7.6: Exercises

7.1. Find the numbers. The sum of two positive number is 20 . Find the numbers

(a) if their product is a maximum

(b) if the sum of their squares is a minimum

(c) if the product of the square of one and the cube of the other is a maximum.

7.2. Distance, velocity and acceleration. A tram ride departs from its starting place at \(t=0\) and travels to the end of its route and back. Its distance from the terminal at time \(t\) can be approximately described by the expression

\[S(t)=4 t^{3}(10-t) \nonumber \]

where \(t\) is in minutes, \(0<t<10\), and \(S\) is distance in meters.

(a) Find the velocity as a function of time.

(b) When is the tram moving at the fastest rate?

(c) At what time does it get to the furthest point away from its starting position?

(d) Sketch the acceleration, the velocity, and the position of the tram on the same set of axes.

7.3. Distance of two cars. At 9A.M., \(\operatorname{car} B\) is \(25 \mathrm{~km}\) west of car \(A\). Car \(A\) then travels to the south at \(30 \mathrm{~km} / \mathrm{h}\) and car \(B\) travels east at \(40 \mathrm{~km} / \mathrm{h}\). When are they closest to each other and what is this distance?

7.4. Cannonball movement. A cannonball is shot vertically upwards from the ground with initial velocity \(v_{0}=15 \mathrm{~m} / \mathrm{sec}\). The height of the ball, \(y\) (in meters), as a function of the time, \(t\) (in sec) is given by

\[y=v_{0} t-4.9 t^{2} \nonumber \]

Determine the following:

(a) the time at which the cannonball reaches its highest point

(b) the velocity and acceleration of the cannonball at \(t=0.5 \mathrm{~s}\), and \(t=\) \(1.5 \mathrm{~s}\)

(c) the time at which the cannonball hits the ground.

7.5. Net nutrient increase rate. In Example 7.2, we considered the net rate of increase of nutrients in a spherical cell of radius \(r\). Here we further explore this problem.

(a) Draw a sketch of \(N(r)\) based on Equation (7.1). Use your sketch to verify that this function has a local maximum.

(b) Use the first derivative test to show that the critical point \(r=2 k_{1} / k_{2}\) is a local maximum.

7.6. Nutrient increase in cylindrical cell. Consider a long skinny cell in the shape of a cylinder with radius \(r\) and a fixed length \(L\). The volume and surface area of such a cell (neglecting endcaps) are \(V=\pi r^{2} L=K\) and \(S=2 \pi r L\)

(a) Adapt the formula for net rate of increase of nutrients \(N(t)\) for a spherical cell Equation (7.1) to the case of a cylindrical cell.

(b) Find the radius of the cylindrical cell that maximizes \(N(t)\). Be sure to verify that you have found a local maximum.

7.7. Cylinder of minimal surface area. In this exercise we continue to explore Example 7.3.

(a) Reason that the surface area of the cylinder, \(S(r)=2 \frac{K}{r}+2 \pi r^{2}\) is a function that has a local minimum using curve-sketching.

(b) Use the first derivative test to show that \(r=\left(\frac{K}{2 \pi}\right)^{1 / 3}\) is a local minimum for \(S(r)\).

(c) Show the algebra required to find the value of \(L\) corresponding to this \(r\) value and show that \(L / r=2\).

7.8. Dimensions of a box. A closed 3-dimensional box is to be constructed in such a way that its volume is \(4500 \mathrm{~cm}^{3}\). It is also specified that the length of the base is 3 times the width of the base.

Determine the dimensions of the box which satisfy these conditions and have the minimum possible surface area. Justify your answer.

7.9. Dimensions of a box. A box with a square base is to be made so that its diagonal has length 1 ; see Figure \(7.9\).

(a) What height \(y\) would make the volume maximal?

(b) What is the maximal volume? (hint: a box having side lengths \(\ell, w, h\) has diagonal length \(D\) where \(D^{2}=\ell^{2}+w^{2}+h^{2}\) and volume \(V=\) \(\ell w h)\)

Figure 7.9: Figure for Exercise 9; box with a square base.

7.10. Minimum distance. Find the minimum distance from a point on the positive \(x\)-axis \((a, 0)\) to the parabola \(y^{2}=8 x\).

7.11. The largest garden. You are building a fence to completely enclose part of your backyard for a vegetable garden. You have already purchased material for a fence of length \(100 \mathrm{ft}\). What is the largest rectangular area that this fence can enclose?

7.12. Two gardens. A fence of length \(100 \mathrm{ft}\) is to be used to enclose two gardens. One garden is to have a circular shape, and the other to be square. Determine how the fence should be cut so that the sum of the areas inside both gardens is as large as possible.

7.13. Dimensions of an open box. A rectangular piece of cardboard with dimension \(12 \mathrm{~cm}\) by \(24 \mathrm{~cm}\) is to be made into an open box (i.e., no lid) by cutting out squares from the corners and then turning up the sides. Find the size of the squares that should be cut out if the volume of the box is to be a maximum.

7.14. Alternate solution to Kepler’s wine barrel. In this exercise we follow an alternate approach to the most economical wine barrel problem posed by Kepler (as in Example 7.4). Through this approach, we find the proportions (height:radius) of the cylinder that minimizes the length \(L\) of the wet rod in Figure \(7.3\) for a fixed volume.

(a) Explain why minimizing \(L\) is equivalent to minimizing \(L^{2}\) in Equation (7.4)

(b) Explain how Equation (7.3) can be used to specify a constraint for this problem. (hint: consider the volume, \(V\) to be fixed and show that you can solve for \(r^{2}\) ).

(c) Use your result in (c) to eliminate \(r\) from the formula for \(L^{2}\). Now \(L^{2}(h)\) depends only on the height of the cylindrical wine barrel.

(d) Use calculus to find any local minima for \(L^{2}(h)\). Be sure to verify that your result is a minimum.

(e) Find the corresponding value of \(r\) using your result in (b).

(f) Find the ratio \(h / r\). You should obtain the same result as in Equation (7.5).

7.15. Rectangle with largest area. Find the side lengths, \(x\) and \(y\), of the rectangle with largest area whose diameter \(L\) is given (hint: eliminate one variable using the constraint. To simplify the derivative, consider that critical points of \(A\) would also be critical points of \(A^{2}\), where \(A=\) \(x y\) is the area of the rectangle. If you have already learned the chain rule, you can use it in the differentiation).

7.16. Shortest path. Find the shortest path that would take a milk-maid from her house at \((10,10)\) to fetch water at the river located along the \(x\)-axis and then to the thirsty cow at \((3,5)\).

7.17. Water and ice. Why does ice float on water? Because the density of ice is lower! In fact, water is the only common liquid whose maximal density occurs above its freezing temperature. This phenomenon favours the survival of aquatic life by preventing ice from forming at the bottoms of lakes. According to the Handbook of Chemistry and Physics, a mass of water that occupies one liter at \(0^{\circ} \mathrm{C}\) occupies a volume (in liters) of

\[V=-a T^{3}+b T^{2}-c T+1 \nonumber \]

at \(T^{\circ} \mathrm{C}\) where \(0 \leq T \leq 30\) and where the coefficients are

\[a=6.79 \times 10^{-8}, b=8.51 \times 10^{-6}, c=6.42 \times 10^{-5} . \nonumber \]

Find the temperature between \(0^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C}\) at which the density of water is the greatest. (hint: maximizing the density is equivalent to minimizing the volume. Why is this?).

7.18. Drug doses and sensitivity. The reaction \(R(x)\) of a patient to a drug dose of size \(x\) depends on the type of drug. For a certain drug, it was determined that a good description of the relationship is:

\[R(x)=A x^{2}(B-x) \nonumber \]

where \(A\) and \(B\) are positive constants. The sensitivity of the patient’s body to the drug is defined to be \(R^{\prime}(x)\).

(a) For what value of \(x\) is the reaction a maximum, and what is that maximum reaction value?

(b) For what value of \(x\) is the sensitivity a maximum? What is the maximum sensitivity?

7.19. Thermoregulation in a swarm of bees. In the winter, honeybees sometimes escape the hive and form a tight swarm in a tree, where, by shivering, they can produce heat and keep the swarm temperature elevated.

Heat energy is lost through the surface of the swarm at a rate proportional to the surface area \(\left(k_{1} S\right.\) where \(k_{1}>0\) is a constant). Heat energy is produced inside the swarm at a rate proportional to the mass of the swarm (which you may take to be a constant times the volume). We assume that the heat production is \(k_{2} V\) where \(k_{2}>0\) is constant.

Swarms that are not large enough may lose more heat than they can produce, and then they die. The heat depletion rate is the loss rate minus the production rate. Assume that the swarm is spherical. Find the size of the swarm for which the rate of depletion of heat energy is greatest.

7.20. Cylinder inside a sphere. Work through the steps for the calculations and classification of critical point(s) in Example G.2, that is, find the dimensions of the largest cylinder that would fit in a sphere of radius \(R\).

7.21. Circular cone circumscribed about a sphere. A right circular cone is circumscribed about a sphere of radius 5 . Find the dimension of this cone if its volume is to be a minimum.

Note: this is a rather challenging geometric problem.

7.22. Optimal reproductive strategy. Animals that can produce many healthy babies that survive to the next generation are at an evolutionary advantage over other, competing, species. However, too many young produce a heavy burden on the parents (who must feed and care for them). If this causes the parents to die, the advantage is lost. Further, competition of the young with one another for food and parental attention jeopardizes the survival of these babies.

Suppose that the evolutionary Advantage \(A\) to the parents of having litter size \(x\) is

\[A(x)=a x-b x^{2} . \nonumber \]

Suppose that the Cost \(C\) to the parents of having litter size \(x\) is

\[C(x)=m x+e . \nonumber \]

The Net Reproductive Gain \(G\) is defined as

\[G=A-C . \nonumber \]

(a) Explain the expressions for \(A, C\) and \(G\).

(b) At what litter size is the advantage, \(A\), greatest?

(c) At what litter size is there least cost to the parents?

(d) At what litter size is the Net Reproductive Gain greatest?

7.23. Behavioural Ecology. Social animals that live in groups can spend less time scanning for predators than solitary individuals. However, they waste time fighting with the other group members over the available food. There is some group size at which the net benefit is greatest because the animals spend the least time on these unproductive activities - and thus can spend time on feeding, mating, etc.

Assume that for a group of size \(x\), the fraction of time spent scanning for predators is

\[S(x)=A \frac{1}{(x+1)} \nonumber \]

and the fraction of time spent fighting with other animals over food is

\[F(x)=B(x+1)^{2} \nonumber \]

where \(A, B\) are constants.

Find the size of the group for which the time wasted on scanning and fighting is smallest.

7.24. Logistic growth. Consider a fish population whose density (individuals per unit area) is \(N\), and suppose this fish population grows logistically, so that the rate of growth \(R\) satisfies

\[R(N)=r N(1-N / K) \nonumber \]

where \(r\) and \(K\) are positive constants.

Figure 7.10: In logistic growth, the population growth rate \(G\) depends on population size \(N\) as shown here.

(a) Sketch \(R\) as a function of \(N\) or explain Figure 7.10.

(b) Use a first derivative test to justify the claim that \(N=K / 2\) is a local maximum for the function \(G(N)\).

7.25. Logistic growth with harvesting. Consider a fish population of density \(N\) growing logistically, i.e. with rate of growth \(R(N)=\) \(r N(1-N / K)\) where \(r\) and \(K\) are positive constants. The rate of harvesting (i.e. removal) of the population is

\[h(N)=q E N \nonumber \]

where \(E\), the effort of the fishermen, and \(q\), the catchability of this type of fish, are positive constants.

At what density of fish does the growth rate exactly balance the harvesting rate? This density is called the maximal sustainable yield: MSY.

7.26. Conservation of a harvested population. Conservationists insist that the density of fish should never be allowed to go below a level at which growth rate of the fish exactly balances with the harvesting rate. At this level, the harvesting is at its maximal sustainable yield. If more fish are taken, the population keeps dropping and the fish eventually go extinct.

What level of fishing effort should be used to lead to the greatest harvest at this maximal sustainable yield?

Note: you should first complete the Exercise 25.

7.27. Optimal foraging. Consider Example \(7.7\) for the optimal foraging model.

(a) Show that the parameter \(k\) in Equation (7.6) is the time at which \(f(t)=\) \(E_{\max } / 2\)

(b) Consider panel (5) of Figure 7.7. Show that a function such as a Hill function would have the shape shown in that sketch. Interpret any parameters in that function.

(c) Use the quotient rule to calculate the derivative of the function \(R(t)\) given by Equation (7.7) and show that you get Equation (7.8).

(d) Fill in the missing steps in the calculation in Equation (7.9) to find the optimal value of \(R(t)\).

7.28. Rate of net energy gain while foraging and traveling. Animals spend energy in traveling and foraging. In some environments this energy loss is a significant portion of the energy budget. In such cases, it is customary to assume that to survive, an individual would optimize the rate of net energy gain, defined as

\[Q(t)=\frac{\text { Net energy gained }}{\text { total time spent }}=\frac{\text { Energy gained }-\text { Energy lost }}{\text { total time spent }} \nonumber \]

Assume that the animal spends \(p\) energy units per unit time in all activities (including foraging and traveling). Assume that the energy gain in the patch ("patch energy function") is given by Equation (7.6). Find the optimal patch time, that is the time at which \(Q(t)\) is maximized in this scenario.

7.29. Maximizing net energy gain: Suppose that the situation requires an animal to maximize its net energy gained \(E(t)\) defined as

\[\begin{aligned} E(t)= & \text { energy gained while foraging } \\ & \text {-energy spent while foraging and traveling. } \end{aligned} \nonumber \]

(This means that \(E(t)=f(t)-r(t+\tau)\) where \(r\) is the rate of energy spent per unit time and \(\tau\) is the fixed travel time).

Assume as before that the energy gained by foraging for a time \(t\) in the food patch is \(f(t)=E_{\text {max }} t /(k+t)\).

(a) Find the amount of time \(t\) spent foraging that maximizes \(E(t)\).

(b) Indicate a condition of the form \(k<?\) that is required for existence of this critical point.