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QuestionGregor drives up a 15∘15^{\circ} hill for 5 km5 \mathrm{~km}. What is the vertical rise? Round to the nearest metre.

Studdy Solution

STEP 1

Assumptions1. Gregor drives up a hill inclined at an angle of 15∘15^{\circ} to the horizontal. . The distance driven is 5 km5 \mathrm{~km}.
3. We are asked to find the vertical distance risen, which is the height of the triangle formed by the inclined hill, the horizontal ground, and the path driven.
4. We assume that the path driven by Gregor forms a straight line, making the situation a right triangle problem.

STEP 2

We can use the sine of the angle to find the vertical distance. In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side (height in this case) to the length of the hypotenuse (distance driven in this case). So we havesin⁡(θ)=HeightHypotenuse\sin(\theta) = \frac{Height}{Hypotenuse}

STEP 3

We rearrange the equation to solve for the heightHeight=sin⁡(θ)×HypotenuseHeight = \sin(\theta) \times Hypotenuse

STEP 4

Now, we substitute the given values into the equation. The angle θ\theta is 15∘15^{\circ} and the hypotenuse (distance driven) is  km \mathrm{~km}.
Height=sin⁡(15∘)× kmHeight = \sin(15^{\circ}) \times \mathrm{~km}

STEP 5

We calculate the sine of 15∘15^{\circ} using a calculator.sin⁡(15∘)≈0.2588\sin(15^{\circ}) \approx0.2588

STEP 6

Substitute the value of sin⁡(15∘)\sin(15^{\circ}) back into the equationHeight=0.2588×5 kmHeight =0.2588 \times5 \mathrm{~km}

STEP 7

Calculate the heightHeight≈1.294 kmHeight \approx1.294 \mathrm{~km}

STEP 8

Convert the height from kilometers to meters because the problem asks for the answer in meters. There are1000 meters in a kilometer, so we multiply by1000.
Height≈1.294 km×1000=1294 mHeight \approx1.294 \mathrm{~km} \times1000 =1294 \mathrm{~m}

STEP 9

Round the height to the nearest meterHeight≈1294 mHeight \approx1294 \mathrm{~m}So, Gregor has risen approximately1294 meters vertically.

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