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6. A 20-foot ramp is used at the loading dock of a factory. If the base of the ramp is placed 19 feet from the base of the dock, how high is the loading dock? 20 19

6. A 20-foot ramp is used at the loading dock of a factory. If the base of the ramp is placed 19 feet from the base of the dock, how high is the loading dock? 20 19

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Problem 6: Calculation of the Loading Dock Height** **Given Problem:** A 20-foot ramp is used at the loading dock of a factory. If the base of the ramp is placed 19 feet from the base of the dock, how high is the loading dock? **Diagram Description:** The provided problem includes a right triangle diagram. The components are: - The hypotenuse (ramp length): 20 feet - The base of the triangle (horizontal distance from the base of the ramp to the dock): 19 feet - The vertical leg (height of the loading dock), denoted as 'x' **Solution Approach:** To solve for the height of the loading dock, we employ the Pythagorean theorem, which states: \[a^2 + b^2 = c^2\] In this context: - \( a \) represents the height of the loading dock \(x\) - \(b\) represents the base of the ramp (19 feet) - \(c\) represents the length of the ramp (20 feet) Applying the values to the Pythagorean theorem: \[x^2 + 19^2 = 20^2\] First, calculate: \[19^2 = 361\] \[20^2 = 400\] So, \[x^2 + 361 = 400\] Isolate \(x^2\) by subtracting 361 from 400: \[x^2 = 400 - 361\] \[x^2 = 39\] Solving for \(x\): \[x = \sqrt{39}\] \[x ≈ 6.24\] **Answer:** The height of the loading dock is approximately 6.24 feet.
Transcribed Image Text:**Problem 6: Calculation of the Loading Dock Height** **Given Problem:** A 20-foot ramp is used at the loading dock of a factory. If the base of the ramp is placed 19 feet from the base of the dock, how high is the loading dock? **Diagram Description:** The provided problem includes a right triangle diagram. The components are: - The hypotenuse (ramp length): 20 feet - The base of the triangle (horizontal distance from the base of the ramp to the dock): 19 feet - The vertical leg (height of the loading dock), denoted as 'x' **Solution Approach:** To solve for the height of the loading dock, we employ the Pythagorean theorem, which states: \[a^2 + b^2 = c^2\] In this context: - \( a \) represents the height of the loading dock \(x\) - \(b\) represents the base of the ramp (19 feet) - \(c\) represents the length of the ramp (20 feet) Applying the values to the Pythagorean theorem: \[x^2 + 19^2 = 20^2\] First, calculate: \[19^2 = 361\] \[20^2 = 400\] So, \[x^2 + 361 = 400\] Isolate \(x^2\) by subtracting 361 from 400: \[x^2 = 400 - 361\] \[x^2 = 39\] Solving for \(x\): \[x = \sqrt{39}\] \[x ≈ 6.24\] **Answer:** The height of the loading dock is approximately 6.24 feet.
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