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Chapter 13: Problem 7

For the final exam in a scuba diving certification course, Karl navigates from one point in a lake to another. Karl begins the test \(x\) meters directly beneath the boat and swims straight down toward the bottom of the lake for 8 meters. He then turns to his right and swims in a straight line parallel to the surface of the lake and swims 24 meters, at which point he swims directly from his location, in a straight line, back to the boat. If the distance that Karl swims back to the boat is 26 meters, what is the value of \(x ?\)

Short Answer

Expert verified
The depth from where Karl initially started from is 10 meters.

Step by step solution

01

Identify Given Values

We know that Karl swims straight down for 8 meters, then parallel to the surface for 24 meters, and finally back to the boat which forms the hypotenuse of 26 meters. We represent the initial depth where Karl started from with the variable \(x\). Hence we have: \[x= ? (unknown)\] \[a = 8 (Height of the triangle)\] \[b = 24 (Base of the triangle)\] \[c = 26 (Hypotenuse of the triangle)\]
02

Applying the Pythagorean theorem

We apply the Pythagorean theorem to solve for the value of \(x\), where \(x\) is represented as length \(a\), but 'downwards'. So, we will use the values of \(b = 24\) and \(c = 26\) where the equation becomes: \[x^2 = c^2 - b^2\] and substituting the given values, we get: \[x^2 = 26^2 - 24^2\]
03

Calculating for \(x\)

We do the subtractions and solve for \(x\). \[x^2 = 676 - 576\] \[x^2 = 100\] Taking the square root on both sides because \(x\) is the square root of 100, we find: \[x = 10\]

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Most popular questions from this chapter

Triangle \(A B C\) is contained within a circle with center \(C\). Points \(A\) and \(B\) lie on the circle. If the area of circle \(C\) is \(25 \pi,\) and the measure of angle \(A C B\) is \(60^{\circ},\) which of the following are possible lengths for side \(A B\) of triangle \(A B C ?\) Indicate all such lengths. a. 3 b. 4 c. 5 d. 6 e. 7

$$\begin{array}{cc}\text { Quantity } \mathbf{A} & \text { Quantity } \mathbf{B} \\ \text {The circumference of a circular region with radius \(r\)} & \text {The perimeter of a square with side \(r\) }\end{array}$$ a. Quantity A is greater. b. Quantity B is greater. c. The two quantities are equal. d. The relationship cannot be determined from the information given.

\((a, 6)\) is a point (not shown) in Quadrant I. \((-6, b)\) is a point (not shown) in Quadrant II. $$\begin{array}{cc}\text { Quantity } \mathbf{A} & \text { Quantity } \mathbf{B} \\ a & b \end{array}$$ a. Quantity A is greater. b. Quantity B is greater. c. The two quantities are equal. d. The relationship cannot be determined from the information given.

Which of the following could be the degree measures of two angles in a right triangle? Indicate all such angles. A. \(20^{\circ}\) and \(70^{\circ}\) B. \(30^{\circ}\) and \(60^{\circ}\) C. \(45^{\circ}\) and \(45^{\circ}\) D. \(55^{\circ}\) and \(55^{\circ}\) E. \(75^{\circ}\) and \(75^{\circ}\)

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