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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 114
Chapter 3, Problem 114

To find the height of a lamppost (see accompanying figure), you stand a 6-ft pole 20 ft from the lamp and measure the length a of its shadow, finding it to be 15 ft, give or take an inch. Calculate the height of the lamppost using the value a = 15 and estimate the possible error in the result.


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Verified step by step guidance
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First, understand the problem setup: You have a lamppost casting a shadow and a smaller pole also casting a shadow. The problem involves similar triangles formed by the lamppost and its shadow, and the pole and its shadow.
Identify the variables: Let h be the height of the lamppost, and a = 15 ft be the length of the shadow of the pole. The height of the pole is 6 ft, and the distance from the pole to the lamppost is 20 ft.
Set up the proportion using similar triangles: The ratio of the height of the lamppost to the length of its shadow is equal to the ratio of the height of the pole to the length of its shadow. This gives the equation: hh-a = 615
Solve for h: Rearrange the equation to solve for h. Multiply both sides by (h - a) to get: h = 615 * (h - 15). Then, solve this equation to find the value of h.
Estimate the error: Consider the possible error in the measurement of a, which is given as 'give or take an inch'. Calculate how this affects the height of the lamppost by considering the derivative of h with respect to a, and use it to estimate the error in h.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Similar Triangles

The concept of similar triangles is fundamental in this problem, as it allows us to relate the height of the lamppost to the height of the pole using their respective shadow lengths. When two triangles are similar, their corresponding sides are proportional, which means we can set up a ratio to find the unknown height of the lamppost based on the known height of the pole and the lengths of their shadows.
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Proportional Relationships

Proportional relationships are essential for solving the problem, as they enable us to express the relationship between the heights of the pole and the lamppost in terms of their shadow lengths. By establishing a proportion, we can derive a formula that allows us to calculate the height of the lamppost based on the known dimensions, ensuring that the ratios remain consistent.
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Error Estimation

Error estimation is crucial in this context, as it helps quantify the uncertainty in the calculated height of the lamppost due to the measurement of the shadow length. By considering the possible error in the shadow measurement (given as 'give or take an inch'), we can calculate a range for the height of the lamppost, providing a more accurate representation of the result and its reliability.
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Related Practice
Textbook Question

The circumference of the equator of a sphere is measured as 10 cm with a possible error of 0.4 cm. This measurement is used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of


a. the radius.

b. the surface area.

c. the volume.

Textbook Question

How accurately should you measure the edge of a cube to be reasonably sure of calculating the cube’s surface area with an error of no more than 2%?

Textbook Question

Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x = 0.