# Using Implicit Differentiation to Solve Related Rates Problems

Implicit differentiation is a powerful technique in calculus that proves invaluable when unraveling the complexities of related rates problems. This method allows us to differentiate equations with implicit dependencies, enabling a nuanced understanding of how variables change with respect to time. Applied extensively in physics, engineering, and various scientific disciplines, implicit differentiation serves as a fundamental tool in modeling dynamic systems. By delving into this approach, we gain a precise and efficient means to address the dynamic relationships between variables, providing a gateway to solving intricate related rates problems with clarity and precision.

Questions

- The radius r of a sphere is increasing at a constant rate of 0.04 centimeters per second.At the time when the radius of the sphere is 10 centimeters, what is the rate of increase of its volume?
- Bases are located on the field 90 feet away from one another Jimmy is running at a speed of 10 ft/sec from second to third base. When Jimmy is halfway to third base, how quickly is the distance between him and home plate decreasing?
- How do you find the rate at which water is pumped into an inverted conical tank that has a height of 6m and a diameter of 4m if water is leaking out at the rate of #10,000(cm)^3/min# and the water level is rising #20 (cm)/min#?
- A cylinder gets taller at a rate of 3 inches per second, but the radius shrinks at a rate of 1 inch per second. How fast is the volume of the cylinder changing when the height is 20 inches and the radius is 10 inches?
- Two boats leave the port at the same time with one boat traveling north at 15 knots per hour and the other boat traveling west at 12 knots per hour. How fast is the distance between the boats changing after 2 hours?
- How do you find y'' by implicit differentiation for #4x^3 + 3y^3 = 6#?
- The sun is shining and a spherical snowball of volume 340 ft3 is melting at a rate of 17 cubic feet per hour. As it melts, it remains spherical. At what rate is the radius changing after 7 hours?
- The angle of elevation of the sun is decreasing by 1/4 radians per hour. How fast is the shadow cast by a building of height 50 meters lengthening, when the angle of elevation of the sun is #pi/4#?
- A cube of ice is melting and the volume is decreasing at a rate of 3 cubic m/s. How fast is the height decreasing when the cube is 6 inches in height?
- A hypothetical cube grows so that the length of its sides are increasing at a rate of 4 m/min. How fast is the volume of the cube increasing when the sides are 7 m each?
- A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 50 ft from the pole?
- If #y=x^3+2x# and #dx/dt=5#, how do you find #dy/dt# when #x=2# ?
- The hands of a clock in some tower are approximately 2m and 1.5m in length. How fast is the distance between the tips of the hands changing at 9:00?
- A farmer wishes to enclose a rectangular field of area 450 ft using an existing wall as one of the sides. The cost of the fence for the other 3 sides is $3 per foot. How do you find the dimensions that minimize the cost of the fence?
- A swimming pool is 25 ft wide, 40 ft long, 3 ft deep at one end and 9 ft deep at the other end. If water is pumped into the pool at the rate of 10 cubic feet/min, how fast is the water level rising when it is 4 ft deep at the deep end?
- A spherical balloon is inflated so that its radius (r) increases at a rate of 2/r cm/sec. How fast is the volume of the balloon increasing when the radius is 4 cm?
- Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 cm^2?
- You throw a rock into a pond and watch the circular ripple travel out in all directions along the surface. If the ripple travels at 1.4 m/s, what is the approximate rate that the circumference is increasing when the diameter of the circular ripple is 6m?
- A point is moving along the curve #y=sqrt(x)# in such a way that its x coordinate id increasing at the rate of 2 units per minute. At what rate is its slope changing (a) when x=1? (b) when x=4?
- How can I solve the mentioned problem?Please,help.