Questions: Alex strings lights on his 6-foot long fireplace mantel from one end to the other end during the holiday season. The curve that the string of lights creates is called a catenary. This curve is modeled by the equation y=10 cosh(x/10)-4.5 on the interval [-3,3], where x and y are measured in feet. At what height is the string of lights closest to the floor? Report your answer to one decimal place.

Solution Steps

To find the height at which the string of lights is closest to the floor, we need to determine the minimum value of the catenary function \( y = 10 \cosh \left(\frac{x}{10}\right) - 4.5 \) over the interval \([-3, 3]\). The function \(\cosh(x)\) is symmetric and has its minimum at \(x = 0\). Therefore, we will evaluate the function at \(x = 0\) to find the minimum height.

The height of the string of lights is modeled by the equation

\[ y = 10 \cosh \left(\frac{x}{10}\right) - 4.5 \]

where \(x\) is the horizontal distance from the center of the mantel.

To find the height closest to the floor, we need to evaluate the function at the critical point \(x = 0\), since the catenary function is symmetric and reaches its minimum at this point.

Substituting \(x = 0\) into the function:

\[ y(0) = 10 \cosh(0) - 4.5 = 10 \cdot 1 - 4.5 = 5.5 \]

Final Answer