Hello,
I got this problem here which I can't really solve

A Contractor is given the task of calculating the cost of a tunnel through a hill. The shape of the tunnel's cross section is made up of two curves as shown in the diagram. The upper curve is parabolic while the lower curve is sinusoidal (half cycle of sine curve 2 m deep). The tunnel is 10 m wide 6 m high at the wiedest points as shown in the diagram.

The Task: Given that the tunnel is 1.25 km long and cast $120/m^3 to excavate, estimate the cost of excavation to the nearest$ 1000.

2. Well, you can find the cross sectional area of the two curves.

The sine curve has equation:

$\displaystyle y = -\sin\left(\dfrac{\pi x}{10}\right)$

The parabola has equation

y = ax(x-12)

Putting (6, 4), we get a = -1/9

Hence, equation of parabola: $\displaystyle y = -\dfrac{x}{9} (x - 12) = -\dfrac{x^2}{9} + \dfrac{4x}{3}$

Find the Area using:

$\displaystyle \displaystyle \int^{12}_0 -(-\sin\left(\dfrac{\pi x}{10}\right)) + (-\dfrac{x^2}{9} + \dfrac{4x}{3})\ dx$

Then, find the volume using

$\displaystyle V = Ad$

A is the cross sectional area and d is the depth of the prism.

The price will become trivial then.

3. Thanks for the help

i understand the fuction for the sine curve but the parabola equation makes no sense to me?! isn't a parabola function y=a(x-h)^2+k

regards

4. That too, but I prefered using the form

$\displaystyle y = a(x-b)(x-c)$

where b and c are the roots of the parabola.

Since the roots occur at x = 0 and x = 12, this becomes y = a(x -0)(x - 12) = ax(x-12)

You can go with y = a(x-h)^2 +k too.

From the graph, we know that h = 6 and k = 4 then we use the point (0, 0)

0 = a(0 - 6)^2 + 4

a = -4/36 = -1/9