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Chapter 11: Problem 58

A 600 pound Sasquatch statue is suspended by two cables from a gymnasium ceiling. If each cable makes a \(60^{\circ}\) angle with the ceiling, find the tension on each cable. Round your answer to the nearest pound.

Short Answer

Expert verified
Each cable's tension is approximately 346 pounds.

Step by step solution

01

Draw a Diagram

First, visualize the scenario by drawing a diagram. The Sasquatch statue is hanging from two cables. Each cable forms a 60掳 angle with the ceiling. This forms two symmetrical triangles where the combined vertical component of the tensions in the cables equals the weight of the statue (600 pounds).
02

Identify Forces and Components

Each cable has a tension force, which can be split into horizontal and vertical components. Given the symmetry, the vertical components of both cables' tensions must sum up to the total weight of the statue.
03

Use Trigonometric Functions

The vertical component of the tension in each cable (T) is found using the sine function. Since the angle with the ceiling is given by 60掳, the angle with the vertical is also 60掳 (considering the symmetrical setup). Therefore, the vertical component is: \[ T_{vertical} = T \cdot \sin(60^\circ) \] Since there are two cables, their combined vertical components must equal the weight: \[ 2 \cdot T \cdot \sin(60^\circ) = 600 \]
04

Solve for Tension (T)

Start by solving for T in the equation: \[ 2 \cdot T \cdot \sin(60^\circ) = 600 \] Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), substitute this value into the equation: \[ 2 \cdot T \cdot \frac{\sqrt{3}}{2} = 600 \]\[ T \cdot \sqrt{3} = 600 \]\[ T = \frac{600}{\sqrt{3}} \] This can be further simplified by multiplying the numerator and denominator by \(\sqrt{3}\): \[ T = \frac{600 \cdot \sqrt{3}}{3} \] \[ T = 200\sqrt{3} \]
05

Calculate and Round Final Answer

Calculate the numerical value for \(200\sqrt{3}\): \[ 200\sqrt{3} \approx 346.41 \] Round this to the nearest pound: 346 pounds. Hence, the tension in each cable is approximately 346 pounds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tension in Cables
When a heavy object, like a Sasquatch statue, is suspended, each supporting cable needs to hold a part of the weight. This tension ensures that the object remains stable. Tension refers to the force exerted along a cable or rope when it is pulled tight by forces acting from opposite ends.

In this specific problem, the tension in each cable is crucial for maintaining balance. Since the cables are at an angle, the tension is not directly equal to the weight of the statue. The cables work together to counteract the gravitational pull downward. This means both cables share the responsibility of supporting the full 600 pounds of the statue.
  • Each cable exerts a force, which we call tension.
  • These tensions are symmetrical due to the equal angles with the ceiling.
  • The goal is to find how much force each cable exerts to maintain balance.
Sine Function
Trigonometry is a powerful mathematical tool used to analyze angles and dimensions. It involves functions that relate the angles of triangles to their side lengths. In this scenario, we specifically use the sine function to understand forces in play.

The sine function helps determine the vertical component of the tension in each cable. When you hear about the 'sine of an angle,' it refers to the ratio of the length of the side opposite the angle to the hypotenuse in a right-angled triangle. Here, the hypotenuse would be the cable's tension, and the opposite side would be the vertical component that balances the weight.
  • Use \( \sin(60^\circ) \) to find the vertical part of the tension.
  • This component must add up to the statue's full weight for it to remain stable.
  • Sine function simplifies the calculation, making it feasible to find the exact tension.
Triangle Forces
Visualizing forces as triangles can greatly aid our understanding of tension in this context. Imagine each cable as the hypotenuse of a triangle where the ceiling is one side and the force downwards due to gravity is another.

Each cable, along with the ceiling and the vertical line, forms a right triangle. The angle between the ceiling and the cable is crucial; it is noted to be 60掳, helping to determine the force components involved.
  • Decompose the tension into perpendicular components: horizontal and vertical.
  • The vertical component is affected by the angle with the ceiling and dictates the force needed to stabilize the statue.
  • Ensuring the sum of these vertical components from both cables equals the total gravitational force on the statue is essential.
Weight Distribution
Weight distribution involves dividing up the weight of the suspended object equally between the supporting cables. In this problem, the symmetry of the angle ensures that each cable carries half of the total weight.

To achieve this balance, the cables' angles and the trigonometric sine function coordinate to produce forces that counterbalance the statue's weight. The result is that no part of the structure is overloaded, ensuring safety and stability.
  • Recognize that effective tension in both cables leads to an equal division of the pointer load.
  • Help students understand that equal angles imply equal distribution of force.
  • This understanding simplifies complex scenarios and assists in solving trigonometry-based problems confidently.

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Most popular questions from this chapter

The goal of this exercise is to use vectors to describe non-vertical lines in the plane. To that end, consider the line \(y=2 x-4 .\) Let \(\vec{v}_{0}=\langle 0,-4\rangle\) and let \(\vec{s}=\langle 1,2\rangle .\) Let \(t\) be any real number. Show that the vector defined by \(\vec{v}=\vec{v}_{0}+t \vec{s}\), when drawn in standard position, has its terminal point on the line \(y=2 x-4\). (Hint: Show that \(\vec{v}_{0}+t \vec{s}=\langle t, 2 t-4\rangle\) for any real number \(t\).) Now consider the non-vertical line \(y=m x+b\). Repeat the previous analysis with \(\vec{v}_{0}=\langle 0, b\rangle\) and let \(\vec{s}=\langle 1, m\rangle .\) Thus any non-vertical line can be thought of as a collection of terminal points of the vector sum of \(\langle 0, b\rangle\) (the position vector of the \(y\) -intercept) and a scalar multiple of the slope vector \(\vec{s}=\langle 1, m\rangle\).

Convert the equation from polar coordinates into rectangular coordinates. $$ 12 r=\csc (\theta) $$

In Exercises \(41-50\), use set-builder notation to describe the polar region. Assume that the region contains its bounding curves. The region inside the circle \(r=5\) but outside the circle \(r=3\).

$$ r=\theta, 0 \leq \theta \leq 12 \pi $$

We know that \(|x+y| \leq|x|+|y|\) for all real numbers \(x\) and \(y\) by the Triangle Inequality established in Exercise 36 in Section 2.2. We can now establish a Triangle Inequality for vectors. In this exercise, we prove that \(\|\vec{u}+\vec{v}\| \leq\|\vec{u}\|+\|\vec{v}\|\) for all pairs of vectors \(\vec{u}\) and \(\vec{v}\). (a) (Step 1) Show that \(\|\vec{u}+\vec{v}\|^{2}=\|\vec{u}\|^{2}+2 \vec{u} \cdot \vec{v}+\|\vec{v}\|^{2}\). 6 (b) (Step 2) Show that \(|\vec{u} \cdot \vec{v}| \leq\|\vec{u}\|\|\vec{v}\| .\) This is the celebrated Cauchy-Schwarz Inequality. (Hint: To show this inequality, start with the fact that \(|\vec{u} \cdot \vec{v}|=|\|\vec{u}\|\|\vec{v}\| \cos (\theta)|\) and use the fact that \(|\cos (\theta)| \leq 1\) for all \(\theta\).) (c) (Step 3) Show that \(\|\vec{u}+\vec{v}\|^{2}=\|\vec{u}\|^{2}+2 \vec{u} \cdot \vec{v}+\|\vec{v}\|^{2} \leq\|\vec{u}\|^{2}+2|\vec{u} \cdot \vec{v}|+\|\vec{v}\|^{2} \leq\|\vec{u}\|^{2}+\) \(2\|\vec{u}\|\|\vec{v}\|+\|\vec{v}\|^{2}=(\|\vec{u}\|+\|\vec{v}\|)^{2}\) (d) (Step 4) Use Step 3 to show that \(\|\vec{u}+\vec{v}\| \leq\|\vec{u}\|+\|\vec{v}\|\) for all pairs of vectors \(\vec{u}\) and \(\vec{v}\). (e) As an added bonus, we can now show that the Triangle Inequality \(|z+w| \leq|z|+|w|\) holds for all complex numbers \(z\) and \(w\) as well. Identify the complex number \(z=a+b i\) with the vector \(u=\langle a, b\rangle\) and identify the complex number \(w=c+d i\) with the vector \(v=\langle c, d\rangle\) and just follow your nose!
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