Solved: ?A stone is dropped into a lake, creating a circular ripple that travels outward

A particle moves according to a law of motion \(s=f(t)\), \(t \geqslant 0\), where \(t\) is measured in seconds and \(s\) in feet. (a) Find the velocity at time \(t\). (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time \(t\) and after 1 second. (h) Graph the position, velocity, and acceleration functions for \(0 \leqslant t \leqslant 6\). (i) When is the particle speeding up? When is it slowing down? \(f(t)=t^{3}-9 t^{2}+24 t\) Equation Transcription: ? ?? Text Transcription: s=f(t) t geqslant 0 t s t t 0 leqslant t leqslant 6 f(t)=t^3-9t^2+24t

A particle moves according to a law of motion \(s=f(t)\), \(t \geqslant 0\), where \(t\) is measured in seconds and \(s\) in feet. (a) Find the velocity at time \(t\). (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time \(t\) and after 1 second. (h) Graph the position, velocity, and acceleration functions for \(0 \leqslant t \leqslant 6\). (i) When is the particle speeding up? When is it slowing down? \(f(t)=\frac{9 t}{t^{2}+9}\) Equation Transcription: ? ?? Text Transcription: s=f(t) t geqslant 0 t s t t 0 leqslant t leqslant 6 f(t)=9t/t^2+9

A particle moves according to a law of motion \(s=f(t)\), \(t \geqslant 0\), where \(t\) is measured in seconds and \(s\) in feet. (a) Find the velocity at time \(t\). (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time \(t\) and after 1 second. (h) Graph the position, velocity, and acceleration functions for \(0 \leqslant t \leqslant 6\). (i) When is the particle speeding up? When is it slowing down? \(f(t)=\sin (\pi t / 2)\) Equation Transcription: ? ?? Text Transcription: s=f(t) t geqslant 0 t s t t 0 leqslant t leqslant 6 f(t)=sin (pi t/2)

A particle moves according to a law of motion \(s=f(t)\), \(t \geqslant 0\), where \(t\) is measured in seconds and \(s\) in feet. (a) Find the velocity at time \(t\). (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time \(t\) and after 1 second. (h) Graph the position, velocity, and acceleration functions for \(0 \leqslant t \leqslant 6\). (i) When is the particle speeding up? When is it slowing down? \(f(t)=t^{2} e^{-t}\) Equation Transcription: ? ?? Text Transcription: s=f(t) t geqslant 0 t s t t 0 leqslant t leqslant 6 f(t)=t^2 e^-t

Graphs of the velocity functions of two particles are shown, where \(t\) is measured in seconds. When is each particle speeding up? When is it slowing down? Explain. Equation Transcription: Text Transcription: t

Graphs of the position functions of two particles are shown, where \(t\) is measured in seconds. When is the velocity of each particle positive? When is it negative? When is each particle speeding up? When is it slowing down? Explain. Equation Transcription: Text Transcription: t

Suppose that the graph of the velocity function of a particle is as shown in the figure, where \(t\) is measured in seconds. When is the particle traveling forward (in the positive direction)? When is it traveling backward? What is happening when \(5<t<7\)? Equation Transcription: Text Transcription: t 5<t<7

For the particle described in Exercise 7, sketch a graph of the acceleration function. When is the particle speeding up? When is it slowing down? When is it traveling at a constant speed?

The height (in meters) of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of 24.5 m/s is \(h=2+24.5 t-4.9 t^{2}\) after t seconds. (a) Find the velocity after 2 s and after 4 s (b) When does the projectile reach its maximum height? (c) What is the maximum height? (d) When does it hit the ground? (e) With what velocity does it hit the ground?

If a ball is thrown vertically upward with a velocity of \(80 \mathrm{ft} / \mathrm{s}\), then its height after \(t\) seconds is \(s=80 t-16 t^{2}\). (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is \(96 \mathrm{ft}\) above the ground on its way up? On its way down? Equation Transcription: Text Transcription: 80 ft/s t s=80t-16t^2 96 ft

If a rock is thrown vertically upward from the surface of Mars with velocity \(15 \mathrm{~m} / \mathrm{s}\), its height after t seconds is \(h=15 t-1.86 t^2\). (a) What is the velocity of the rock after \(2 \mathrm{~s}\)? (b) What is the velocity of the rock when its height is \(25 \mathrm{~m}\) on its way up? On its way down?

A particle moves with position function \(s=t^{4}-4 t^{3}-20 t^{2}+20 t \quad t \geqslant 0\) (a) At what time does the particle have a velocity of \(20 \mathrm{~m} / \mathrm{s}\)? (b) At what time is the acceleration 0 ? What is the significance of this value of \(t\) ? Equation Transcription: ? Text Transcription: s=t^4-4t^3-20t^2+ 20t t geqslant 0 20 m/s t

(a) A company makes computer chips from square wafers of silicon. A process engineer wants to keep the side length of a wafer very close to 15 mm and needs to know how the area A(x) of a wafer changes when the side length x changes. Find \(A^{\prime}(15)\) and explain its meaning in this situation. (b) Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount \(\Delta x\). How can you approximate the resulting change in area \(\Delta A\) if \(\Delta x\) is small?

(a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dV/dx when x = 3 mm and explain its meaning. (b) Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. Explain geometrically why this result is true by arguing by analogy with Exercise 13(b).

(a) Find the average rate of change of the area of a circle with respect to its radius r as r changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when r = 2. (c) Show that the rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount \(\Delta r\). How can you approximate the resulting change in area \(\Delta A\) if \(\Delta r\) is small?

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of \(60 \mathrm{~cm} / \mathrm{s}\). Find the rate at which the area within the circle is increasing after (a) \(1 \mathrm{~s}\), (b) \(3 \mathrm{~s}\), and (c) \(5 \mathrm{~s}\). What can you conclude? Equation Transcription: Text Transcription: 60 cm/s 1 s 3 s 5 s

A spherical balloon is being inflated. Find the rate of increase of the surface area \(\left(S=4 \pi r^{2}\right)\) with respect to the radius \(r\) when \(r\) is (a) \(1 \mathrm{ft}\), (b) \(2 \mathrm{ft}\), and (c) \(3 \mathrm{ft}\). What conclusion can you make? Equation Transcription: Text Transcription: (S=4r^2) r r 1 ft 2 ft 3 ft

(a) The volume of a growing spherical cell is \(V=\frac{4}{3} \pi r^{3}\), where the radius \(r\) is measured in micrometers \(\left(1 \mu \mathrm{m}=10^{-6} \mathrm{~m}\right)\). Find the average rate of change of \(V\) with respect to \(r\) when\(r\) changes from (i) \(5 \ to \ 8 \mu \mathrm{m}\) (ii) \(5 \ to \ 6 \mu \mathrm{m}\) (iii) \(5 \ to \ 5.1 \mu \mathrm{m}\) (b) Find the instantaneous rate of change of \(V\) with respect to \(r\) when \(r=5 \mu \mathrm{m}\). (c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise 15 (c). Equation Transcription: Text Transcription: V=4/3 pi r^3 r 1 mu m=10^-6 m) V r r 5 to 8 mu m 5 to 6 mu m 5 to 5.1 mu m V r r=5 mu m

The mass of the part of a metal rod that lies between its left end and a point \(x\) meters to the right is \(3 x^{2} \mathrm{~kg}\). Find the linear density (see Example 2) when \(x\) is (a) \(1 \mathrm{~m}\), (b) \(2 \mathrm{~m}\), and (c) \(3 \mathrm{~m}\). Where is the density the highest? The lowest? Equation Transcription: Text Transcription: x 3x^2 kg x 1 m 2 m 3 m

If a cylindrical water tank holds 5000 gallons, and the water drains from the bottom of the tank in 40 minutes, then Torricelli's Law gives the volume \(V\) of water remaining in the tank after \(t\) minutes as \(V=5000\left(1-\frac{1}{40} t\right)^{2} \quad 0 \leqslant t \leqslant 40\) Find the rate at which water is draining from the tank after (a) \(5 \mathrm{~min}\), (b) \(10 \mathrm{~min}\), (c) \(20 \mathrm{~min}\), and (d) \(40 \mathrm{~min}\). At what time is the water flowing out the fastest? The slowest? Summarize your findings. Equation Transcription: ?? Text Transcription: V t V=5000(1-1/40t)^2 0 leqslant t leqslant 40 5 min 10 min 20 min 40 min

The quantity of charge \(Q\) in coulombs (C) that has passed through a point in a wire up to time \(t\) (measured in seconds) is given by \(Q(t)=t^{3}-2 t^{2}+6 t+2\). Find the current when (a) \(t=0.5 \mathrm{~s}\) and (b) \(t=1 \mathrm{~s}\). (See Example 3. The unit of current is an ampere \([1 \mathrm{~A}=1 \mathrm{C} / \mathrm{s}]\).) At what time is the current lowest? Equation Transcription: Text Transcription: Q t Q(t)=t^3-2t^2+6t+2 t=0.5 s t=1 s [1 A=1 C/s]

Newton's Law of Gravitation says that the magnitude \(F\) of the force exerted by a body of mass \(m\) on a body of mass \(M\) is \(F=\frac{G m M}{r^{2}}\) where \(G\) is the gravitational constant and \(r\) is the distance between the bodies. (a) Find \(d F / d r\) and explain its meaning. What does the minus sign indicate? (b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of \(2 \mathrm{~N} / \mathrm{km}\) when \(r=20,000 \mathrm{~km}\). How fast does this force change when \(r=10,000 \mathrm{~km}\) ? Equation Transcription: Text Transcription: F m M F=GmM/r^2 G r dF/dr 2 N/km r=20,000 km r=10,000 km

The force \(F\) acting on a body with mass \(m\) and velocity \(v\) is the rate of change of momentum: \(F=(d / d t)(m v)\). If \(m\) is constant, this becomes \(F=m a\), where \(a=d v / d t\) is the acceleration. But in the theory of relativity, the mass of a particle varies with \(v\) as follows: \(m=m_{0} / \sqrt{1-v^{2} / c^{2}}\) where \(m_{0}\) is the mass of the particle at rest and \(c\) is the speed of light. Show that \(F=\frac{m_{0} a}{\left(1-v^{2} / c^{2}\right)^{3 / 2}}\) Equation Transcription: Text Transcription: F v F=(d/dt)(mv) m F=ma a=dv/dt v m=m_0/ sqrt 1-v^2/c^2 m_0 c F=m_0a/(1-v^2/c^2)^3/2

Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about \(2.0 \mathrm{~m}\) and at high tide it is about \(12.0 \mathrm{~m}\). The natural period of oscillation is a little more than 12 hours and on a day in June, high tide occurred at 6 : 45 AM. This helps explain the following model for the water depth D (in meters) as a function of the time t (in hours after midnight) on that day: \(D(t)=7+5 \cos [0.503(t-6.75)]\) How fast was the tide rising (or falling) at the following times? (a) 3: 00 AM (b) 6:00 AM (c) 9:00 AM (d) Noon

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: \(P V=C\). (a) Find the rate of change of volume with respect to pressure. (b) A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain. (c) Prove that the isothermal compressibility (see, Example 5) is given by \(\beta=1 / P\). Equation Transcription: Text Transcription: PV=C beta=1/P

If, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value [A] = [B] = a moles/L, then, \([\mathrm{C}]=a^{2} k t /(a k t+1)\) where k is a constant. (a) Find the rate of reaction at time t. (b) Show that if x = [C] then \(\frac{d x}{d t}=k(a-x)^{2}\) (c) What happens to the concentration as \(t \rightarrow \infty\) ? (d) What happens to the rate of reaction as \(t \rightarrow \infty\) ? (e) What do the results of parts (c) and (d) mean in practical terms?

In Example 6 we considered a bacteria population that doubles every hour. Suppose that another population of bacteria triples every hour and starts with 400 bacteria. Find an expression for the number n of bacteria after t hours and use it to estimate the rate of growth of the bacteria population after 2.5 hours.

The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function \(n=f(t)=\frac{a}{1+b e^{-0.7 t}}\) where \(t\) is measured in hours. At time \(t=0\) the population is 20 cells and is increasing at a rate of \(12 \ \mathrm{cells}\) / hour. Find the values of \(a\) and \(b\). According to this model, what happens to the yeast population in the long run? Equation Transcription: Text Transcription: n=f(t)=a/1+be^-0.7t t t=0 1 12 cells/ hour a b

The table gives the world population \(P(t)\), in millions, where \(t\) is measured in years and \(t=0\) corresponds to the year 1900 . (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a graphing calculator or computer to find a cubic function (a third-degree polynomial) that models the data. (c) Use your model in part (b) to find a model for the rate of population growth. (d) Use part (c) to estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a). (e) In Section 1.1 we modeled ????(????) with the exponential function \(f(t)=\left(1.43653 \times 10^{9}\right) \cdot(1.01395)^{t}\) Use this model to find a model for the rate of population growth. (f ) Use your model in part (e) to estimate the rate of growth in 1920 and 1980. Compare with your estimates in parts (a) and (d). (g) Estimate the rate of growth in 1985. Equation Transcription: Text Transcription: P(t) t t=0 f(t)= (1.43653 x 10^9) dot (1.01395)^t

The table shows how the average age of first marriage of Japanese women has varied since 1950. (a) Use a graphing calculator or computer to model these data with a fourth-degree polynomial. (b) Use part (a) to find a model for \(A^{\prime}(t)\). (c) Estimate the rate of change of marriage age for women in 1990. (d) Graph the data points and the models for \(A\) and \(A^{\prime}\). Equation Transcription: Text Transcription: A'(t) A A'

Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius \(0.01 \mathrm{~cm}\), length \(3 \mathrm{~cm}\), pressure difference 3000 dynes \(/ \mathrm{cm}^{2}\), and viscosity \(\eta=0.027\) (a) Find the velocity of the blood along the centerline \(r=0\), at radius \(r=0.005 \mathrm{~cm}\), and at the wall \(r=R=0.01 \mathrm{~cm}\) (b) Find the velocity gradient at \(r=0, \ r=0.005\), and \(r=0.01\). (c) Where is the velocity the greatest? Where is the velocity changing most? Equation Transcription: 3000 dynes . Text Transcription: 0.01 cm 3 cm 3000 dynes /cm^2 Eta = 0.027 r=0 r=0.005 cm r=R=0.01 cm r=0, r=0.005 r=0.01

The frequency of vibrations of a vibrating violin string is given by \(f=\frac{1}{2 L} \sqrt{\frac{T}{\rho}}\) where L is the length of the string, T is its tension, and \(\rho\) is its linear density. [See Chapter 11 in D. E. Hall, Musical Acoustics, 3rd ed. (Pacific Grove, CA, 2002).] (a) Find the rate of change of the frequency with respect to (i) the length (when T and \(\rho\) are constant), (ii) the tension (when L and \(\rho\) are constant), and (iii) the linear density (when L and T are constant). (b) The pitch of a note (how high or low the note sounds) is determined by the frequency f. (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates, (ii) when the tension is increased by turning a tuning peg, (iii) when the linear density is increased by switching to another string.

Suppose that the cost (in dollars) for a company to produce \(x\) pairs of a new line of jeans is \(C(x)=2000+3 x+0.01 x^{2}+0.0002 x^{3}\) (a) Find the marginal cost function. (b) Find \(C^{\prime}(100)\) and explain its meaning. What does it predict? (c) Compare \(C^{\prime}(100)\) with the cost of manufacturing the 101st pair of jeans. Equation Transcription: Text Transcription: x C(x)=2000+3x+0.01x^2+0.0002x^3 C?(100)

The cost function for a certain commodity is \(C(q)=84+0.16 q-0.0006 q^{2}+0.000003 q^{3}\) (a) Find and interpret \(C^{\prime}(100)\). (b) Compare \(C^{\prime}(100)\) with the cost of producing the 101st item. Equation Transcription: . Text Transcription: C(q)=84+0.16q?0.0006q^2+0.000003q^3

If \(p(x)\) is the total value of the production when there are \(x\) workers in a plant, then the average productivity of the workforce at the plant is \(A(x)=\frac{p(x)}{x}\) (a) Find \(A^{\prime}(x)\). Why does the company want to hire more workers if \(A^{\prime}(x)>0\)? (b) Show that \(A^{\prime}(x)>0\) if \(p^{\prime}(x)\) is greater than the average productivity. Equation Transcription: Text Transcription: p(x) x A(x)=p(x)/x A?(x) A?(x)>0 p?(x)

If ???? denotes the reaction of the body to some stimulus of strength ????, the sensitivity ???? is defined to be the rate of change of the reaction with respect to ????. A particular example is that when the brightness ???? of a light source is increased, the eye reacts by decreasing the area ???? of the pupil. The experimental formula has been used to model the dependence of ???? on x when ???? is measured in square millimeters and ???? is measured in appropriate units of brightness. (a) Find the sensitivity. (b) Illustrate part (a) by graphing both ???? and ???? as functions of ????. Comment on the values of ???? and ???? at low levels of brightness. Is this what you would expect?

Patients undergo dialysis treatment to remove urea from their blood when their kidneys are not functioning properly. Blood is diverted from the patient through a machine that filters out urea. Under certain conditions, the duration of dialysis required, given that the initial urea concentration is c>1, is given by the equation \(t=\ln \left(\frac{3 c+\sqrt{9 c^2-8 c}}{2}\right)\)

Patients undergo dialysis treatment to remove urea from their blood when their kidneys are not functioning properly. Blood is diverted from the patient through a machine that filters out urea. Under certain conditions, the duration of dialysis required, given that the initial urea concentration is c>1, is given by the equation \(t=\ln \left(\frac{3 c+\sqrt{9 c^2-8 c}}{2}\right)\) Calculate the derivative of t with respect to c and interpret it.

The gas law for an ideal gas at absolute temperature \(T\) (in kelvins), pressure \(P\) (in atmospheres), and volume \(V\) (in liters) is \(P V=n R T\), where \(n\) is the number of moles of the gas and \(R=0.0821\) is the gas constant. Suppose that, at a certain instant, \(P=8.0 \mathrm{~atm}\) and is increasing at a rate of \(0.10 \mathrm{~atm} / \mathrm{min}\) and \(V=10 \mathrm{~L}\) and is decreasing at a rate of \(0.15 \mathrm{~L} / \mathrm{min}\). Find the rate of change of \(T\) with respect to time at that instant if \(n=10 \mathrm{~mol}\). Equation Transcription: Text Transcription: T P V PV=nRT n R=0.0821 P=8.0 atm 0.10 atm/min V=10 L 0.15 L/min T n=10 mol

In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation \(\frac{d P}{d t}=r_{0}\left(1-\frac{P(t)}{P_{c}}\right) P(t)-\beta P(t)\) where \(r_{0}\) is the birth rate of the fish, \(P_{c}\) is the maximum population that the pond can sustain (called the carrying capacity), and \(\beta\) is the percentage of the population that is harvested. (a) What value of \(d P / d t\) corresponds to a stable population? (b) If the pond can sustain 10,000 fish, the birth rate is \(5 \%\), and the harvesting rate is \(4 \%\), find the stable population level. (c) What happens if \(\beta\) is raised to \(5 \%\)? Equation Transcription: Text Transcription: dP/dt=r_0 (1-P(t)/Pc) P(t)- beta P(t) r_0 Pc beta dP/dt 5% 4%

In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by \(W(t)\), and caribou, given by \(C(t)\), in northern Canada. The interaction has been modeled by the equations \(\frac{d C}{d t}=a C-b C W \quad \frac{d W}{d t}=-c W+d C W\) (a) What values of \(d C / d t\) and \(d W / d t\) correspond to stable populations? (b) How would the statement “The caribou go extinct” be represented mathematically? (c) Suppose that \(a=0.05, \ b=0.001, \ c=0.05\), and \(d=0.0001\). Find all population pairs \((C, W)\) that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct? Equation Transcription: Text Transcription: W(t) C(t) dC/dt=aC-bCW dW/dt=-cW+dCW dC/dt dW/dt a=0.05,b=0.001,c=0.05 d=0.0001 (C,W)

Hermann Ebbinghaus (1850 –1909) pioneered the study of memory. A 2011 article in the Journal of Mathematical Psychology presents the mathematical model \(R(t)=a+b(1+c t)^{-\beta}\) for the Ebbinghaus forgetting curve, where \(R(t)\) is the fraction of memory retained \(t\) days after learning a task; \(a, b\), and \(c\) are experimentally determined constants between \(0\) and \(1 ; \beta\) is a positive constant; and \(R(0)=1\). The constants depend on the type of task being learned. (a) What is the rate of change of retention ???? days after a task is learned? (b) Do you forget how to perform a task faster soon after learning it or a long time after you have learned it? (c) What fraction of memory is permanent? Equation Transcription: Text Transcription: R(t) = a + b(1+ct)^-beta R(t) t a,b c 1;beta R(0)=1

The difficulty of “acquiring a target” (such as using a mouse to click on an icon on a computer screen) depends on the ratio between the distance \(D\) to the target and width \(W\) of the target. According to Fitts’ law, the index \(I\) of difficulty is modeled by \(I=\log _{2}\left(\frac{2 D}{W}\right)\) This law is used for designing products that involve human–computer interactions. (a) If \(W\) is held constant, what is the rate of change of \(I\) with respect to \(D\)? Does this rate increase or decrease with increasing values of \(D\)? (b) If \(D\) is held constant, what is the rate of change of \(I\) with respect to \(W\)? What does the negative sign in your answer indicate? Does this rate increase or decrease with increasing values of \(W\)? (c) Do your answers to parts (a) and (b) agree with your intuition? Equation Transcription: Text Transcription: I= log_2(2D/W) I D W