spring to double its natural length?

• 26L ft-lb

• 25L ft-lb

• 24L ft-lb

• 20L ft-lb

A cylindrical tank of radius 3 ft and height 10 ft stands on a platform 50 ft high. How much work will be done in filling the tank by pumping the water from the ground level?

• $4,590w\pi\ \left(ft-lb\right)$

• $4,050w\pi\ \left(ft-lb\right)$

• $4,950w\pi\ \left(ft-lb\right)$

• $4,955w\pi\ \left(ft-lb\right)$

The horizontal cross-section of a well-containing mineral water is a circle of radius 4 ft. The cost of pumping the water to an outlet at the top of the well is 2 centavos per ft-lb of work done. If the mineral water weighs and is sold for 20 pesos per cu. ft., find the depth to which the water is to be pumped out to realize a maximum profit.

• 16 ft

• 18 ft

• 12 ft

• 10 ft

A horizontal cylindrical tank 4 ft in diameter and 10 ft long is full of water. Find the work done in pumping the water to a point 4 ft above the top of the tank.

• $280\pi w\ \left(ft-lb\right)$

• $260\pi w\ \left(ft-lb\right)$

• $250\pi w\ \left(ft-lb\right)$

• $240\pi w\ \left(ft-lb\right)$

A tank full of water is 12 ft long, 4 ft wide, and 6 ft deep. Find how much the surface of the water is lowered when one-third of the necessary work has been done to pump the water to the top of the tank.

• 3.4641 ft

• 2.4641 ft

• 4.4641 ft

• 8.4641 ft