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Given an equation, find a rate (related rates problem)

Calculus Related Rates Problem: Given an equation, find the rate. Not all Related Rates problems are word problems. Here’s a different type that came from a student. The problem gives you an equation, and then asks you to find a rate: If $y = x^3 + 2x$ and $\dfrac{dx}{dt} = 6$, find $\dfrac{dy}{dt}$ when $x=5.$… [read more]

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snowball melts, area decreases at given rate (related rates problem)

Calculus Related Rates Problem: As a snowball melts, its area decreases at a given rate. How fast does its radius change? A spherical snowball melts symmetrically such that it is always a sphere. Its surface area decreases at the rate of $\pi$ in$^2$/min. How fast is its radius changing at the instant when $r =… [read more]

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shadow lamp post (related rates problem)

Calculus Related Rates Problem: Lamp post casts a shadow of a man walking. A 1.8-meter tall man walks away from a 6.0-meter lamp post at the rate of 1.5 m/s. The light at the top of the post casts a shadow in front of the man. How fast is the “head” of his shadow moving… [read more]

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How to Solve Optimization Problems in Calculus

Need to solve Optimization problems in Calculus? Let’s break ’em down and develop a strategy that you can use to solve them routinely for yourself. Overview Optimization problems will always ask you to maximize or minimize some quantity, having described the situation using words (instead of immediately giving you a function to max/minimize). Typical phrases… [read more]

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garden fence (optimization problem)

Calculus Optimization Problem: What dimensions minimize the cost of a garden fence? Sam wants to build a garden fence to protect a rectangular 400 square-foot planting area. His next-door neighbor agrees to pay for half of the fence that borders her property; Sam will pay the rest of the cost. What are the dimensions of… [read more]

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Optimization Problem Solving Strategy

Calculus Optimization Problem Solving Strategy We will use the steps outlined below to solve each Calculus Optimization problem on this site, step-by-step, every single time. We hope that this will help you see the strategy we’re using so you can learn it too, and then be able to apply it to all of your problems,… [read more]

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printed poster (optimization problem)

Calculus Optimization Problem: What are the dimensions of the poster with the smallest total area? A rectangular poster must have a printed area of 320 cm$^2$. It will have top and bottom margins that are 5 cm each, and side margins that are 4 cm. What are the dimensions of the poster with the smallest… [read more]

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least expensive open-topped can (optimization problem)

Calculus Optimization Problem: What dimensions minimize the cost of an open-topped can? An open-topped cylindrical can must contain V cm$^3$ of liquid. (A typical can of soda, for example, has V = 355 cm$^3$.) What dimensions will minimize the cost of metal to construct the can? Calculus Solution We’ll use our standard Optimization Problem Solving… [read more]

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how fast is the ladder’s top sliding (related rates problem)

Calculus Related Rates Problem: How fast is the ladder’s top sliding? A 10-ft ladder is leaning against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house. When the base has slid to 8 ft from the house, it… [read more]

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water drains from a cone (related rates problem)

Calculus Related Rates Problem: How fast is the water level falling as water drains from the cone? An inverted cone is 20 cm tall, has an opening radius of 8 cm, and was initially full of water. The water drains from the cone at the constant rate of 15 cm$^3$ each second. The water’s surface… [read more]

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