Assign variables and restate the following problem in terms of known and unknown derivatives (but do not solve it): How fast is the volume of a cube increasing if its side increases at a rate of 0.5 cm/s?

Q: assume that an airplane is flying toward you at a constant height, h miles, and at a speed dx/dt.…

A: assume that an airplane is flying toward you at a constant height, h miles, and at a speed dx/dt.…

Q: f. A truck travelling southwards at 40 kph and a car traveling eastward at a rate of 60 kph are…

A:

Q: Let l be the length of a diagonal of a rectangle whose sides have lengths x and y, and assume that x…

A: Given Data The length of the rectangle is y=9 ft. The width of the rectangle is x=7 ft. The rate of…

Q: Two cars start moving from the same point. One travels south at 24 mi/h and the other travels west…

A:

Q: When air expands adiabatically (without gaining or losing heat), its pre and volume V are related by…

A: For decreasing rate apply negative sign

Q: A balloon leaves the ground 30 m from an observer. If it rises at the rate of 2 m/s, how fast is it…

A:

Q: The acceleration of a Maserati is proportional to the difference between 250 km/h and the velocity…

A:

Q: A cube is heated in such a way that the volume increases at a rate of 1 mm/s. The rate of change of…

A: A cube is heated in way that volume increases at the rate of 1mm/s. the side of the cube is 7mm…

Q: A hot-air balloon is 140 ft above the ground when a motorcycle (traveling in a straight line on a…

A: Given the height in which the hot air balloon placed above the ground is 140 ft. Velocity of the…

Q: With snow on the ground and falling at a constant rate, a snowplow began plowing down a long…

A:

Q: A spherical snowball is melting in such a way that its volume is decreasing at a rate of 2 cm / min.…

A: Given-  dVdt=-2cm3/min (negative sign indicates that it is decreasing). To find- The rate of radius…

Q: The rate at which bacteria multiply is proportional to the instantaneous N numbers present. If the…

A: Let y(t) represents the number of bacteria present in the culture after t hours. Let the initial…

Q: 7. A 6ft tall human is walking away from a 20ft lamppost. Given that the length of the shadow of the…

A: Let the distance of the person from the bottom of the light post be x. Let the length of the shadow…

Q: In physics, the kinetic energy, K, of an object of mass m, moving at a velocity v, is given by the…

A: Given that kinetic energy K for an object of mass m and velocity V is given by K=(1/2)mv2. If mass…

Q: Suppose that the acceleration of a vehicle at time t (in s) is given by a(t) = 2t+ 1 (mph/s).If the…

A:

Q: A substance cools in moving air at a rate proportional to the difference between the temperature of…

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Q: A metallic ball with a diameter of 12 cm is covered with a layer of ice, so if the ice begins to…

A:

Q: A cone-shaped tank of water is leaking water at a constant rate of 3 m³/hour. If the base radius of…

A: Using volume  of cone and applications of derivative idea we solve this

Q: Let  x(V) = V1/3  be the length of the side of a cube in terms of its volume. As V decreases, does…

A: V is decreasing that is dv/dt is negative  Let dv/dt =-k where k is constant.

Q: The height of a triangle is increasing at a rate of 4 cm/s while the area of the triangle is…

A: Let the height, base and area of a triangle be 'h', 'b' and 'A' Given, dhdt=4 cm/sdAdt=56 cm2/s

Q: Suppose y varies at a constant rate of change of 4 with respect to x (so that Δy=4⋅Δx ), and when…

A: topic - linear equation in two variables

Q: The lower base of a trapezoid is 8 m. If its height is increasing at the rate of 0.5m/sec while its…

A:

Q: 1. A car is travelling at 50 mi/hr when the breaks are fully applied, producing a constant…

A:

Q: Let l be the length of a diagonal of a rectangle whose sides have lengths x and y, and assume that x…

A: Now in triangle ABC, by using Pythagorean theorem, we get

Q: Let I be the length of a diagonal of a rectangle whose sides have lengths a and y, and assume that a…

A:

Q: Water is discharged from an inverted cone-shaped tank with a height of 12 meters and a radius of 6…

A: Given: The height of the inverted cone-shaped tank, h = 12m. Also, its radius, r = 6m. From the…

Q: Assume that a snowball melts so that its volume decreases at a rate proportional to its surface…

A: Let V denote the volume and A denote surface area. Rate of melting is proportional to its surface…

Q: After 2 days, 10g of radioactive chemical is present. Three days later, 5g is present. How much of…

A: Given:      After  2 days, 10g of radioactive chemical is present. Three days later5 g chemical is…

Q: When air expands adiabatically (without gaining or losingheat), its pressure P and volume V are…

A: Pressure and Volume are related by equation At a instant

Q: A mold grows at a rate that is proportional to the amount present. Initially there is 6oz of this…

A: The amount of mold at time 0 is m0. The amount of mold present after t hours is m. The rate of…

Q: snowball melts so that its surface area decreases at a rate of 2 cm2/min, find the rate (in cm/min)…

A: Given query is to find the rate at which diameter of snowball is decreasing.

Q: The volume (v) of a sphere varies directly with the cube of its radius (r). If k is the constant of…

A:

Q: A mold grows at a rate that is proportional to the amount present. Initially, there is 5 oz of this…

A:

Q: Two airplanes are flying in the air at the same height. Airplane A is flying east at 494 mi/h and…

A: Point A represents position of airplane A. Point B represents position of airplane B.Point C…

Q: If the brakes on a car can give the car a constant negative acceleration of 2k m/2, where k is a…

A: Use simple velocity, distance, acceleration formula.

Q: A man of height 1.9 meters walks away from a 5-meter lamppost at a speed of 1.1 m/s. Find the rate…

A: Given Data: The height of the man is: h=1.9 m The height of the lamppost is: H=5 m The increasing…

Q: If V(t) is the volume of water in a reservoir at time t (seconds), v(t) is the rate of change,  then…

A: Rate Change: Rate change is defined as the change in one quantity with respect to any other quantity…

Q: As an orange grows, the rate of change of the radius r of the orange is inversely proportional to…

A: Given query is to find differential equation.

Q: A snowball melts in such a way that the rate of change in its volume is proportional to its surface…

A:

Q: A car traveling at 60 mi/h (88 ft/s) skids 176 ft after its brakes are suddenly applied. Under the…

A:

Q: The base x of a triangle is decreasing at the rate of 5 in./sec. and the height y of the triangle is…

A: Given Base = x  Height = y Rate of change of base = dx/dt = - 5 in/s Rate of change of height =…

Q: A searchlight, 1/2 mi from a straight shore, rotates at the rate of 2 rev/min. How fast is the spot…

A: Let the angle formed by line AB with the line AC is x  in the triangle ABC,   So, from the tangent…

Q: Initially filled with water, a conical reservoir is 24 ft deep and 12 ft across the top. Water goose…

A: Solution: Consider the given that conical reservoir 24ft (h) deep and 12ft across the top that is…

Q: What is the velocity (in ft/s) of a sandbag 3.0 s after it is released from a hot-air balloon that…

A: According to Questions  Initial velocity (u)=0 ft/s  time=t=3s

Q: If the rate of change in life expectancy I at birth for persons born in the United States can be…

A:

Q: A body weighing 64 Ib is dropped from a height of 100 ft with an initial velocity of 10 ft/sec.…

A:

Q: The area of a rectangle is increasing at a rate of 4 cm/sec ? while the width of the rectangle…

A: The area of a rectangle is increasing at a rate of 4 cm/sec ? while the width of the rectangle…

Q: A 15-foot ladder is leaning against a vertical wall when Josiah begins pulling the foot of the…

A: This question is based on application of Derivative and Pythagoras theorem.

Q: Let l be the length of a diagonal of a rectangle whose sides have lengths x and y, and assume that x…

A: Given that,  'L' be the length of a diagonal of a rectangle whose sides have lengths x and y, and x…

Q: Ship A is 70 km west of ship B and is sailing south at the rate of 25 kph. Ship B is sailing north…

A: At t = 0 { initial position } , distance between ships.Let the distance between ship A and ship B be…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.