MATH 107
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CALCULUS I WITH
ANALYTIC GEOMETRY
WEEK: _10_
AY 2021 -2022
STUDENT’S NAME:
__________________________________________
YEAR AND SECTION:
__________________________________________
DATE RECEIVED:
__________________________________________
TEACHER’S NAME:
__EDDIE L. FERRER________________________
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RELATED RATES
I. Overview
In this module we will study related rates problems. In such problems one tries to find the rate at which some
quantity is changing by relating the quantity to other quantities whose rates of change are known.
II. Learning Outcomes
At the end of the week, the pre-service teacher (PST) should be able to:
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solve derivative-related problems (rates of change and related rates).
III.Discussion and
Self-Assessment
Activities(SAA)
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IV. Summary/Key
Points
V. End of Module
Assessment (EMA)
1–2 Both x and y denote functions of t that are related by the given equation. Use this equation and the given
derivative information to find the specified derivative.
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2. Equation: 𝑥 2 + 𝑦 2 = 2𝑥 + 4𝑦.
(a) Given that 𝑑𝑥/𝑑𝑡 = −5, find 𝑑𝑦/𝑑𝑡 when (𝑥, 𝑦) = (3,1).
(b) Given that 𝑑𝑦/𝑑𝑡 = 6, find 𝑑𝑥/𝑑𝑡 when (𝑥, 𝑦) = (1 + √2, 2 + √3).
3. Let A be the area of a square whose sides have length 𝑥, and assume that 𝑥 varies with the same 𝑡.
(a) Draw a picture of the square with the labels A and 𝑥 placed appropriately.
(b) Write an equation that relates A and 𝑥.
(c) Use the equation in part (b) to find an equation that relates 𝑑𝐴/𝑑𝑡 and 𝑑𝑥/𝑑𝑡.
(d) At a certain instant the sides are 3 ft long and increasing at a rate of 2 ft/min. How fast is the area
increasing at that instant?
4. In parts (a)–(d), let A be the area of a circle of radius r, and assume that r increases with the time t.
(a) Draw a picture of the circle with the labels A and 𝑟 placed appropriately.
(b) Write an equation that relates A and 𝑟.
(c) Use the equation in part (b) to find an equation that relates 𝑑𝐴/𝑑𝑡 and 𝑑𝑟/𝑑𝑡.
(d) At a certain instant the radius is 5 cm and increasing at the rate of 2 cm/s. How fast is the area increasing
at that instant?
5. A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 3
ft/s. How rapidly is the area enclosed by the ripple increasing at the end of 10 s?
VI. Looking Ahead
Coffee is poured at a uniform rate of 20 cm3/s into a cup whose inside is shaped like a truncated cone (see the
accompanying figure). If the upper and lower radii of the cup are 4 cm and 2 cm and the height of the cup is 6
cm, how fast will the coffee level be rising when the coffee is halfway up? [Hint: Extend the cup downward to
form a cone.]
VII. Self-learning Module
Evaluation
Rate your learning experience in using this module according to the following scale.
Put a check mark on your response.
4 – I learned a lot from this module.
3 – I learned just right.
2 – I still need guidance on certain topics.
1 – I did not understand anything.
VIII. References
Anton, H., Bivens, I.C. & Davis, S. (2011). Calculus Early Transcendentals (10th Ed.). Wiley.
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