A bottle of juice at the tuckshop cost R9.55 each and you must buy 9. Determine approximately how much change you will get if you have R100.​

The Option C is correct that Negative 3 less-than x less-than-or-equal-to 2 by solving inequality using an algebraic expression.

The equation-like form of the formula 5x 4 > 2x + 3 has an arrowhead in place of the equals sign. It is an illustration of inequity. The left half, 5x 4, is larger than the right half, 2x + 3, as evidenced by this.

To solve the inequality, we need to isolate the variable, x, in the middle of the inequality.

Starting with:

-4 < 3x + 5 ≤ 11

Taking out 5 from each component of the inequality:

-4 - 5 < 3x + 5 - 5 ≤ 11 - 5

Simplifying:

-9 < 3x ≤ 6

Dividing by 3 (and remembering to reverse the direction of the inequality if we divide by a negative number):

-3 < x ≤ 2

Therefore, the solution to the inequality is:

-3 < x ≤ 2

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The Option C is correct that Negative 3 less-than x less-than-or-equal-to 2 by solving inequality using an algebraic expression.

The equation-like form of the formula 5x 4 > 2x + 3 has an arrowhead in place of the equals sign. It is an illustration of inequity. The left half, 5x 4, is larger than the right half, 2x + 3, as evidenced by this.

To solve the inequality, we need to isolate the variable, x, in the middle of the inequality.

Starting with:

-4 < 3x + 5 ≤ 11

Taking out 5 from each component of the inequality:

-4 - 5 < 3x + 5 - 5 ≤ 11 - 5

Simplifying:

-9 < 3x ≤ 6

Dividing by 3 (and remembering to reverse the direction of the inequality if we divide by a negative number):

-3 < x ≤ 2

Therefore, the solution to the inequality is:

-3 < x ≤ 2

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For the original function f(x) = x2 + 9:
Domain: Since x2 is defined for all real numbers, the domain is all real numbers, or (-, ).
Range: Since x2 is always non-negative and we add 9 to it, the range is [9, ].

To find the inverse function of f(x) = x^2 + 9, follow these steps:

1. Replace f(x) with y: y = x^2 + 9
2. Swap x and y: x = y^2 + 9
3. Solve for y to get the inverse function:

Subtract 9 from both sides:
x - 9 = y^2

Take the square root of both sides (considering only the positive square root as the original function has a non-negative output):
y = sqrt(x - 9)

The inverse function is f^(-1)(x) = sqrt(x - 9).

Now let's find the domain and range:

For the original function f(x) = x^2 + 9:
- Domain: Since x^2 is defined for all real numbers, the domain is all real numbers or (-∞, ∞).
- Range: Since x^2 is always non-negative and we add 9 to it, the range is [9, ∞).

For the inverse function f^(-1)(x) = sqrt(x - 9):
- Domain: The square root function is defined only for non-negative numbers. So, the domain is [9, ∞).
- Range: The square root function has a non-negative output. So, the range is [0, ∞).

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If a researcher conducted a 2-tailed, non-directional test with an alpha level of .04, then the corresponding critical value Z-score(s) for a 2-tailed, non-directional test with an alpha level of 0.04 would be -2.06 and 2.06, respectively.

Explanation:

To find the critical value z-scores for a 2-tailed, non-directional test with an alpha level of 0.04, you can follow these steps:

Step 1. Divide the alpha level by 2, since it's a 2-tailed test: 0.04 / 2 = 0.02.

Next, we can use a standard normal distribution table or a Z-score calculator to find the Z-score(s) that correspond to an area of 0.02 in the tail(s) of the standard normal distribution.

For a 2-tailed test, we need to find two critical values, one for each tail. Since the standard normal distribution is symmetric, the critical values will be the same in magnitude but opposite in sign. So, we need to find the Z-score that corresponds to an area of 0.02 in the lower tail and the Z-score that corresponds to an area of 0.02 in the upper tail.

Step 2. Use a z-score table or online calculator to find the z-score corresponding to an area of 0.98 (1 - 0.02) in the standard normal distribution.

Therefore, the corresponding critical value Z-score(s) for a 2-tailed, non-directional test with an alpha level of 0.04 would be -2.06 and 2.06, respectively.

The correct answer is:
a. +2.06 and -2.06

These z-scores represent the critical values, with 2% of the area in each tail of the distribution.

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To find the net change in velocity from time t=1 to time t=5, we need to integrate the acceleration function a(t) from t=1 to t=5. The net change in velocity from time t=1 to time t=5 is approximately 37.539 units (rounded to three decimal places).

To find the net change in velocity from time t=1 to time t=5, we need to find the definite integral of the acceleration function a(t) = ln(3t^4) with respect to time over the interval [1, 5]. To do this, we integrate a(t) with respect to t:∫[1 to 5] ln(3t^4) dtLet's call the antiderivative of a(t) as v(t), which represents the velocity function:v(t) = ∫ln(3t^4) dtNow, to find the net change in velocity, we evaluate v(t) at t=5 and t=1, and subtract the results:Net change in velocity = v(5) - v(1)Once you compute this, you will have the net change in velocity from time t=1 to time t=5.

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C  is  the true mean is contained in it.

Since we have a 98% confidence interval, we expect 98% of the intervals generated from the 1200 samples to contain the true mean. Therefore, we expect 0.98 x 1200 = 1176 intervals to contain the true mean.

Since we have a 98% confidence interval, we expect 2% of the intervals generated from the 300 samples to not contain the true mean. Therefore, we expect 0.02 x 300 = 6 intervals to not contain the true mean.

We cannot determine with certainty whether the true mean is contained in the given interval, but we can say that there is a 98% probability that the true mean falls within the interval

Therefore, the answer is C.

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Using a standard normal distribution table or the cumulative distribution function (CDF), the area under the standard normal curve to the right of z=−1.5 is approximately 0.0668, rounded to four decimal places.

Explanation:

To find the area under the standard normal curve to the right of z=−1.5, Follow these steps:

Step 1: To find the area under the standard normal curve to the right of z=−1.5, we need to use a standard normal distribution table or calculator.

Using a standard normal distribution table, we can find the area to the right of z=−1.5 is 0.0668 (rounded to four decimal places).

Step 2: Alternatively, we can use a calculator or statistical software to find the area using the cumulative distribution function (CDF) of the standard normal distribution. Using a calculator or software, we get the same result of 0.0668.

Therefore, the area under the standard normal curve to the right of z=−1.5 is approximately 0.0668, rounded to four decimal places.

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Answer:

8

Step-by-step explanation:

The first step is to simplify the fraction inside the parentheses,

using a^b / a^c = a^(b-c)

6^7  3^3             6^(7-6)   3^(3-4)   = 6^1  3^-1   =  6/3  =2

-----------------  =                                                    

6^6    3^4          

Now we take care of the outside parentheses

3^2 = 2*2*2 = 8                                                            

The solution to the differential equation dx/dt = x*sin(t), with the initial condition x(0) = 1, is x(t) = e^(-cos(t)).

To solve the differential equation:

dx/dt = x*sin(t)

We can separate the variables and integrate both sides:

1/x dx = sin(t) dt

Integrating both sides gives:

ln|x| = -cos(t) + C

Where C is a constant of integration. Solving for x, we have:

|x| = e^(-cos(t)+C) = e^C * e^(-cos(t))

Since x(0) = 1, we can substitute t=0 and x=1 into the solution to find C:

|1| = e^C * e^(-cos(0))

So e^C = 1, and C=0. Substituting this value of C back into the solution, we have:

|x| = e^(-cos(t))

Since the initial condition x(0) = 1, we take the positive value of the absolute value:

x(t) = e^(-cos(t))

Therefore, the solution to the differential equation dx/dt = x*sin(t), with the initial condition x(0) = 1, is x(t) = e^(-cos(t)).

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The period of simple harmonic motion of the mass-spring system can be found using the formula: T = 2π√(m/k) where T is the period, m is the mass, and k is the spring constant.

In this case, the mass of the object is 2 N, but we need to convert this to kilograms by dividing by the acceleration due to gravity: m = 2 N / 9.8 m/s^2 = 0.204 kg The spring constant is given as 4 N/m. Plugging in these values to the formula, we get: T = 2π√(0.204 kg / 4 N/m) = 2π√(0.051 m) ≈ 0.804 s .

Therefore, the period of simple harmonic motion for this mass-spring system is approximately 0.804 seconds. Now, we can find the period using the mass (0.204 kg) and the spring constant (4 N/m). T = 2π √(0.204 kg / 4 N/m) T ≈ 2π √(0.051) T ≈ 1.42 s The period of simple harmonic motion is approximately 1.42 seconds.

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The acceleration of the particle with position vector r(t) = (g sin t)i + (g cos t)j + htk can be expressed in the form [tex]aT^T + aN^N[/tex] without finding T and N.  

The tangential component of the acceleration is given by aT = g cos t i - g sin t j, while the normal component is aN = -h k. This means that the particle is undergoing a uniform circular motion with radius g and angular velocity dθ/dt = g/h.

The tangential component of the acceleration is responsible for changing the speed of the particle, while the normal component is responsible for changing the direction of the velocity vector.

In other words, the tangential component is perpendicular to the normal component and together they form a right-angled triangle with hypotenuse equal to the acceleration vector.

Therefore, by expressing the acceleration vector in terms of its tangential and normal components, we can better understand the motion of the particle without explicitly calculating the unit tangent and normal vectors.

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Answer:

The total surface area of a cube can be found using the formula 6s^2, where s is the length of an edge.

In this case, s = 3 meters, so the surface area of one face is 3^2 = 9 square meters.

There are 6 faces in a cube, so the total surface area that needs to be painted is:

6 x 9 = 54 square meters

Therefore, the correct answer is 54 m².

The lim(x→∞) x*sin(6π/x) does not exist.

The limit you want to find is lim(x→∞) x*sin(6π/x). To solve this limit, we will use l'Hospital's Rule, which is applicable when the limit takes the indeterminate form 0*∞.

Step 1: Rewrite the limit as a fraction:
lim(x→∞) (sin(6π/x)) / (1/x)

Step 2: Apply l'Hospital's Rule by differentiating both the numerator and the denominator:
Numerator: d(sin(6π/x))/dx = (6π*cos(6π/x)) * (-1/x²)
Denominator: d(1/x)/dx = -1/x²

Step 3: Simplify the limit:
lim(x→∞) [(6π*cos(6π/x)) * (-1/x²)] / [-1/x²] = lim(x→∞) 6π*cos(6π/x)

Step 4: Evaluate the limit:
Since cos(6π/x) oscillates between -1 and 1 as x approaches infinity, the limit does not exist.

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The linearization of the function z=xy√ at the point (-2, 9) is: L(x,y) = -6 + 3√(x+2) - 2√(y-9)

The linearization of a function f(x,y) at a point (a,b) is given by:

L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)

where fx and fy are the partial derivatives of f with respect to x and y, evaluated at (a,b).

In this case, we have f(x,y) = xy√ and (a,b) = (-2,9). We need to find fx and fy at this point:

fx(x,y) = y√

fy(x,y) = x√

Evaluating these at (-2,9), we get:

fx(-2,9) = 3√

fy(-2,9) = -2√

So the linearization of f at (-2,9) is:

L(x,y) = f(-2,9) + fx(-2,9)(x+2) + fy(-2,9)(y-9)

= -6 + 3√(x+2) - 2√(y-9)

Therefore, the linearization of the function z=xy√ at the point (-2, 9) is:

L(x,y) = -6 + 3√(x+2) - 2√(y-9).

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To solve for n in the given equation:

2 - (1/2n) = 3n + 16/(2 - 1/n)

First, we can simplify the right-hand side of the equation by finding a common denominator for the fraction:

2 - (1/2n) = (3n(2n - 1) + 16n)/(2n - 1)

Next, we can simplify the left-hand side of the equation by combining like terms:

(4n - 1)/2n = (3n(2n - 1) + 16n)/(2n - 1)

We can then cross-multiply and simplify:

(4n - 1)(2n - 1) = 3n(2n - 1) + 16n

8n^2 - 6n + 1 = 6n^2 + 11n

2n^2 - 17n + 1 = 0

Using the quadratic formula, we can solve for n:

n = (17 ± sqrt(17^2 - 4(2)(1)))/(2(2))

n = (17 ± sqrt(281))/4

Therefore, the two solutions for n are:

n = (17 + sqrt(281))/4 or n = (17 - sqrt(281))/4

Both solutions are real numbers, but they are not integers.

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The equal-weighted average, the market capitalization-weighted average, and the volume-weighted average.

The type of average that sums the price of each stock and divides the total by a divisor is called a "price-weighted average". In this type of average, the price of each stock is used as a weight to determine its contribution to the overall index.

For example, suppose we have an index with three stocks: A, B, and C. The price of each stock is $10, $20, and $30, respectively. To calculate the price-weighted average of this index, we would add up the prices of each stock and divide by a divisor, which is usually adjusted for changes in the stock prices or for the addition or removal of stocks from the index. In this case, the calculation would be:

($10 + $20 + $30) / 3 = $20

So the price-weighted average of this index is $20.

One drawback of price-weighted averages is that they are sensitive to changes in the prices of higher-priced stocks, since those stocks have a greater weight in the index. This can lead to distortions in the index if the prices of the higher-priced stocks change significantly. Additionally, price-weighted averages do not take into account the market capitalization or trading volume of each stock, which may not accurately reflect the overall market or sector performance.

Other types of averages that address these limitations include the equal-weighted average, the market capitalization-weighted average, and the volume-weighted average.

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The given sequence 1, 4/3, 5/3, 2, ... is neither an arithmetic nor a geometric sequence.

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is called the "common difference" and is denoted by the letter "d". The first term of an arithmetic sequence is usually denoted by "a".

The general form of an arithmetic sequence can be written as:

a, a + d, a + 2d, a + 3d, ...

According to the given information:

The given sequence 1, 4/3, 5/3, 2, ... is neither an arithmetic nor a geometric sequence.

An arithmetic sequence is a sequence in which the difference between any two consecutive terms is constant. In other words, if you subtract any term from its adjacent term, you will always get the same value. Similarly, a geometric sequence is a sequence in which the ratio between any two consecutive terms is constant. In other words, if you divide any term by its adjacent term, you will always get the same value.

Let's check the given sequence to see if it satisfies the conditions for arithmetic or geometric sequences:

1 - 4/3 = -1/3

4/3 - 5/3 = -1/3

5/3 - 2 = -1/3

As we can see, the differences between consecutive terms are not constant, so the given sequence is not an arithmetic sequence.

1 / (4/3) = 3/4

(4/3) / (5/3) = 4/5

(5/3) / 2 = 5/6

As we can see, the ratios between consecutive terms are not constant, so the given sequence is not a geometric sequence.

Therefore, the given sequence 1, 4/3, 5/3, 2, ... is neither an arithmetic nor a geometric sequence.

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Answer:

Arithmetic

Step-by-step explanation:

4/3-1= 1/3

1+1/3=4/3

4/3+1/3=5/3

5/3+1/3=6/3

6/3=2

So that means this is an arithmetic sequence.

The probability of drawing at least one red card is (d) 60/90.

The probability of drawing at least one red card can be found by finding the probability of drawing two yellow cards and subtracting that from 1.

The probability of drawing a yellow card on the first draw is 6/10. The probability of drawing a yellow card on the second draw, without replacement, is 5/9 (since there are only 9 cards left). So the probability of drawing two yellow cards in a row is:

(6/10) * (5/9) = 30/90

To find the probability of drawing at least one red card, we can subtract this from 1:

1 - 30/90 = 60/90

So the answer is (d) 60/90.

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The solution to the differential equation [tex]dp[/tex] = aP - b, where a and b are nonzero constants, is P(t) = (b/a) + Ce^(at), where C is a constant of integration.

To solve the differential equation, we can start by separating the variables and integrating both sides. This gives us:

∫ 1/P dP = ∫ a dt - ∫ b dt

Simplifying the integrals and taking antiderivatives, we get:

ln|P| = at - bt + C

where C is a constant of integration. Exponentiating both sides gives us:

|P| = e^(at-bt+C)

Since a and b are nonzero constants, we can write this as:

|P| = e^C * e^(at) * e^(-bt)

Using the absolute value notation is not necessary because the exponential function is always positive, but it is included here for completeness. We can rewrite this as:

P(t) = ± e^C * e^(at) * e^(-bt)

We can simplify this expression by setting the constant of integration to C = ln(b/a), which gives us:

P(t) = (b/a) * e^(at) + De^(-bt)

where D is a constant of integration. We can simplify this further by combining the constants of integration, giving us:

P(t) = (b/a) + Ce^(at)

where C = De^(-bt) is another constant of integration. This is the final solution to the differential equation [tex]dp[/tex] = aP - b.

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The Pigeonhole Principle states that if there are more pigeons than pigeonholes, then there must be at least one pigeonhole with two or more pigeons. In this case, there are 7 days of the week (pigeonholes) and 110 days (pigeons) to choose from.

Therefore, if we divide the 110 days into 7 groups based on the day of the week, the largest group can have at most ⌊110/7⌋ = 15 days. But since we have 7 groups, by the Pigeonhole Principle, at least one group must have more than ⌊110/7⌋ = 15 days. Thus, at least 16 of any 110 days chosen must fall on the same day of the week.

In simpler terms, if you have 110 days to choose from and only 7 days of the week, it is inevitable that some days will have to overlap.

In fact, at least one day of the week must have more than 15 days chosen, which means at least 16 days must fall on that day of the week. This principle can be applied to many situations where there are more items to choose from than categories to put them in.

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Answer:

The correct answer is d. average.

Step-by-step explanation:

The mean is a measure of central tendency in statistics and is often referred to as the average of a data set. It is calculated by adding up all the values in the data set and dividing by the total number of values. The mean is a common way to summarize a data set and provides a single value that represents the "typical" value of the data. It is not the same as the mode, which is the most frequently occurring value in the data set, or the range, which is the difference between the largest and smallest values in the data set

The total number of fish caught over the first 14 years is1,554,393 fish over the first 14 years.

The number of fish caught each year decreases by 3%, which means the company catches 97% of the previous year's total.

Therefore, the number of fish caught each year can be calculated as follows:

Year 1: 130,000

Year 2: 130,000 x 0.97 = 126,100

Year 3: 126,100 x 0.97 = 122,243

Year 4: 122,243 x 0.97 = 118,419

Year 5: 118,419 x 0.97 = 114,627

Year 6: 114,627 x 0.97 = 110,867

Year 7: 110,867 x 0.97 = 107,138

Year 8: 107,138 x 0.97 = 103,441

Year 9: 103,441 x 0.97 = 99,775

Year 10: 99,775 x 0.97 = 96,140

Year 11: 96,140 x 0.97 = 92,535

Year 12: 92,535 x 0.97 = 88,960

Year 13: 88,960 x 0.97 = 85,416

Year 14: 85,416 x 0.97 = 81,902

Therefore, the total number of fish caught over the first 14 years is:

130,000 + 126,100 + 122,243 + 118,419 + 114,627 + 110,867 + 107,138 + 103,441 + 99,775 + 96,140 + 92,535 + 88,960 + 85,416 + 81,902 = 1,554,393

Rounded to the nearest whole number, the company caught a total of 1,554,393 fish over the first 14 years.

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Answer:

Step-by-step explanation:

So 20% of the science club students is 10.

I am trying to find 100% as this equals ALL the students on the science club trip.

20% = 10

100% / 20% = 5

This means I need to multiply both sides by 5 to get to 100%
20% = 10

(Multiply both sides by 5)

100% = 50

Therefore there are 50 students in the science club (C)

if y = 4 x^3 - 5x and dx/dt=4, by Using the chain rule dy/dt = 172 when x = 2.

Given the function y = 4x^3 - 5x and we need to find dy/dt when x = 2 and dx/dt = 4. We can do this using the following steps:

Step 1: Differentiate the function y with respect to x to find dy/dx.
Thus, First, we find f'(x) by taking the derivative of y with respect to x:

dy/dx = d(4x^3 - 5x)/dx = 12x^2 - 5

Step 2: To find dy/dt, we need to find dy/dx and substitute x = 2 into the resulting expression, along with dx/dt = 4. Thus, substitute the given value of x = 2 into the expression for dy/dx.
dy/dx = 12(2)^2 - 5 = 12(4) - 5 = 48 - 5 = 43

Step 3: Use the chain rule to find dy/dt, which states that dy/dt = dy/dx * dx/dt.

Step 4: Finally, we use the chain rule formula to find dy/dt when x = 2:

Substitute the values of dy/dx and dx/dt into the chain rule equation.
dy/dt = 43 * 4 = 172

So, when x = 2 and dx/dt = 4, dy/dt = 172.

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Answer:

8. Image below!

9. 134 feet

10. Below.

11. The answers represent the timings at which the rocket was at height 164 feet.

12. 5.702s

Step-by-step explanation:

8. Image below

9. [tex]f(1)= -16(1)^2+100(1)+50 = 134[/tex]

I'm doing this in a rush but I can tell you what you can do for the rest.

10. Use the quadratic formula

[tex]164 = -16x^2+100x+50\\\\0 = -16x^2+100x-214\\\\[/tex]

And set a= -16, b=100, c=-214

You'll get two x-values.

11. The answers represent the timings at which the rocket was at height 164 feet.

12. Hitting the ground means the height is equal to 0. So...

[tex]0 = -16x^2+100x+50\\[/tex]

Using the quadratic formula you get

x=5.702s

and

x= 0.548s

This means the rocket hits the ground at 5.702s.

Answer:

Step-by-step explanation:

To calculate the y-component, we need to determine the y-component for each of these masses:

If m1 is at the origin, it is at (0,0). This means that it is at y=0.

If m2 is at (4.0,5.4), it is at y = 5.4.
If m3 is at (1.0, 2.8), it is at y = 2.8.

Thus, we can use the equation for finding equilibrium, which is each mass x position, divided by all the masses:

(m1 * 0 + m2 * 5.4 + m3 * 2.8) / (m1+m2+m3) = 2.53 (3 sig figs)

The series is convergent because sn can be expressed as a telescoping sum, which means that the series will approach a finite value as n approaches infinity.

To determine whether the series [infinity] cos 4n − cos 4n/2n=1 is convergent or divergent by expressing sn as a telescoping sum, we can rewrite the terms using the identity cos 2x = 2cos²ˣ − 1:

cos 4n − cos 4n/2n=1 = 2cos^24n/2 − 1 − 2cos^24n/2n+1 + 2cos^24n+2/2n+2 − 1

This expression has a telescoping sum because each term cancels with the previous and next terms. So we can simplify it as:

s_n = (2cos² 2n − 1) − (2cos² 2n+1 − 1)

s_n = 2(cos² 2n − cos² 2n+1)

s_n = −2(cos² 2n+1 − cos² 2n)

Therefore, the series is convergent because sn can be expressed as a telescoping sum, which means that the series will approach a finite value as n approaches infinity.

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To find the volume, you need to multiply all the values together.

1/3 x 5/6 x 2/3 = 5/27

Assume that the linear system [tex]Ax = b[/tex] has more than one solution, and let [tex]x1[/tex] and [tex]x2[/tex] be two distinct solutions. Let [tex]x3 := ux1+7x2[/tex], where µ, [tex]7 E IR[/tex] with the property that [tex]u+n = 1.[/tex]

Then we have: [tex]Ax1 = b and Ax2 = b[/tex] since x1 and x2 are solutions.

Subtracting the second equation from the first, we get: [tex]A(x1 - x2) = 0.[/tex]

Since [tex]x1[/tex] and [tex]x2[/tex]are distinct solutions, we know that [tex]x1 - x2 ≠ 0[/tex].

Therefore,[tex]A(x1 - x2) = 0[/tex] this implies that the columns of A are linearly dependent. That is, there exist scalars [tex]c1, c2, ..., cn[/tex] (not all zero) such that

[tex]c1a1 + c2a2 + ... + cnan = 0,[/tex]

where [tex]a1, a2, ...,[/tex]and an are the columns of A.

Let x be any solution of Ax = b. Then we have:[tex]A(x + tx3) = Ax + tAx3 = b + tAx3[/tex]

where t is any scalar. But we know that [tex]Ax3 = A(ux1 + 7x2) = uAx1 + 7Ax2 = ub + 7b = 8b,[/tex] since [tex]Ax1 = Ax2 = b.[/tex]

Therefore, we have: [tex]A(x + tx3) = b + t(8b) = (1 + 8t)b.[/tex]

Thus, [tex]x + tx3[/tex] is a solution of [tex]Ax = b[/tex] for any scalar t.

In particular, if we take [tex]t = 1/n,[/tex] where n is any nonzero integer, we get:

[tex]x + (1/n)x3 = (1 - 1/n)x + (1/n)ux1 + (7/n)x2.[/tex]

Since [tex]u + 7 = 1,[/tex] we have:[tex](1/n)ux1 + (7/n)x2 = (1/n)((1 - u)x1 + ux1 + 7x2) = (1/n)x1 + (7/n)x2.[/tex]

Therefore, we can write:[tex]x + (1/n)x3 = (1 - 1/n)x + (1/n)x1 + (7/n)x2.[/tex]

This shows that [tex]x + (1/n)x3[/tex] is another solution of Ax = b for any nonzero integer n. Since we can find infinitely many integers n such that 1/n is nonzero, we conclude that there are infinitely many solutions of .

Therefore, if the linear system [tex]Ax = b[/tex] has more than one solution, then it must have infinitely many solutions.

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The profit is maximum when 40 lawns are mowed.

Given that Kieran is starting a lawn-mowing buisness in her neighborhood. She creates a graph to help her determine what to charge customers per lawn to maximize her profits. She uses {c} to represent the number of lawns she mows and {y} to represent her profit in dollars.

The profit is maximum when 40 lawns are mowed as at this point the the peak of the parabola occurs.

Therefore, the profit is maximum when 40 lawns are mowed.

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The solution of the given differential equation is :

p(t) = A*e^(k*sin(t)), where A is a constant value.

The differential equation given is:

dp/dt = (k*cos(t))p, where k is a positive constant.

This equation models a population that undergoes yearly fluctuations. To find the solution of this equation, we can use the method of separation of variables.

First, separate the variables by dividing both sides by p and multiplying both sides by dt:

(dp/p) = (k*cos(t))dt

Now, integrate both sides with respect to their respective variables:

∫(1/p)dp = ∫(k*cos(t))dt

Upon integrating, we get:

ln|p| = k*sin(t) + C

To solve for p, take the exponent of both sides:

p(t) = e^(k*sin(t) + C)

Since e^C is also a constant, we can write the solution as:

p(t) = A*e^(k*sin(t))

Here, A is a constant that depends on the initial conditions of the problem. This solution represents the population that undergoes yearly fluctuations based on the given differential equation.

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