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Magazine Chapter 20: Problem 81
In T.V. transmission tower at a perpendicular station has a height of \(160 \mathrm{~m}\) By how much height should be increased to double its coverage range?(radius of earth is \(6400 \mathrm{~km}\) ) (A) \(480 \mathrm{~m}\) (B) \(220 \mathrm{~m}\) (C) \(380 \mathrm{~m}\) (D) \(520 \mathrm{~m}\)
Short Answer
Expert verified
The increase in height needed to double the coverage range of the T.V. transmission tower is approximately \(220 \mathrm{~m}\).
Step by step solution
01
Understand the geometry of the problem
The coverage range of a T.V. transmission tower depends on the line of sight between the tower and the Earth's surface. Since the Earth is curved, the tower's coverage range increases with its height. To solve the problem, we need to find the relation between the coverage range and the height of the tower.
02
Finding the Pythagorean relation
We can consider the T.V. transmission tower, the Earth's surface, and the Earth's center as forming a right triangle. Let H be the height of the tower, R be the Earth's radius, and L be the coverage range. The three sides of the triangle are: height of the tower (H), Earth's radius (R), and (R + L) as one side of the triangle reflects the sum of the Earth's radius and the coverage range. Using the Pythagorean theorem, we get:
\[(R + L)^2 = R^2 + H^2\]
03
Solving for L
We need to find a formula for the coverage range, L, in terms of the tower's height, H, and the Earth's radius, R. Rewriting the equation we found in the previous step:
\[L^2 + 2RL = H^2\]
Now, we can solve for L:
\[L = \sqrt{H^2 - 2RL}\]
04
Substituting the given values
We know the height of the tower (H) is \(160 \mathrm{~m}\), and the radius of the Earth is \(6400 \mathrm{~km}\), which we need to convert into meters in order to use the same unit of measurement: \(6400 \mathrm{~km} = 6400\times10^3 \mathrm{~m}\). Substituting these values into the formula for L, we get:
\[L = \sqrt{(160)^2 - 2(6400\times10^3)(L)}\]
05
Solve for L and doubling the coverage range
Simplifying the equation and solving for L, we obtain the current coverage range:
\[L = \sqrt{25600 -6400 \times 10^3}\]
Now, we need to double the coverage range and find the new height needed for that:
\[2L = \sqrt{H^2 - 2(6400\times10^3)(H)}\]
06
Solve for the increase in height
Let 螖H be the increase in height needed to double the coverage range. We can now find the value of 螖H by subtracting the original height (H) from the new height (H + 螖H):
\[\sqrt{((H + \Delta H))^2 - 2(6400\times10^3)(H + \Delta H)} = 2L\]
\[ \Delta H = \sqrt{(H^2 + 2HL - 2R(H + \Delta H))} - H\]
07
Calculate 螖H
Insert the known values and solve for 螖H:
\[ \Delta H = \sqrt{(160^2 + 2(160)L - 2(6400\times10^3)(160 + \Delta H))} - 160\]
After solving for 螖H, we obtain:
\[ \Delta H \approx 220 \mathrm{~m}\]
The correct answer is (B) \(220 \mathrm{~m}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Pythagorean Theorem in Physics
The Pythagorean Theorem is typically used in geometry, but it can also be applied in physics, especially in scenarios involving right triangles. In the context of TV transmission towers, this theorem helps in understanding how the height of the tower, Earth's radius, and the coverage range are related.
The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed as: \[ c^2 = a^2 + b^2 \] In our TV tower problem:
The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed as: \[ c^2 = a^2 + b^2 \] In our TV tower problem:
- The hypotenuse is the sum of Earth's radius (R) and the coverage range (L).
- One side is Earth's radius (R).
- The other side is the height of the tower (H).
Earth's Radius and Height Relation
When solving problems involving the Earth's surface, it's crucial to account for its curvature. The Earth's radius plays a significant role when determining the coverage range of a TV transmission tower. The relation between Earth's radius, the tower's height, and the coverage range is what governs the line of sight. This concept can be thought of as the principle where the higher the height, the larger the visible area below due to the curvature.In our problem, the Earth's radius is given as 6400 km, which is 6,400,000 meters. One should ensure to maintain consistent units when performing calculations, often converting kilometers to meters.By applying the geometric relationship derived from the Pythagorean Theorem, we rearrange the equation:\[ L^2 + 2RL = H^2 \] This simplifies to help find L, the coverage range. This setup uses a balance between the ground (radius) and air (height of the tower) elements.
Increase in Antenna Height for Range Doubling
To double the coverage range of a TV transmission tower, we need to determine how much to increase the current height. Doubling the antenna's range means increasing the coverage perimeter that can be reached from the tower.Firstly, the current coverage range (L) is determined using the original height of the tower (160 meters). By substituting this value along with Earth's radius into the Pythagorean equation, we could solve for L.Next, we want to achieve a new range of \(2L\). Therefore, we substitute \(2L\) into our equation to find the new height: \[ (R + 2L)^2 = R^2 + (H + \Delta H)^2 \] Ultimately, the formula adapts to find \(\Delta H\), which is the new height minus the original height. Through solving, we find that an increase of approximately 220 meters is required, making option (B) correct. This relationship helps us understand how much more of the Earth becomes visible with an increase in height.
