A road rises 1m for every 15m measured along the road.
a. What is the angle of inclination of the road to the nearest degree?
b. How far does a car travel horizontally when it travels 15m along the road? Give the answer to the nearest tenth of a metre.
This question was confusing, but I came up along with help from Google calculator and an inverse trigonometric calculator, that (A) is around 4 degrees and (B) is around 15 meters (to the hundredth degree is 14.96).
a. Well, it seems like the road has got some ambitions to be a mountain! To find the angle of inclination, we need to use some trigonometry. The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, the opposite side is 1m (the rise) and the adjacent side is 15m (the measured distance). So, the tangent of the angle is 1/15. Using inverse tangent (arctan) function on a calculator, we find that the angle of inclination is approximately 3.8 degrees.
b. Ah, the car is going on a scenic route, I see! To find how far the car travels horizontally, we can use trigonometry once again. Since we know the angle of inclination is 3.8 degrees, we can use the cosine function, which relates the length of the adjacent side to the hypotenuse. The hypotenuse represents the total distance traveled, which is given as 15m. So, the horizontal distance traveled is 15m multiplied by the cosine of 3.8 degrees. Calculating that gives us approximately 14.9m, rounded to the nearest tenth of a meter.
To find the angle of inclination of the road, we can use the tangent function. The tangent of an angle is equal to the opposite side (1m) divided by the adjacent side (15m).
a. Angle of inclination ≈ tan^(-1) (1/15)
Using a scientific calculator, we can find the angle of inclination to the nearest degree:
Angle of inclination ≈ 3.81 degrees
b. To find how far a car travels horizontally when it travels 15m along the road, we need to find the adjacent side of the triangle formed. Using the cosine function, we can calculate it.
Adjacent side = 15m * cos(3.81 degrees)
Using a scientific calculator, we can find the adjacent side value:
Adjacent side ≈ 14.9979m ≈ 15m (rounded to the nearest tenth of a meter)
To find the angle of inclination of the road, you can use trigonometry. The angle of inclination is the angle between the road and the horizontal plane. Let's call this angle θ.
a. To find θ, we can use the tangent function. Tangent is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side. In this case, the side opposite θ is the rise of 1m, and the side adjacent to θ is the 15m distance along the road.
tan(θ) = opposite / adjacent
tan(θ) = 1 / 15
To solve for θ, take the inverse tangent (also called arctan or tan^(-1)) of both sides:
θ = arctan(1 / 15)
Using a calculator, the angle of inclination θ is approximately 3.8 degrees (rounded to the nearest degree).
b. To find how far the car travels horizontally, we need to find the length of the horizontal distance (adjacent side) traveled when it goes 15m along the road.
We know that the angle θ is 3.8 degrees, and we want to find the length of the side adjacent to this angle. We can use trigonometry again, this time using the cosine function.
cos(θ) = adjacent / hypotenuse
cos(3.8°) = adjacent / 15
Rearranging the equation to solve for the adjacent side (horizontal distance traveled):
adjacent = cos(3.8°) * 15
Using a calculator, the car travels approximately 14.9 meters horizontally (rounded to the nearest tenth of a meter).