First we draw a picture: Letting \(c\) represent the cost of the lateral side, we can write an expression for the cost of materials: Since we know that \(V=\pi r^2h\), we can use this relationship to eliminate \(h\) (we could eliminate \(r\), but it’s a little easier if we eliminate \(h\), which appears in only one place in the above formula for cost). We find \begin{align*} C(r)&=2c\pi r\frac {V}{\pi r^2}+2Nc\pi r^2\\ &=\frac {2cV}{r}+2Nc\pi r^2. \end{align*}

Since we have expressed the cost as a function of a single variable, the problem becomes a“calculus problem”: Find the minimum of the function \(C\) on the interval \((0,\infty )\).

Although the function \(C\) is continuous on its domain, its domain is an open, unbounded interval and the Extreme Value Theorem does not apply; there is no guarantee that the minimum even exists. But we don’t give up! Our strategy will be to search for a critical number and check if the minimum is attained there. By solving the equation

we find that the point where \(r=\sqrt [3]{V/(2N\pi )}\) is the only critical number of \(C\). Since \(C''(r)=4cV/r^3+4Nc\pi \) is positive when \(r\) is positive, there is a local minimum at the critical number, and hence a global minimum since there is only one critical number.

Another way to reach this conclusion is to examine the sign of the derivative \(C'(r)\) on the interval \((0,\infty )\):

From this factorization it is clear that \(C'(r)<0\) on \((0,\sqrt [3]{V/(2N\pi )})\), and \(C'(r)>0\) on \((\sqrt [3]{V/(2N\pi )}, \infty )\). This means that the function \(C\) is decreasing on \((0,\sqrt [3]{V/(2N\pi )})\) and increasing on \((\sqrt [3]{V/(2N\pi )}, \infty )\), and, therefore, has a global minimum at the critical number.

Finally, since \(h=V/(\pi r^2)\), \begin{align*} \frac {h}{r}&=\frac {V}{\pi r^3}\\ &=\frac {V}{\pi (V/(2N\pi ))}\\ &=\answer [given]{2N}, \end{align*}

so the minimum cost occurs when the height \(h\) is \(2N\) times the radius. If, for example, there is no difference in the cost of materials, the height is twice the radius.

The first step is to convert the problem into a function maximization problem. The revenue for selling \(n\) items at \(x\) dollars is given by and the cost of producing \(n\) items is given by However, from the problem we see that the number of items sold is itself a function of \(x\), So profit is give by: \begin{align*} P(x) &= r(x) - c(x)\\ &= nx - (2000+0.5 n)\\ &=-10000x^2+25000x-12000. \end{align*}

We want to know the maximum value of the function \(P\) on the interval \([0,1.5]\). The Extreme Value Theorem now guarantees that the maximum is attained, since the function \(P\) is continuous on the closed interval, \([0,1.5]\). We have to find the critical numbers:

or which implies that the only critical number occurs at \(x=1.25\). Now we have to evaluate the function at the critical number and both endpoints, and compare the values:

\(P(0)=-12000\), \(P(1.25)=3625\), and \(P(1.5)=3000\). Note that \(P(1.25)\) is the maximum of these. Thus the maximum profit is $\(3625\), attained when we set the price at $\(1.25\) and sell \(7500\) items.

Alternately, we could apply the Second Derivative Test at the critical number and conclude that there must be a local maximum there, since \(P''(x)=-20000<0\). Since this is the only critical number, it must be a global maximum as well. We can confirm our results by looking at the graph of \(y=P(x)\):

Let \(R\) be the radius of the sphere, and let \(r\) and \(h\) be the base radius and height of the cone inside the sphere. Our goal is to maximize the volume of the cone: \(V_c=\pi r^2h/3\). The largest \(r\) could be is \(R\) and the largest \(h\) could be is \(2R\).

Notice that the function we want to maximize, \(\pi r^2h/3\), depends on two variables. Our next step is to find the relationship between \(r\) and \(h\) and use it to solve for one of the variables in terms of the other, so as to have a function of only one variable to maximize. In this problem, the condition is apparent in the figure below.

Apply the Pythagorean Theorem to the right triangle in the figure above to obtain

Solve for \(r^2\), since \(r^2\) is found in the formula for the volume of the cone, and find that Substitute this into the formula for the volume of the cone to find that By simplifying, we were able to express the volume of the cone as a function of a single variable, \(h\),

We want to maximize the function \(V_{\text {c}}\) on the interval \([0,2R]\). Since this is a continuous function on a closed interval, the maximum is attained either at a critical number or an end point. By solving the equation

we find that the function has at \(h=4R/3\) its only critical number, since \(h=0\) is an end point. We evaluate the function at all the critical numbers and end points,

The maximum is obviously attained at \(h=4R/3\) . Since the volume of the sphere is \((4/3)\pi R^3\), the fraction of the sphere occupied by the cone is

There are two immediate solutions that we could consider, each of which we will reject through “common sense.” First, we could minimize the distance by directly connecting the two locations with a straight line. However, this requires that all the wire be laid underwater, the most costly option. Second, we could minimize the underwater length by running a wire all 5000 ft. along the beach, directly across from the offshore facility. This has the undesired effect of having the longest distance of all, probably ensuring a nonminimal cost.

The optimal solution likely has the line being run along the ground for a while, then underwater, as the figure implies. We need to label our unknown distances: the distance run along the ground and the distance run underwater. Recognizing that the underwater distance can be measured as the hypotenuse of a right triangle, we can label our figure as follows

By choosing \(x\) as we did, we make the expression under the square root simple. We now create the cost function: So we have This function only makes sense on the interval \([0,5000]\). While we are fairly certain the endpoints will not give a minimal cost, we still evaluate \(c(x)\) at each to verify. We now find the critical numbers of \(c(x)\). We compute \(c'(x)\) as Recognize that this is never undefined. Setting \(c'(x)=0\) and solving for \(x\), we have: \begin{align*} -50+\frac {130x}{\sqrt {x^2+1000^2}} &= 0 \\ \frac {130x}{\sqrt {x^2+1000^2}} &= 50\\ \frac {130^2x^2}{x^2+1000^2} &= 50^2\\ 130^2x^2 &= 50^2(x^2+1000^2) \\ 130^2x^2-50^2x^2 &= 50^2\cdot 1000^2\\ (130^2-50^2)x^2 &= 50000^2\\ x^2 &= \frac {50000^2}{130^2-50^2}\\ x &= \frac {50000}{\sqrt {130^2-50^2}}\\ x &= \frac {50000}{120}, \end{align*}

Evaluating \(c(x)\) at \(x=416.67\) gives a cost of about \(\$370000\). The distance the power line is laid along land is \(5000-416.67 = 4583.33\mathrm {ft}\) and the underwater distance is \(\sqrt {416.67^2+1000^2} \approx 1083\mathrm {ft}\). We can confirm our results by looking at the graph of \(y=c(x)\):

Let \(x\) be the distance short of \(C\) where you turn off, the distance from \(B\) to \(C\). We want to minimize the total travel time. Recall that when traveling at constant velocity, time is distance divided by velocity.

You travel the distance from \(D\) to \(B\) at speed \(v\), and then the distance from \(B\) to \(A\) at speed \(w\). The distance from \(D\) to \(B\) is \(a-x\). By the Pythagorean theorem, the distance from \(B\) to \(A\) is

Hence the total time for the trip is We want to find the minimum value of \(T\) when \(x\) is between 0 and \(a\). As usual we set \(T'(x)=0\) and solve for \(x\). Write We find that Notice that \(a\) does not appear in the last expression, but \(a\) is not irrelevant, since we are interested only in critical values that are in \([0,a]\), and \(wb/\sqrt {v^2-w^2}\) is either in this interval or not. If it is, we can use the second derivative to test it: Since this is always positive there is a local minimum at the critical number, and so it is a global minimum as well.

If the critical value is not in \([0,a]\) it is larger than \(a\). In this case the minimum must occur at one of the endpoints. We can compute \begin{align*} T(0)&={a\over v}+{b\over w} \\ T(a)&={\sqrt {a^2+b^2}\over w} \end{align*}

but it is difficult to determine which of these is smaller by direct comparison. If, as is likely in practice, we know the values of \(v\), \(w\), \(a\), and \(b\), then it is easy to determine this. With a little cleverness, however, we can determine the minimum in general. We have seen that \(T''(x)\) is always positive, so the derivative \(T'(x)\) is always increasing. We know that at \(wb/\sqrt {v^2-w^2}\) the derivative is zero, so for values of \(x\) less than that critical value, the derivative is negative. This means that \(T(0)>T(a)\), so the minimum occurs when \(x=a\).

So the upshot is this: If you start farther away from \(C\) than \(wb/\sqrt {v^2-w^2}\) then you always want to cut across the sand when you are a distance \(wb/\sqrt {v^2-w^2}\) from point \(C\). If you start closer than this to \(C\), you should cut directly across the sand.