a basketball player pushes down with a force of 50 n on a basketball that is inflated to a gauge pressure of 8.0 x 10^4. what is the diameter of contact between the ball and the floor?

This statement is an example of a dichotomy, where two options are presented as the only possibilities.

The statement presents two mutually exclusive options - that light consists of either particles or waves. This is not an argument, but a statement of possible explanations for the nature of light. It is not deductive or inductive reasoning, but rather a scientific hypothesis that can be tested through experimentation and observation.

In conclusion, the statement that either light consists of tiny particles or it consists of waves is not an argument, but rather a dichotomy of possible explanations for the nature of light. It is up to scientific experimentation and observation to determine which explanation is most accurate.

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A spring-loaded gun is cocked by compressing a short, strong spring by a distance d. It fires a signal flare of mass m directly upward. The flare has speed v0 as it leaves the spring and is observed to rise to a maximum height h above the point where it leaves the spring.

After it leaves the spring, effects of drag force by the air on the flare are significant. The work done on a spring by compressing or stretching it is given by:W = (1/2)kx²where,W is the work donek is the force constantx is the distance by which the spring is compressed or stretchedTherefore, work done on the spring during compression,W = (1/2) k d² ...(1) From the work done on the spring,W = (1/2) k d²Using this formula, the force constant can be calculated,k = 2W/d² ...(2)

The total mechanical energy of the flare when it is fired from the spring,Em = (1/2)mv₀²where,m is the mass of the flarev₀ is the speed of the flare when it leaves the spring When the flare reaches its maximum height h, all of its kinetic energy is converted into potential energy. Thus,mgh = (1/2)mv₀²i.e.,gh = (1/2)v₀² ...(3)The amount of mechanical energy dissipated into thermal energy is equal to the initial mechanical energy minus the mechanical energy at maximum height. Thus, Ethermal = Em - mgh Ethermal = (1/2)mv₀² - mgh Substituting the value of v₀² from equation (3),Ethermal = (1/2)m(2gh) - mgh Ethermal = mgh .

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Based on the data, it is unclear whether the two methods provide the same mean value for natural vibration frequency.

In order to determine whether the two methods provide the same mean value for natural vibration frequency, we would need to conduct a hypothesis test. Specifically, we would need to conduct a two-sample t-test, comparing the mean natural vibration frequency for the two methods. The null hypothesis would be that the means are equal, while the alternative hypothesis would be that they are not equal.

Unfortunately, the question does not provide us with enough information to conduct this test. We do not know the sample size or standard deviation for each method, nor do we know the difference in means between the two methods. Therefore, we cannot determine whether the two methods provide the same mean value for natural vibration frequency based on the data given.

In conclusion, we cannot draw any conclusions about whether the two methods provide the same mean value for natural vibration frequency based on the information provided. More data is needed in order to conduct a hypothesis test and determine whether there is a significant difference between the means of the two methods.

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Apply these steps to each p-value (column) to determine whether to reject or fail to reject the null hypothesis.

Based on your question, I understand that you want to know whether to reject or fail to reject the null hypothesis for each two-tailed p-value using the p < .05 criterion. Since you didn't provide specific p-values, I will explain the concept for you to apply to your data:

For a two-tailed test with a significance level (α) of 0.05, you will follow these steps:

1. Compare the p-value to the significance level (α = 0.05).
2. If the p-value is less than α (p < 0.05), you will reject the null hypothesis.
3. If the p-value is greater than or equal to α (p ≥ 0.05), you will fail to reject the null hypothesis.

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Using the calculus methods and the concept of center of mass, the moment of inertia of the rod can be determined.

Consider a rod of length L with non-uniform density. To obtain its moment of inertia with respect to an axis passing through the pivot at its center, we may proceed as follows; The rod is divided into infinitesimal small masses. Let 'x' be the distance of a small mass element from the center. Then the mass density at that point will be (3x2+1). Let 'm' be the mass of this small element.

Then, using calculus, we can find that the total mass of the rod is 300 kg. The moment of inertia of the rod is obtained by integrating the product of the mass element, the square of the distance from the pivot and the mass density over the length of the rod. This integral can be evaluated using standard calculus techniques.

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The question states that Shelia's measured glucose level one hour after a sugary drink follows a normal distribution with a mean (µ) of 131 mg/dl and a standard deviation (s) of 10.9 mg/dl.

A normal distribution is a probability distribution that is symmetric and bell-shaped, where the majority of the data falls near the mean. The mean is the central tendency of the distribution, while the standard deviation measures the spread or variability of the data.

In this case, we know that Shelia's glucose level is normally distributed with a mean of 131 mg/dl and a standard deviation of 10.9 mg/dl. This means that most of the time, her glucose level will fall within one standard deviation of the mean, which is between 120.1 mg/dl (131 - 10.9) and 141.9 mg/dl (131 + 10.9).

Knowing the mean and standard deviation of Shelia's glucose levels can be helpful in predicting her glucose levels in the future. If we assume that her glucose levels continue to follow a normal distribution, we can estimate the probability of her glucose level falling within a certain range. Additionally, monitoring her glucose levels over time can help identify any patterns or trends that may require intervention or management.

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Assuming the rope and pulleys are massless and frictionless, the tension in the rope is the same throughout. Let's call this tension T. Since the block is at rest, the forces in the vertical direction must balance. The weight of the block is pulling down with a force of W, and the tension in the rope is pulling up with a force of T. Therefore, T = W.

Now let's look at the spring scale. The spring scale is connected to the rope on one side and the ceiling on the other. The tension in the rope is transmitted through the spring scale to the ceiling.

Therefore, the reading on the spring scale is also T, which we just found to be W. So the answer is W, or in other words, the weight of the block.

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When the hunter pulls the polar bear to him, the polar bear will move:: B. 3.3m

To solve this problem, we need to use the concept of conservation of momentum. Since the ice is frictionless, the total momentum before and after the hunter pulls the rope will be the same.

Initially, both the hunter and the polar bear are stationary, so the total momentum is 0. When the hunter pulls the polar bear, the magnitudes of their momenta will be equal and opposite, thus conserving momentum. We can calculate the distances each moves by using the ratio of their masses.

Let x be the distance the hunter moves and y be the distance the polar bear moves. Since their momenta are equal and opposite, we have:

(640 N)x = (3200 N)y

The sum of these distances is the initial separation of 20 m:

x + y = 20 m

Now, substitute the first equation into the second equation to solve for y:

y = (640 N / 3200 N)x

x + (640 N / 3200 N)x = 20 m

x(1 + 640 N / 3200 N) = 20 m

x = 20 m / (1 + 640 N / 3200 N)

x ≈ 16 m

Since x is the distance the hunter moves, y will be the distance the polar bear moves:

y = 20 m - 16 m = 4 m

As 4 m is not one of the options given, the closest answer would be: B. 3.3m

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The two numbers are 50 and -50, whose difference is 100 and whose product (-50 * 50 = -2500) is a minimum.

To find two numbers whose difference is 100 and whose product is a minimum, we can set up a system of equations using the given conditions. Let x and y be the two numbers, then:

1) x - y = 100
2) We want to minimize the product: P(x, y) = xy

From equation 1, we can write x as x = y + 100. Now, substitute this into equation 2 to get:

P(y) = (y + 100)y

To minimize the product, we can use calculus. Differentiate P(y) with respect to y:

dP/dy = 2y + 100

Set the derivative equal to zero and solve for y:

0 = 2y + 100
y = -50

Now, find x using the x = y + 100 equation:

x = -50 + 100
x = 50

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Of the following is the basis of the current standard for the meter the correct statemnt is The current standard for the meter is defined as the distance that light travels in a specified time interval.

The meter is currently defined based on the speed of light in a vacuum. It is defined as the distance traveled by light in 1/299,792,458 of a second. This definition was established by the International Committee for Weights and Measures (CIPM) and is commonly known as the "speed of light in a vacuum" definition. This definition provides a precise and universal standard for the meter, as the speed of light is a fundamental constant of nature. It allows for accurate and consistent measurements of length across different regions and time periods. The other options listed in the question, such as the scratch marks on a platinum-iridium bar, a strand of carbon fiber, wavelengths of light emitted by krypton-86, or the relationship with the inch, are not the current basis for the standard meter. These were historical or alternative methods of defining the meter, but the current standard is based on the speed of light.

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When k2s is added to a saturated solution of pbs, the concentrations of pb2 and s2− will change according to the solubility product constant (Ksp) for pbs. Ksp is the product of the concentrations of the ions in a saturated solution at equilibrium. In this case, adding k2s will introduce additional s2− ions, which will react with pb2 ions to form more pbs and decrease the concentration of pb2 ions. This is because the reaction will shift towards the product side to maintain equilibrium.

The overall effect on the concentration of s2− ions will depend on the magnitude of the Ksp for pbs and the amount of k2s added. If the Ksp for pbs is small, the addition of k2s may have a negligible effect on the concentration of s2− ions. However, if the Ksp for pbs is large and the amount of k2s added is significant, the concentration of s2− ions may increase as the equilibrium shifts towards the reactant side to maintain Ksp.

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The null hypothesis for this test would be that there is no linear relationship between the desire to have cosmetic surgery and the level of body satisfaction. The alternative hypothesis, on the other hand, would be that there is a linear relationship, and that as level of body satisfaction increases, desire for cosmetic surgery decreases.  To conduct this test, you could use a linear regression analysis to see if there is a significant negative slope between the two variables. You would also want to calculate the correlation coefficient and its associated p-value to determine the strength and significance of the relationship.

Assuming a significance level of 0.05, if the p-value is less than 0.05, we would reject the null hypothesis and conclude that there is evidence of a negative linear relationship between the desire for cosmetic surgery and the level of body satisfaction. If the p-value is greater than 0.05, we would fail to reject the null hypothesis and conclude that there is not enough evidence to support a linear relationship between the two variables.

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The equation of the line tangent to the curve at a given point can be found using the derivative of the curve.

To find the equation of the line tangent to the curve when x has the first value, you will need to take the derivative of the curve first. Once you have the derivative, plug in the x value of the point where you want to find the tangent line. This will give you the slope of the tangent line at that point.

Next, use the point-slope form of the equation of a line to find the equation of the tangent line. You will need to plug in the coordinates of the point where the tangent line touches the curve as well as the slope of the tangent line that you just found with the derivative.

To summarize, finding the equation of the line tangent to the curve involves taking the derivative of the curve, plugging in the x value of the point to find the slope of the tangent line, and using the point-slope form of the equation of a line to find the equation of the tangent line.

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The frequency of the light in the medium is the same as in air.

When a light beam passes through a medium with a different refractive index than the medium it was in before, its speed changes. The speed of light in a vacuum is always constant, but it can slow down or speed up when it enters a medium with a different refractive index.

The frequency of light does not change as it passes from one medium to another because the number of wave crests per unit time is always the same. The wavelength, on the other hand, changes when a light wave passes from one medium to another with a different refractive index. This results in a change in the direction of the light wave or in a phenomenon known as refraction, as well as a change in the speed of the light wave.

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Walking has a fuel economy of 1300 MPG equivalent, while cycling has an MPG equivalent of 913.33.

Walking has a fuel economy of 1300 MPG equivalent because gasoline produces about 1.30 x 10⁸ J/gal. If a walker uses about 220 kcal/h to travel at 3.08 mi/h, the walker would use 220 kcal/4184 J ≈ 52.56 J. Then, multiply this number by 3600 s/h, divide 3.08 mi/h by 52.56 J/s, and convert the resulting value to miles per gallon equivalent to get 1300 MPG.

For cycling, a person travelling at 10.5 mi/h expends about 400 kcal/h above the resting metabolic rate. To calculate the energy cost of cycling in J/s, convert the kilocalories expended per hour to joules and divide by 3600. You can then calculate the fuel economy by dividing the distance travelled (10.5 miles/hour) by the energy cost in J/s. This gives an equivalent fuel economy of 913.33 MPG.

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When plating a 10-7 dilution, we can use the following formula to calculate the expected number of PFU: Expected number of PFU = Titer x Dilution Factor .

The titer of the bacteriophage sample is given as 2.4 x 109 PFU/ml. This means that there are 2.4 x 109 bacteriophages in one milliliter of the sample. To calculate the expected number of PFU in a 10-7 dilution, we first need to determine the dilution factor. A 10-7 dilution means that we are diluting the original sample by a factor of 107 (or 10,000,000).  So, the dilution factor = 107 = 10,000,000 .

TFTC stands for "too few to count." In this case, with 24 PFU, the number is within the countable range, which is typically between 30 and 300 PFU. Since the count is below 30, it is considered TFTC. If the count was above 300, it would be TNTC (too numerous to count).

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One possible sketch of such a function f could be a cubic function friction that intersects the x-axis at the inflection point, as shown below.

A cubic function has an odd degree, which means that it must cross the x-axis at least once. If the inflection point is at the x-axis, then the function must change from concave down to concave up or vice versa at that point, which means it has an inflection point but no local minima or maxima. To ensure continuity, we can choose the coefficients of the cubic function such that it passes through the inflection point smoothly, without any kinks or jumps. For example, we could choose a function like f(x) = x^3 - 3x, which has an inflection point at (0,0) and no local extrema, as shown below:

The inflection point of this function occurs at x = 0, where f''(x) = 6x changes sign from negative to positive. The function is decreasing on (-∞,0) and increasing on (0,∞), so it has no local maxima or minima. The graph of this function looks like a "S" curve, with the inflection point at the bottom.
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the phasor representation of an inductance corresponds to a vector that is perpendicular to the voltage phasor in an AC circuit. phasors are used to simplify complex AC circuits by representing sinusoidal voltages and currents vectors that vary in magnitude and phase angle.

In the case of an inductor, the phasor voltage leads the phasor current by 90 degrees, which means that the phasor representing the inductance is oriented perpendicular to the voltage phasor. This phasor relationship allows for easy analysis of circuit behavior and simplification of complex calculations involving multiple components. The phasor representation of an inductance corresponds to a complex impedance.  In phasor representation, an inductance corresponds to a complex impedance with a purely imaginary part.

Understand that impedance is a combination of resistance and reactance, where reactance can be either inductive or capacitive.  For an inductor, the reactance (X_L) is calculated as X_L = 2 * π * f * L, where f is the frequency and L is the inductance  In phasor representation, the impedance (Z) of an inductor is represented as a complex number, with the real part representing the resistance (which is usually very small or zero for an ideal inductor) and the imaginary part representing the inductive reactance. So, Z = R + jX_L, where R is the resistance and j is the imaginary unit.  The phasor representation of an inductance corresponds to a complex impedance, highlighting the imaginary part that represents the inductive reactance in the system.

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When an object is subjected to an impulse, the change in the object's momentum can be determined by using the formula:Δp = FΔtwhere Δp is the change in momentum, F is the force applied, and Δt is the time during which the force is applied.

The object's speed and direction after the impulse will depend on the direction and magnitude of the force applied. If the force is applied in the same direction as the object's initial velocity, it will cause the object to speed up in the same direction. If the force is applied in the opposite direction as the object's initial velocity, it will cause the object to slow down or even change direction.

In order to determine the object's speed and direction after the impulse, the direction and magnitude of the force must be known. Without this information, a specific cannot be given.

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The potential difference between the plates is 3.56×10^3 V.

The potential difference between the plates can be calculated using the formula for kinetic energy, which is KE = 1/2mv^2. Since the electron has a very small mass, we can assume that its kinetic energy is equal to the electrical potential energy gained by moving through the electric field. Therefore, we can use the formula for electrical potential energy, which is PE = qV, where q is the charge of the electron and V is the potential difference between the plates.

We know that the electron acquired 5.70×10−16 JJ of kinetic energy, which is equal to the electrical potential energy gained by moving through the electric field. Thus, we can substitute the given values into the formula for electrical potential energy to find the potential difference between the plates.

PE = qV
5.70×10−16 J = (1.602×10−19 C)V
Solving for V gives:
V = 3.56×10^3 V

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The potential energy of the system can be calculated using Coulomb's Law and the principle of superposition.

Coulomb's Law states that the force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. The principle of superposition states that the total force on a charge due to a group of other charges is the vector sum of the individual forces on the charge due to each of the other charges.

To calculate the potential energy of the system, we need to first calculate the total force on each charge due to the other two charges. Using Coulomb's Law and the principle of superposition, we can then calculate the work done in bringing the charges from infinity to their current positions.

The potential energy of the system is equal to the negative of the work done in bringing the charges together. Taking the potential energy of the three charges as zero when they are infinitely far apart, we can calculate the potential energy of the system as the negative of the work done in bringing the charges together.

The potential energy of the system can be calculated using Coulomb's Law and the principle of superposition, and is equal to the negative of the work done in bringing the charges together from infinity to their current positions.

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The current rating that the fuse in the primary circuit should have is 2.5 A. A fuse is a device used in an electric circuit to protect the circuit from an overcurrent condition.

The fuse is the weakest link in the circuit, which means that it should have a current rating that is less than the maximum current that can flow through the circuit. If the current flowing through the circuit exceeds the rating of the fuse, the fuse will blow, which will break the circuit and protect the components from damage. In this case, we need to determine the current rating of the fuse in the primary circuit.

The primary circuit is the part of the circuit that connects the AC power source to the transformer. A transformer is a device that is used to change the voltage level of the AC power. The current rating of the fuse in the primary circuit should be less than the maximum current that can flow through the primary circuit. This is typically determined by the size of the transformer.

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The differential equation for da/dt is da/dt = -720qsin(2t) = -1440qsin(t)cos(t). To find the differential equation for da/dt, we need to use the equation q=CV. We can differentiate this equation with respect to time to get dq/dt = C(dV/dt).

Using the given values, we have C=2F and V(t) = 90cos(2t), so dV/dt = -180sin(2t). Substituting these values into the equation, we get dq/dt = -360sin(2t).   Next, we need to express dq/dt in terms of q. We can do this by using Ohm's Law, V=IR, where I is the current in the circuit. Rearranging this equation, we have I = V/R.

Using the given values, we have R=65 ohms and V(t) = 90cos(2t), so I(t) = 90cos(2t)/65. Substituting this into the equation for dq/dt, we get dq/dt = -360sin(2t) = -180I(t)sin(2t). Finally, we can express dq/dt in terms of q by substituting q=CV, which gives dq/dt = C(dV/dt) = -360Csin(2t) = -720qsin(2t).

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The cost of the window heat loss for a utility rate of 0.18$/kW.h is 0.915 $/day. The heat loss due to convection and radiation is 211.85W.

From the given,

T₀ = 15°C

Ts = 0°C

A = l×b = 1×1.8 m = 1.8 m

ε = 0.94

R = 0.18 $/kW.h

For air, T = 280K

v = 14.11 ×10⁻⁶ m²/s

α = 19.86×10⁻⁶ m²/s

Pr = 0. 71

k = 0.0247 W/m.k

a) The window would show a frost layer at the base rather than at the top, The window layer is the thinnest at the top of the window, and the heat flux from the warmer air passes through it increases. Also, at the bottom of the floor, the air is more stratified and cooler.

b) the heat loss,

Q(rad)= q(conv) + q(rad)

         = A[h(T₀ - Ts) + εσ(T₀⁴ - Ts⁴)]

Rα = gβΔΤL³/vα

     = 9.8×1/280×(15-0)×(1.8)³/14.11 ×10⁻⁶×19.86×10⁻⁶

     = 7284157065

Q(loss) = (1.18×3.138×(15-0)×0.94×5.67×10⁻⁸×[(288)⁴-(273)⁴]

           = 211.854W

Thus, the heat loss is 211.854W.

c) Cost = Q(loss)×R×24

           = 211.854×0.18/1000×24

           = 0.915$/kW.h

Thus, the cost of window heat loss is 0.915 $/ day.

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The acceleration function can be found by taking the second derivative of the position function, which is a(t) = -2.56cos(8t). This is a simple harmonic motion with amplitude of 0.04m and a period of T=pi/4s.

At t=0 s, the block will be at its maximum displacement from equilibrium, 0.04 m to the right. At t=rts, where r is the square root of the ratio of the mass to the spring constant, the block will pass through the equilibrium position and continue to oscillate back and forth. At t=2rt s, the block will be back at its maximum displacement, 0.04 m to the left.

The velocity function can be found by taking the derivative of the position function, which is x(t) = 0.04cos(8t). The velocity function is therefore v(t) = -0.32sin(8t).

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Velocity and speed are two concepts that are often used interchangeably, but they actually have different meanings. Speed refers to how fast an object is moving, while velocity refers to both the object's speed and the direction in which it is moving.

For example, a car traveling at 60 miles per hour north has a velocity of 60 miles per hour north, while a car traveling at 60 miles per hour east has a velocity of 60 miles per hour east. In other words, velocity takes into account the object's speed and the direction in which it is moving. On the other hand, speed only refers to how fast the object is moving, regardless of its direction.

In summary, velocity is a vector quantity that includes both speed and direction, while speed is a scalar quantity that only refers to how fast an object is moving.

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To develop arguments for and against people working from home to reduce their energy consumption with the restrictions explained in the question, you should do structured research on the topic to create your opinion.

Regarding the cost, we could argue in favor that this work would reduce costs of fuel, public transport, parking, etc. As for the arguments against, we could argue that this could increase domestic costs with electricity and computer maintenance.

So you can follow this example to make your own case for working from home and such restrictions, doing research on each to identify what the advantages and disadvantages are.

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There are 1,048,576 possible outcomes (microstates) when flipping 20 fair coins. The probability of getting the sequence "HTHHTTTHTHHHTHHHHTHT" in exactly that order is approximately 9.5367e-07.

a) There are 2 possible outcomes (heads or tails) for each coin flip, and since there are 20 coin flips, the total number of possible outcomes, or microstates, is given by 2²⁰

Answer: 2²⁰= 1,048,576 possible outcomes.

b) To calculate the probability of getting the sequence "HTHHTTTHTHHHTHHHHTHT" in exactly that order, we need to determine the probability of obtaining each individual outcome (head or tail) and multiply them together.

Since each coin flip is independent and has a 1/2 chance of resulting in either heads or tails (assuming the coins are fair), the probability of obtaining the desired sequence is (1/2)²⁰

Answer: (1/2)²⁰≈ 9.5367e-07

c) To calculate the probability of getting exactly 12 heads and 8 tails in any order, we need to determine the number of ways to arrange 12 heads and 8 tails within the 20 coin flips.

This can be calculated using the binomial coefficient, also known as "n choose k." The formula for the binomial coefficient is:

C(n, k) = n! / (k! * (n-k)!)

Where n is the total number of coin flips and k is the number of heads.

Using this formula, the probability can be calculated as follows:

P(12 heads and 8 tails) = C(20, 12) * (1/2)^20

Calculating C(20, 12):

C(20, 12) = 20! / (12! * (20-12)!)

          = 20! / (12! * 8!)

          = (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13) / (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)

          = 125,970

P(12 heads and 8 tails) = 125,970 * (1/2)^20

Answer: P(12 heads and 8 tails) ≈ 0.12013435364 (approximately)

a) There are 1,048,576 possible outcomes (microstates) when flipping 20 fair coins.

b) The probability of getting the sequence "HTHHTTTHTHHHTHHHHTHT" in exactly that order is approximately 9.5367e-07.

c) The probability of getting exactly 12 heads and 8 tails in any order is approximately 0.12013435364.

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The resultant velocity of the airplane, taking into account both its airspeed and the wind velocity, can be determined by vector addition. The airplane is traveling N60°W at an airspeed of 600 km/h, while encountering a wind blowing from a bearing of 200° at 70 km/h.

To find the resultant velocity, we can break down the given velocities into their components. The airspeed of 600 km/h at an angle of N60°W can be resolved into two components: 300 km/h towards the west (N90°W) and 519.62 km/h towards the south (S30°W). Similarly, the wind velocity of 70 km/h at a bearing of 200° can be resolved into two components: 34.04 km/h towards the west (W) and 60.32 km/h towards the north (N).

Adding the corresponding components together, we get a resultant velocity of 266.04 km/h towards the west (W) and 459.62 km/h towards the south (S). Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity as approximately 539.37 km/h. Finally, we can determine the direction of the resultant velocity using trigonometry, finding an angle of approximately S59.49°W (or N59.49°E).

In summary, the resultant velocity of the airplane is approximately 539.37 km/h towards S59.49°W (or N59.49°E).

[tex]Resulatant\,velocity=\sqrt{(west\,\,component)^2+(south\,\,component)^2} =\sqrt{300^2+519.62^2} =539.37km/h[/tex]

The direction of the resultant velocity can be determined using the formula:

[tex]\[\theta = \arctan\left(\frac{{\text{{south component}}}}{{\text{{west component}}}}\right) = \arctan\left(\frac{{519.62 \text{{ km/h}}}}{{300 \text{{ km/h}}}}\right) \approx 59.49°\][/tex]

Since the airplane is traveling N60°W, we subtract the angle obtained from 180° to get the final direction:

[tex]\[\text{{Final direction}} = 180° - 59.49° \approx 120.51°\][/tex]

Therefore, the resultant velocity of the airplane is approximately 539.37 km/h towards S59.49°W (or N59.49°E).

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The exact length of the curve is 105.98.

First, we will use the formula to find the arc length of the curve which is given as:
`L = int_a^b sqrt[1 + (dy/dx)^2]dx`
Here, `a = 0` and `b = 2`. Therefore, we can write:
`L = int_0^2 sqrt[1 + (dy/dx)^2]dx`
We will now find `dy/dx` by differentiating `x` and `y` with respect to `t`.
`x = et − 4t`
Therefore, `dx/dt = e^t - 4`.
`y = 8et⁄2`
Therefore, `dy/dt = 4e^t`.
We can now write `dy/dx` as `dy/dt * dt/dx`. This gives us:
`dy/dx = dy/dt * dx/dt^-1 = 4e^t / (e^t - 4)`
We can now substitute this value into the formula for `L` to obtain:
`L = int_0^2 sqrt[1 + (4e^t / (e^t - 4))^2]dx`
After integrating and simplifying, we get:
`L = (1/2) [5e^2 - 2 ln(2e^2 - 4) - 5]`
Evaluating this expression, we get `L = 105.98` (approx).

Therefore, the exact length of the curve is 105.98.

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