Question

course challenge
an airplane takeoff, levels off, and lands according to the diagram shown below.
all measurements listed are in kilometers.
what is the horizontal distance from the planes starting position to its finishing position?
round your answer to the nearest tenth of a kilometer.
kilometers
show calculations

Explanation:

Step1: Analyze the left triangle

We have a right triangle with hypotenuse \( 5 \) km and height \( 3 \) km. Using the Pythagorean theorem \( a = \sqrt{c^2 - b^2} \), where \( c = 5 \) and \( b = 3 \). So \( x = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \) km.

Step2: Analyze the right triangle

We have a right triangle with hypotenuse \( 7 \) km and height \( 3 \) km. Using the Pythagorean theorem \( a = \sqrt{c^2 - b^2} \), where \( c = 7 \) and \( b = 3 \). So \( y = \sqrt{7^2 - 3^2} = \sqrt{49 - 9} = \sqrt{40} \approx 6.3246 \) km.

Step3: Calculate the total horizontal distance

The middle horizontal segment is \( 3 \) km. So total horizontal distance \( = x + 3 + y = 4 + 3 + \sqrt{40} \approx 4 + 3 + 6.3246 = 13.3246 \) km. Rounding to the nearest tenth gives \( 13.3 \) km. Wait, no, wait. Wait, maybe I misread the diagram. Wait, looking again, maybe the left triangle: hypotenuse 5, height 3, so horizontal leg \( x = \sqrt{5^2 - 3^2} = 4 \). Then the middle part is 3, then the right triangle: hypotenuse 7, height 3, so horizontal leg \( y = \sqrt{7^2 - 3^2} = \sqrt{40} \approx 6.32 \). Then total horizontal distance is \( 4 + 3 + 6.32 = 13.32 \), which rounds to 13.3? Wait, no, maybe the diagram has the middle horizontal part as 3, and the left horizontal segment is x, middle is 3, right is y. Wait, maybe the left triangle: vertical side 3, hypotenuse 5, so horizontal x = 4. Then middle horizontal is 3, then right triangle: vertical side 3, hypotenuse 7, so horizontal y = sqrt(7^2 - 3^2) = sqrt(40) ≈ 6.3246. Then total horizontal distance is 4 + 3 + 6.3246 ≈ 13.3246, which rounds to 13.3? Wait, but maybe I made a mistake. Wait, let's check again. Wait, the left triangle: legs x and 3, hypotenuse 5. So x = sqrt(5^2 - 3^2) = 4. Correct. The right triangle: legs y and 3, hypotenuse 7. So y = sqrt(7^2 - 3^2) = sqrt(40) ≈ 6.3246. Then the middle horizontal part is 3. So total horizontal distance is 4 + 3 + 6.3246 ≈ 13.3246, which rounds to 13.3? Wait, but maybe the diagram is different. Wait, maybe the middle part is 3, and the left horizontal is x, middle is 3, right is y. So total is x + 3 + y. So 4 + 3 + 6.3246 ≈ 13.3. Wait, but let's confirm. Alternatively, maybe the middle segment is 3, and the left and right triangles have height 3. So yes, that's the approach.

Answer:

\( 13.3 \)