A submarine at the surface of the ocean makes an emergency dive, its path making an angle of 21deg with the surface.
a. If it goes for 300 meters along its download path, how deep will it be? What horizontal distance is it from its starting point?
b. How many meters must it go along its downward path to reach a depth of 1,000 meters?
The downward path can be represented by the hypotenuse of a rt. triangle
with an angle of 21 deg. between the
hypotenuse and hor side.
a. Sin21 = Depth/300
Depth = 300 * Sin21 = 107.5m.
Hor. Dist. = 300 * Cos21 = 280.1m.
b. Dist. = 1000m / Sin21 = 2790.4m.
what if instead of 21 its 24?
a. Well, the submarine certainly knows how to make a grand entrance! To find out how deep it will be, we can use some trigonometry. Since the submarine is making an angle of 21 degrees with the surface, we can use the sine function to relate the angle to the depth. The formula is sin(angle) = opposite/hypotenuse. In this case, the opposite side is the depth and the hypotenuse is the distance traveled along the downward path. So, sin(21) = depth/300 meters. Solving for depth, we have depth = 300 meters * sin(21 degrees). Plug in those numbers and you'll know how deep the submarine will be.
Now, to find the horizontal distance it is from its starting point, we can use the cosine function. The formula is cos(angle) = adjacent/hypotenuse. In this case, the adjacent side is the horizontal distance and the hypotenuse is still the distance traveled along the downward path. So, cos(21) = horizontal distance/300 meters. Solve for the horizontal distance, and voila, you'll have your answer!
b. Ah, the submarine is going deep-sea diving this time! To reach a depth of 1,000 meters, we can use the same sine function as before. Now, we need to solve for the hypotenuse (distance traveled along the downward path). Rearranging the formula sin(angle) = opposite/hypotenuse, we have hypotenuse = opposite/sin(angle). In this case, the opposite side is 1,000 meters and the angle is still 21 degrees. So, hypotenuse = 1,000 meters/sin(21 degrees). Calculate that, and you'll know how many meters the submarine needs to go along its downward path to reach a depth of 1,000 meters.
a. To find out how deep the submarine will be, we can use trigonometry.
First, let's define the given information:
- Angle of descent: 21 degrees
- Distance along the downward path: 300 meters
Using the concept of trigonometry, we can establish the relationship between the distance along the downward path, the depth, and the angle of descent.
We can use the tangent function (tan) to find the depth:
tan(θ) = opposite/adjacent,
where θ is the angle of descent, opposite is the depth, and adjacent is the horizontal distance from the starting point.
Given that θ = 21 degrees and the distance along the downward path is 300 meters, we can solve for the depth (opposite).
tan(21) = opposite/300,
opposite = 300 * tan(21).
Using a calculator, we find opposite ≈ 114.57 meters.
Therefore, the submarine will be approximately 114.57 meters deep.
To find the horizontal distance from the starting point (adjacent), we can use the cosine function (cos).
cos(θ) = adjacent/hypotenuse,
where θ is the angle of descent, adjacent is the horizontal distance, and hypotenuse is the distance along the downward path.
Given that θ = 21 degrees and the distance along the downward path is 300 meters, we can solve for the horizontal distance (adjacent).
cos(21) = adjacent/300,
adjacent = 300 * cos(21).
Using a calculator, we find adjacent ≈ 277.24 meters.
Therefore, the submarine will be approximately 277.24 meters horizontally from its starting point.
b. To find out how many meters the submarine must travel along its downward path to reach a depth of 1,000 meters, we can use trigonometry again.
Using the same trigonometric relationships as in part a, we can solve for the distance along the downward path (hypotenuse) when the depth (opposite) is 1,000 meters.
tan(21) = 1,000/hypotenuse,
hypotenuse = 1,000/tan(21).
Using a calculator, we find hypotenuse ≈ 2,600.54 meters.
Therefore, the submarine must travel approximately 2,600.54 meters along its downward path to reach a depth of 1,000 meters.
To solve this problem, we can break it down into components: the vertical component and the horizontal component.
a. Let's start with the first question: if the submarine goes for 300 meters along its downward path, how deep will it be? And what horizontal distance is it from its starting point?
To find the depth, we need to find the vertical component of the distance. We can use the sine function to calculate this.
First, let's find the vertical component:
Vertical component = 300 * sin(21°)
Note: We convert the angle from degrees to radians since most trigonometric functions require radians as input.
To find the horizontal distance, we need to find the horizontal component of the distance. We can use the cosine function to calculate this.
Horizontal component = 300 * cos(21°)
Using a scientific calculator or trigonometric table, we can find the values of sin(21°) and cos(21°). Let's substitute these values into our equations.
Vertical component = 300 * sin(0.366519)
Horizontal component = 300 * cos(0.366519)
Calculating these values, we get:
Vertical component = 300 * 0.359172 = 107.751 meters (rounded to 3 decimal places)
Horizontal component = 300 * 0.93358 = 280.074 meters (rounded to 3 decimal places)
So, the submarine will be approximately 107.751 meters deep, and it will be approximately 280.074 meters horizontally from its starting point.
b. Now, let's move on to the second question: How many meters must it go along its downward path to reach a depth of 1,000 meters?
In this case, we need to find the distance traveled in the vertical component. We can rearrange the formula we used earlier to solve for the distance:
Distance = Depth / sin(21°)
Substituting the values:
Distance = 1,000 / sin(0.366519)
Calculating this, we get:
Distance = 1,000 / 0.359172 = 2784.963 meters (rounded to 3 decimal places)
So, the submarine must travel approximately 2784.963 meters along its downward path to reach a depth of 1,000 meters.