Calculus - Finding the Linear Equation which equals area

Question

I am really stuck on this question... I think it involves finding integration but am struggling to understand the concepts involved. I have attempted the equation $y=70$ through trial and error, integrated that, then found the area between the curves from $0-378.2$ (the point at which the "earth" changes from being removed to being injected) and $378.2-1000$ and have found areas which are not equal? I'm wondering whether I have to purely try another method or continue with trial and error?

Question is as follows:

A road is to implemented at $0$ degree elevation. The graph attached displays the "hills" located across the predicted road.

$(a)$ A linear function of $y=x$ needs to be found so that it divides the following function into equal area so that the "earth" taken out of the hills equals the "earth" needed to fill the valleys.

$(b)$ The government decides that if they extend the elevation to no more then $10$ degree (HINT: Can be less) then it can save in "earth" removal. Define this function

Answer 1

a) Your problem amounts to finding a value $ c $ such that $$ \int_0^{1000} (f(x) - c)\, \mathrm d x = 0, $$ $ f $ being the function describing the road profile. It should be a simple task to work out the definite integral; you are then left with a trivial equation that has to be solved for $ c $.

b) This is a tad trickier, and I'm not actually sure I understood it right, so I'll proceed in the assumption that this is indeed an optimization problem. The road is now allowed to slope and is therefore described by the equation $ y = a x + b $, so you now need to fix two parameters as opposed to just one. The equations you want to solve are thus two: the first one is simply a restatement of the one above, namely $$ \int_0^{1000} (f(x) - ax - b)\, \mathrm d x = 0, $$ which gives you a relation between $ a $ and $ b $. The requirement that the digging be minimized is equivalent to minimizing the $ \mathrm L^2 $ norm $$ \int_0^{1000} (f(x) - ax - b)^2\, \mathrm d x = 0 $$ under the restriction $ |a| \le \tan 10° \approx 0.176 $. I've fed this info into Mathematica and found values $$ \begin{cases}a \approx -0.06\\b = 500\, a + 45 + 10 \log 11 + \frac{\sin 20°}{2}. \end{cases} $$ The exact expression of $ a $ is too hideous to type out explicitly. Also, I didn't check these results for correctness, but at least the method of solution should suffice.

Answer 2

To find a minimal-digging line $g(x)=a x+b$, build a linear system \begin{align} \int_{0}^{1000}\frac{\partial}{\partial a} \left((f(x)-(a x+b))^2\right) dx &=0 \\ \int_{0}^{1000}\frac{\partial}{\partial b} \left((f(x)-(a x+b))^2\right) dx &=0 \end{align}

which has a solution for $a$ and $b$: \begin{align} a &= 0.1097-0.072 \ln(11)+0.003\sin(20)+0.0003\cos(20) \\ &\approx -0.060087 \\ b &= 46 \ln(11)-\sin(20)-\frac{197}{20}-\frac{3}{20}\cos(20) \\ &\approx 99.479025. \end{align}

The slope is $\arctan(a)\frac{180}{\pi}\approx-3.44$°: