Q1. A baseball diamond is a square with side length 90 ft. A runner is sprinting from home plate toward first base at a constant 22 ft/s. At the instant when the runner is 30 ft from home plate, determine:

• a) the rate at which the runner’s distance from second base is changing,

• b) the rate at which the runner’s distance from third base is changing.

Background

Topic: Related Rates

This problem tests your ability to apply related rates to a geometric situation involving a moving object (the runner) and changing distances on a baseball diamond.

Key Terms and Formulas

• Related rates: Using derivatives to relate the rates at which quantities change.

• Pythagorean Theorem: for right triangles.

• Let be the runner's distance from home plate, the distance from first base, and the distance from second base.

Step-by-Step Guidance (for part a)

1. Draw a diagram of the baseball diamond, labeling home plate, first base, second base, and the runner's position feet from home plate along the baseline.

2. Express the runner's distance from second base () in terms of using the Pythagorean Theorem. Recall that second base is diagonally across from home plate.

3. Write the equation relating and : .

4. Differentiate both sides with respect to time to relate and .

5. Plug in the known values: ft, ft/s, and solve for , but stop before the final calculation.

Try solving on your own before revealing the answer!

Q2. A particle moves along the curve . When it passes through the point (5, 2), the y-coordinate is decreasing at 4 cm/s. How fast is the x-coordinate changing at that instant?

Background

Topic: Related Rates

This question tests your ability to relate the rates of change of and for a particle constrained to move along a given curve.

Key Terms and Formulas

• Implicit differentiation: Differentiating both sides of an equation with respect to time .

• Given:

• Given: cm/s at

Step-by-Step Guidance

1. Differentiate both sides of with respect to to relate and .

2. Apply the product rule: .

3. Plug in the values , , and cm/s.

4. Set up the equation to solve for , but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q3. An inverted circular conical tank has base radius 3 m and height 6 m. Water is being pumped in at a rate of 3 m3/min. Find the rate at which the water level is rising when the water is 4 m deep.

Background

Topic: Related Rates (Conical Tank)

This problem involves relating the rate of change of the water volume to the rate of change of the water height in a cone.

Key Terms and Formulas

• Volume of a cone:

• Similar triangles: Relate and using the tank's dimensions.

• Given: m3/min, m

Step-by-Step Guidance

1. Express the radius of the water surface in terms of the water height using similar triangles: .

2. Substitute in the volume formula to write as a function of only.

3. Differentiate with respect to to relate and .

4. Plug in m and m3/min, and set up the equation for , but do not solve for it yet.

Try solving on your own before revealing the answer!

Q4. Use a linear approximation (or differentials) to estimate the value of .

Background

Topic: Linear Approximation / Differentials

This question tests your understanding of using the tangent line (linearization) to approximate function values near a known point.

Key Terms and Formulas

• Linear approximation: for near .

• Here, , , .

Step-by-Step Guidance

1. Find and for at .

2. Compute and evaluate at .

3. Set up the linear approximation formula: .

4. Plug in the values, but do not compute the final result yet.

Try solving on your own before revealing the answer!

Q5. Find where the normal line to the curve at the point (1, 1) meets the curve a second time.

Background

Topic: Tangent and Normal Lines to Implicit Curves

This problem tests your ability to find the equation of the normal line to an implicitly defined curve and determine its intersection with the curve.

Key Terms and Formulas

• Implicit differentiation: To find for the curve.

• Normal line: Perpendicular to the tangent line; its slope is the negative reciprocal of the tangent slope.

• Equation of a line:

Step-by-Step Guidance

1. Implicitly differentiate to find at .

2. Find the slope of the normal line (negative reciprocal of the tangent slope).

3. Write the equation of the normal line passing through .

4. Substitute the normal line equation into the original curve equation to find the coordinates where they intersect again. Set up the resulting equation, but do not solve for the intersection point yet.

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Extra Credit. At the beginning of an experiment, Georg Cantor is at the origin (0, 0) and begins running due East along the x-axis at a constant speed of 6 m/s. At the same instant, a UFO is directly above the origin at a height of 55 m and rises vertically at a constant rate of 2.5 m/s. Compute the rate at which the distance between Cantor and UFO is increasing when they are 100 m apart.

Background

Topic: Related Rates (Distance in 3D)

This problem involves finding the rate of change of the distance between two moving objects in perpendicular directions (one along the x-axis, one vertically).

Key Terms and Formulas

• Let be Cantor's horizontal position, the UFO's height.

• Distance formula:

• Given: m/s, m/s, m

Step-by-Step Guidance

1. Write the distance formula: .

2. Differentiate both sides with respect to to relate , , and .

3. Express and in terms of and the given rates, using the fact that m at the instant of interest.

4. Set up the equation for , but do not compute the final value yet.

Try solving on your own before revealing the answer!