Multivariable Calculus, Linear Algebra, and Differential Equations by Grossman, Stanley I | 9780123043801 | 123043808

Multivariable Calculus, Linear Algebra, and Differential Equations [EXP-51866]

437 SOLVED PROBLEMS

Question: 12.6.2

(A Competitive Model) Consider the system x1′(t) = 3×1(t) – x2(t), x2′(t) = -2×1(t) + 2×2(t). Here, an increase in the population of one species causes a decline in the growth rate of another. Suppose that the initial populations are x1(0) = 90 and x2(0) = 150. Find the populations of both species

Verified Answer:

We have

A=\left(\begin{array}{rr}3 & -1...

Question: 11.4.5

(A Mixing Problem) Consider a tank holding 100 gallons of water in which is dissolved SO pounds of salt. Suppose that 2 gallons of brine, each containing 1 pound of dissolved salt, run into the tank per minute, and the mixture, kept uniform by highspeed stirring, runs out of the tank at the rate of

Verified Answer:

Let

x(t)

be the number of pounds of...

Question: 12.6.4

(A Model of Species Cooperation-Symbiosis) Consider the symbiotic model governed by the system X1′(t) = -1/2 x1(t) + x2(t), x2′(t) = 1/4 x1(t) – 1/2 x2(t). In this model the population of each species increases proportionally to the population of the other and decreases proportionally to its own

Verified Answer:

Here

A=\left(\begin{array}{rr}-\frac{1}{2} ...

Question: 12.6.3

(A Predator-Prey Model) Consider the predator-prey model governed by the system X1′(t) = X1(t) + X2(t), X2′(t) = -X1(t) + X2(t). If the initial populations are x1(0) = x2(0) = 1000, determine the populations of the two species for t > 0.

Verified Answer:

Here

A=\left(\begin{array}{rr}1 & 1 \\ ...

Question: 11.2.6

(Carbon Dating) Carbon dating is a technique used by archaeologists, geologists, and others who want to estimate the ages of certain artifacts and fossils they uncover. The technique is based on certain properties of the carbon atom. In its natural state the nucleus of the carbon atom 12c has 6

Verified Answer:

Let

M(t)

denote the mass of

...

Question: 11.2.4

(Newton’s Law of Cooling) Newton’s law of cooling states that the rate of change of the temperature difference between an object and its surrounding medium is proportional to the temperature difference. If D(t) denotes this temperature difference at time t and if a denotes the constant of

Verified Answer:

The original value is

D_{0}

. We are...

Question: 11.2.3

(Population Growth) A bacterial population is growing continuously at a rate equal to 10% of its population each day. Its initial size is 10,000 organisms. How many bacteria are present after 10 days? After 30 days?

Verified Answer:

Since the percentage growth of the population is [...

Question: 2.7.5

A 1000-kg rocket carrying 2000 kg of fuel is motionless in space. Its engine starts and its exhaust velocity is 0.7 km/sec. What is its speed when all its fuel is consumed?

Verified Answer:

Here

\mathbf{v}_{0}=\mathbf{0}

. The...

Question: 2.8.5

A 1500-kg race car is driven at a speed of 150 km/hr on a circular race track of radius 50 m. What frictional force must be exerted by the tires on the road surface to keep the car from skidding?

Verified Answer:

The frictional force exerted by the tires must be ...

Question: 2.7.2

A cannon whose muzzle is tilted upward at an angle of 30° shoots a ball at an initial velocity of 600 m/sec.(a) Find the position vector at all times t >= 0.(b) How much time does the ball spend in the air?(c) How far does the cannonball travel? (d) How high does the ball get? (e) How far from the

Verified Answer:

(a) From Example 2.1.6 we find, resolving the init...