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(a) A water trough is 1000 cm (i.e., 1 m) long and 50 cm high. The cross section of the trough is in the shape of an isosceles trapezoid that is 30 cm wide at the bottom and 80 cm wide at the top. If water is being pumped into the trough at a rate of 200000 cm³/min, how fast is the water level rising when the water is 30 cm deep?

(a) A water trough is 1000 cm (i.e., 1 m) long and 50 cm high. The cross section of the trough is in the shape of an isosceles trapezoid that is 30 cm wide at the bottom and 80 cm wide at the top. If water is being pumped into the trough at a rate of 200000 cm³/min, how fast is the water level rising when the water is 30 cm deep?

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4. Related Rates. (a) A water trough is 1000 cm (i.e., 1 m) long and 50 cm high. The cross section of the trough is in the shape of an isosceles trapezoid that is 30 cm wide at the bottom and 80 cm wide at the top. If water is being pumped into the trough at a rate of 200000 cm³/min, how fast is the water level rising when the water is 30 cm deep? cm/min (b) Two people start walking from the same point at 12pm. The first person walks due south at 5 miles per hour and the second walks thirty degrees north of due east at 4 miles per hour. How fast is the distance between them changing at 2pm? miles/hour
Transcribed Image Text:4. Related Rates. (a) A water trough is 1000 cm (i.e., 1 m) long and 50 cm high. The cross section of the trough is in the shape of an isosceles trapezoid that is 30 cm wide at the bottom and 80 cm wide at the top. If water is being pumped into the trough at a rate of 200000 cm³/min, how fast is the water level rising when the water is 30 cm deep? cm/min (b) Two people start walking from the same point at 12pm. The first person walks due south at 5 miles per hour and the second walks thirty degrees north of due east at 4 miles per hour. How fast is the distance between them changing at 2pm? miles/hour
(c) A rocket takes off at time t = 0 and a TV camera a feet away from the blast-off point stays focused on the rocket as it ascends. If the rocket rises at the constant speed v ft/sec, how fast must the camera increase its angle to stay focused on the rocket (in terms of u, a and the height h of the rocket only)? What should happen to your answer as h→ ∞o? What does happen to your answer as h→ ∞o? (d) The energy of an object is given by 1 dh E = mgh += m(² radians/sec where m is the (constant) mass of the object, g is constant, and h is the height of the object. If the energy of the object is increasing at a rate of 50 ft.lbs./sec, find the accelaration of the object the instant its velocity is 10 ft/sec. Express your answer in terms of m and g. Hint: Remember that acceleration is the second derivative of height. m/sec²
Transcribed Image Text:(c) A rocket takes off at time t = 0 and a TV camera a feet away from the blast-off point stays focused on the rocket as it ascends. If the rocket rises at the constant speed v ft/sec, how fast must the camera increase its angle to stay focused on the rocket (in terms of u, a and the height h of the rocket only)? What should happen to your answer as h→ ∞o? What does happen to your answer as h→ ∞o? (d) The energy of an object is given by 1 dh E = mgh += m(² radians/sec where m is the (constant) mass of the object, g is constant, and h is the height of the object. If the energy of the object is increasing at a rate of 50 ft.lbs./sec, find the accelaration of the object the instant its velocity is 10 ft/sec. Express your answer in terms of m and g. Hint: Remember that acceleration is the second derivative of height. m/sec²
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