Question

$$ \begin{array}{l}{\text { Tunnel Construction Your engineering firm is bidding for }} \\ {\text { the contract to construct the tunnel shown on the next page. The }} \\ {\text { tunnel is } 300 \mathrm{ft} \text { long and } 50 \mathrm{ft} \text { wide at the base. The cross }} \\ {\text { section is shaped like one arch of the curve } y=25 \cos (\pi x / 50)}\end{array} $$ $$ \begin{array}{l}{\text { Upon completion, the tunnel's inside surface (excluding the }} \\ {\text { roadway) will be treated with a waterproof sealer that costs }} \\ {\$ 1.75 \text { per square foot to apply. How much will it cost to apply }} \\ {\text { the sealer? } }\end{array} $$

Short answer

The total cost to apply the sealer can be computed by first calculating the area under the curve \(y=25 \cos(\pi x/50)\), then multiplying the area by the cost per square foot, and finally by the length of the tunnel. The total cost is then given by \(Total Cost = A * 300 * 1.75\), where \(A\) is the area under the curve.

Step-by-step solution

Setting Up the Integral

First, it is required to identify the shape of the tunnel’s cross-section as the curve \(y=25 \cos(\pi x/50)\). Now compute the area under this curve across its length, -25 to 25, using the formula for the area under a curve: \[A=\int_a^b|f(x)| dx\] The absolute value is used because distance is always positive. In this case, \(f(x)=y=25 \cos(\pi x/50)\) and the limits of integration are \(a=-25\) and \(b=25\), which correspond to half the width of the tunnel on either side of its centerline.

Computing the integral

Now compute the integral. Due to the function being symmetric about \(x=0\), the calculation can be simplified by using the property of definite integrals: \[\int_{-a}^{a}f(x) dx=2\int_{0}^{a}f(x) dx\]. Thus, \[A=2\int_{0}^{25}|25 \cos(\pi x/50)| dx = 50\int_{0}^{25}|\cos(\pi x/50)| dx\] This integral can be solved using regular methods or tools to get the total area.

Computing the total cost

Having found the total area of one cross section of the tunnel, we now need to find the total surface area of the tunnel, excluding the floor. The total surface area is just the area of one cross section (excluding the flat bottom part), times the length of the tunnel: \(Total Area = Area of Cross section * length = A*300\). Now multiply this surface area by the cost per square foot to find the total cost: \(Total Cost = Total Area * cost per square foot = A * 300 * 1.75\).