Line plots are useful to show large quantities of data. True or false

The value of constant of proportionality k = 12.  

The constant of proportionality is a value that relates two directly proportional quantities. It is the number that appears in front of the variable in an equation that represents a direct proportionality between two variables. In other words, if y is directly proportional to x, the equation that relates them is of the form:

y = kx

where k is the constant of proportionality.

If y varies directly with x, then we can write the equation:

y = kx

where k is the constant of proportionality. To solve for k, we can use the given information that when y = 36, x = 3. Substituting these values into the equation, we get:

36 = k(3)

Simplifying, we get:

12 = k

Therefore, The value of  constant of proportionality k = 12.  

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The complete question is:

Assume y varies directly as x. If x=4 when y=36, what is the value of k?

Answer:

$128.37

Step-by-step explanation:

Find 33% of 389:

389(0.33) = 128.37

The amount of the discount is $128.37

The given Fahrenheit temperatures evaluated to their Celsius equivalents are: 68°F = 20°C, 32°F = 0°C, 23°F = -5°C, -4°F = -20°C, and -40°F = -40°C.

The expression C = 5/9(F - 32) is used to convert temperatures in degrees Fahrenheit to degrees Celsius. To evaluate the expression for a given Fahrenheit temperature, we simply substitute the value of F into the formula and perform the necessary calculations.

For example, when F = 68, we substitute the value of F into the formula and simplify as follows:

C = 5/9 (68 - 32)

C = 5/9 * 36

C = 20

Here are the evaluations of the given Fahrenheit temperatures in degrees Celsius using the formula C = 5/9(F - 32):

F= 68: C = 5/9(68-32) = 20

F= 32: C = 5/9(32-32) = 0

F= 23: C = 5/9(23-32) = -5

F= -4: C = 5/9(-4-32) = -20

F= -40: C = 5/9(-40-32) = -40

Matching the correct Celsius temperatures to their respective Fahrenheit temperatures:

F= 68: C = 20

F= 32: C = 0

F= 23: C = -5

F= -4: C = -20

F= -40: C = -40

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Given a temperature F in degrees Fahrenheit, the expression C = 5/9(F - 32) gives the temperature in degrees Celsius. Evaluate the following Expressions and Match the correct answer to each the degrees listed.

1. F= 68  

2. F= 32

3. F = 23

4. F = -4

5. F = -40

The general equation for a circle with center (a,b) and radius r is:

[tex](x-a)^2 + (y-b)^2 = r^2[/tex]. By substituting the values and simplifying the equation, we will get 0.

What is circle ?

In math, a circle is a closed, two-dimensional shape made up of all elements that are spaced uniformly apart from the centre. The radius of something like the circle refers to this distance. Often, the letter "O" or "" is used to represent a circle. The circle is really a fundamental geometric shape and includes a number of significant characteristics, including its radius, which is the distance it around circle, or its area, which is the volume of the circle's interior. A circle's area is equal to 2 times its radius square, and its circumference is equal to 2 times its radius.

To find the equation for the circle that passes through the points (-5,0), (0,4), and (2,4), we need to find the center (a,b) and radius r.

First, we find the midpoint of the line segment between (-5,0) and (0,4), which is:

[tex]((-5+0)/2, (0+4)/2) = (-2.5,2)[/tex]

Next, we find the midpoint of the line segment between (0,4) and (2,4), which is:

[tex]((0+2)/2, 4/2) = (1,2)[/tex]

Since the circle passes through all three points, its center must be equidistant from all three points. Using the distance formula, we can set up three equations:

[tex](-2.5 - a)^2 + (2 - b)^2 = r^2[/tex]

[tex]a^2 + (4 - b)^2 = r^2[/tex]

[tex](1 - a)^2 + (2 - b)^2 = r^2[/tex]

Simplifying the equations and setting them equal to each other, we get:

[tex](-2.5 - a)^2 + (2 - b)^2 = a^2 + (4 - b)^2 = (1 - a)^2 + (2 - b)^2[/tex]

Expanding the squares and simplifying, we get:

[tex]6a - 10 = 0[/tex]

[tex]2b - 12 = 0[/tex]

Solving for a and b, we get:

a = 5/3

b = 6

Now we can find the radius of the circle by plugging in the values of a and b into one of the equations we set up earlier:

[tex]a^2 + (4 - b)^2 = r^2[/tex]

[tex](5/3)^2 + (4 - 6)^2 = r^2[/tex]

[tex]25/9 + 4 = r^2[/tex]

[tex]r^2 = 61/9[/tex]

Therefore, the equation of the circle is:

[tex](x - 5/3)^2 + (y - 6)^2 = 61/9[/tex]

Simplifying, we get:

[tex]4x^2 + 4y^2 - 40x + 48y + 167 = 0.[/tex]

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Answer:

5,5 m and 12 m

Step-by-step explanation:

Let's assume, the length of the rectangle is x and the width is y meters

Let's write 2 equations and put them in a system:

{x × y = 66,

{x = 2y + 1;

Let's replace x in the 1st equation with its value from the 2nd equation:

(2y + 1) × y = 66

[tex]2 {y}^{2} + y = 66[/tex]

[tex]2 {y}^{2} + y - 66 = 0[/tex]

a = 2, b = 1, c = -66

Let's solve this quadratic equation:

[tex]d = {b}^{2} - 4ac = {1}^{2} - 4 \times 2 \times ( - 66) = 1 + 528 = 529 > 0[/tex]

[tex]y1 = \frac{ - b - \sqrt{d} }{2a} = \frac{ - 1 - 23}{2 \times 2} = \frac{ - 24}{4} = - 6[/tex]

Since y has to be a natural number (side lengths cannot have negative units), we cannot take this as an answer

[tex]y2 = \frac{ - b + \sqrt{d} }{2a} = \frac{ - 1 + 23}{2 \times 2} = \frac{22}{4} = 5.5[/tex]

y = 5,5

x = 2 × 5,5 + 1 = 12

the dimensions of the rectangle are 5.5 meters by 12 meters.

The question states that the area of a rectangle is 66 m², and the length of the rectangle is 1 meter more than twice the width. We need to find the dimensions of the rectangle.

Let’s assume the width of the rectangle as “x”.

Then, according to the given information, the length will be (2x + 1).

Area of the rectangle = Length × Width

66 = (2x + 1) × x ⇒ 2x² + x - 66 = 0

Now we have to solve this quadratic equation. We can solve it by factoring or using the quadratic formula. Let’s use the quadratic formula:

x = [ -b ± √(b² - 4ac) ] / 2aHere, a = 2, b = 1, and c = -66

Therefore,x = [ -1 ± √(1² - 4(2)(-66)) ] / 2(2)x = [ -1 ± √(1 + 528)) ] / 4x = [ -1 ± √529 ] / 4

When we solve this, we get two possible solutions:

x = (-1 + 23) / 4 or x = (-1 - 23) / 4x = 22/4 or x = -24/4x = 5.5 or x = -6

The width of the rectangle cannot be negative. Therefore, the width of the rectangle is 5.5 meters.Length of the rectangle = 2 × 5.5 + 1= 11 + 1= 12 meters

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The correct representation of the inequality are:

1 ≥ 2x

6x ≥ 3 + 8x – 4

A point is at 0.5 and a bold line starts at 0.5 and is pointing to the left.

6x ≥ 3 + 4(2x – 1)?

6x ≥ 3 + 8x - 4  ---- correct

6x - 8x ≥ 3 - 4

-2x ≥ - 1

1 ≥ 2x ----- correect

The number line would be A point is at 0.5 and a bold line starts at 0.5 and is pointing to the left.

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The probability of selecting a point

1) in ΔADN is also 1/8.

2) in ΔADM but not in ΔADN is 0.

3) lies neither in ΔADM nor in ΔADN is 3/4.

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.

To solve this problem, we need to use the fact that the diagonals of a square bisect each other and are perpendicular. Let's denote the midpoint of AB as P and the midpoint of BC as Q.

To find the probability that a point selected randomly in the square lies in ΔADM, we need to find the ratio of the area of ΔADM to the area of the square ABCD.

Since M is the midpoint of AD, we have DM = AM = 1/2 * AB. Similarly, since N is the midpoint of CD, we have DN = CN = 1/2 * BC.

Thus, the area of ΔADM is (1/2 * DM * AM) = (1/2 * 1/2 * AB * 1/2 * AB) = 1/8 * AB². The area of the square is AB². Therefore, the probability of selecting a point in ΔADM is (1/8 * AB²) / AB² = 1/8.

To find the probability that a point selected randomly in the square lies in ΔADM but not in ΔADN, we need to subtract the probability of selecting a point in ΔADN from the probability of selecting a point in ΔADM. Since N is the midpoint of CD, we have AN = 1/2 * AB, and therefore DN = 1/2 * BC = 1/2 * AB.

Thus, the area of ΔADN is (1/2 * DN * AN) = (1/2 * 1/2 * AB * 1/2 * AB) = 1/8 * AB², which is the same as the area of ΔADM.

Therefore, the probability of selecting a point in ΔADN is also 1/8.

The probability of selecting a point in ΔADM but not in ΔADN is (1/8 * AB²) - (1/8 * AB²) = 0.

To find the probability that a point selected randomly in the square lies neither in ΔADM nor in ΔADN, we need to subtract the probability of selecting a point in ΔADM from the probability of selecting a point in the square and then subtract the probability of selecting a point in ΔADN from that result.

The probability of selecting a point in the square is 1.

The probability of selecting a point in ΔADM is 1/8, and the probability of selecting a point in ΔADN is also 1/8.

Therefore, the probability of selecting a point that lies neither in ΔADM nor in ΔADN is 1 - (1/8 + 1/8) = 3/4.

Hence, the probability of selecting a point

1) in ΔADN is also 1/8.

2) in ΔADM but not in ΔADN is 0.

3) lies neither in ΔADM nor in ΔADN is 3/4.

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The scatter plot suggests that there is a relationship between the number of chinups and push-ups, but it is not a perfect relationship

What is Scatter plot ?

A scatter plot is a type of graph that is used to display the relationship between two variables. It is a visual representation of a set of data points, where each point represents the values of two variables for a single observation.

Based on the scatter plot, it appears that there is a positive correlation between the number of chin-ups and push-ups several students did. This means that as the number of chin-ups increase, the number of push-ups also tend to increase.

We can see this trend in the general upward slope of the points in the scatter plot. However, there is also a fair amount of variability in the data, as some students did more push-ups than others even when they did a similar number of chin-ups.

Therefore,  the scatter plot suggests that there is a relationship between the number of chin-ups and push-ups, but it is not a perfect relationship

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The number 5 (68) denotes how much Jenna's mother must spend on all of the park entrance tickets along with the full-day food and beverage package.

the total cost for Jenna's water park birthday celebration:Total cost = 5 x 68 + 0.18z

.Equations Let's break down the expression 5 (68):

The number 68 represents the cost of one park entrance ticket with the full-day food and drink package.The number 5 represents the total number of people who need a park entrance ticket with the full-day food and drink package, which includes Jenna and her three friends plus Jenna's mom.So, we can write the following equation to represent the total cost of the park entrance tickets with the full-day food and drink package:

Total cost of park entrance tickets with the full-day food and drink package = 5 x 68Now let's look at the rest of the expression:The number 0.18 represents the cost per mile for gas.The variable z represents the total number of miles that Jenna's mom will drive to take each friend to and from the water park.

So, we can write the following equation to represent the total cost of gas for the trip:Total cost of gas = 0.18zFinally, we can add these two expressions together to get the total cost for Jenna's water park birthday celebration:Total cost = 5 x 68 + 0.18z

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Answer:

V = (l)(w)(h)

Step-by-step explanation:

This is essentially volume equals length times width times height.

V: Volume

L: Length

W: Width

H: Height

This means, that to find the volume of a rectangular prism, you multiply length times width times height. This also works if you have the values of volume and 2 values from length, width, or height, because you can rearrange the equation.

the 3D shape created by rotating the given rectangle around the X-axis will be a cylinder with its base centered at (-4,1.5), a radius of 1, and a height of 3.

What are 3D shapes?

The given coordinates (-3,0),(-5,0),(-5,3), and (-3,3) form a 2D rectangle lying in the XY-plane. When this rectangle is rotated around the X-axis, it will sweep out a 3D shape known as a cylinder.

A cylinder is a three-dimensional shape with a circular base and straight parallel sides. The base of the cylinder is formed by the 2D rectangle in the XY-plane, while the height of the cylinder is determined by the distance between the parallel sides of the rectangle. When the rectangle is rotated around the X-axis, it generates a circular cross-section at each height, resulting in a cylinder.

The center of the base of the cylinder will be located at (-4,1.5) on the XY-plane, which is the midpoint of the line joining the points (-3,0) and (-5,3). The radius of the base will be half of the width of the rectangle, which is (5-3)/2 = 1. The height of the cylinder will be the distance between the parallel sides of the rectangle, which is 3.

Therefore, the 3D shape created by rotating the given rectangle around the X-axis will be a cylinder with its base centered at (-4,1.5), a radius of 1, and a height of 3.

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To find the residual for each data point, we subtract the predicted sales from the actual sales. The residuals for the model are 1, 0, 1, and 1 for x=0, 1, 2, and 3 respectively.

To find the residual for each data point, we need to subtract the predicted sales from the actual sales:

For x=0: Residual = Actual sales - Predicted sales = 22 - 21 = 1

For x=1: Residual = Actual sales - Predicted sales = 33 - 33 = 0

For x=2: Residual = Actual sales - Predicted sales = 36 - 35 = 1

For x=3: Residual = Actual sales - Predicted sales = 28 - 27 = 1

Therefore, the residuals for the model are 1, 0, 1, and 1 for x=0, 1, 2, and 3 respectively.

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The possible lowest diameter of Katie's cylinder is approximately 5.26 cm.

To find the possible lowest diameter of Katie's cylinder, we'll use the concepts of normal distribution, mean, and standard deviation.
1. Recall the given information:
- Mean diameter (µ) = 5 cm
- Standard deviation (σ) = 0.2 cm
- Katie's cylinder is in the upper 10% in size (diameter).
2. Determine the z-score that corresponds to the upper 10% of the distribution:
To find the z-score for the upper 10%, you can use a z-table or a calculator with a built-in function to find the inverse of the cumulative distribution function (commonly known as the "inverse CDF" or "quantile" function). The z-score corresponding to the upper 10% is approximately 1.28.
3. Calculate the possible lowest diameter of Katie's cylinder using the z-score formula:
z = (X - µ) / σ
where:
- z is the z-score
- X is the diameter we want to find
- µ is the mean diameter
- σ is the standard deviation
4. Rearrange the formula to solve for X:
X = z * σ + µ
5. Plug in the values and solve for X:
X = (1.28 * 0.2) + 5
X = 0.256 + 5
X ≈ 5.26 cm

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Answer:

The last option is correct

Step-by-step explanation:

As a result, the horizontal component of velocity has a magnitude of 11.8 m/s\ while the vertical component of velocity has a value of 8.0 m/s.

The pace at which an item changes its position is described by a vector quantity called velocity. It is described as the rate at which displacement changes in relation to time The amount and direction of velocity are both present. Speed refers to the velocity's magnitude.

The football's velocity in this scenario is 32 miles per hour, or around 14.3 meters per second.

We can use trigonometry to determine the sizes of a velocity vector's horizontal and vertical components.

Vₓ = V cos (theta), where V is the magnitude of the velocity vector and theta is the angle that the velocity vector makes with the horizontal axis, gives the horizontal component of velocity.

Vy = V sin (theta), where V is the magnitude of the velocity vector and theta is the angle that the velocity vector makes with the horizontal axis, gives the vertical component of velocity.

In this instance, we are aware of the football's velocity, which is 32 miles per hour, or around 14.3 meters per second6. It makes a 34 degree 6 angle with the ground.

Therefore,

Vₓ= V cos (theta) * cos (34 degrees) = 14.3 m/s * 11.8 m/s

Vy = V sin (theta) * sin (34 degrees) = 14.3 m/s * 8.0 m/s

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Answer:

Step-by-step explanation:

The area of the rectangle garden can be expressed as:

Area = Length × Width

Since one side is 6 feet longer than the other, we can express the length as:

Length = Width + 6

Substituting this expression for length into the area formula, we get:

Area = (Width + 6) × Width

Simplifying this expression, we get:

Area = Width² + 6Width

The cost to install synthetic grass is given as $8 per square foot, so the total cost y can be expressed as:

y = 8(Area)

Substituting the expression for area that we found earlier, we get:

y = 8(Width² + 6Width)

Simplifying this expression, we get:

y = 8Width² + 48Width

Therefore, the equation that correctly shows the relationship between x (the length of the shorter side of the garden) and y (the total cost to install synthetic grass) is:

y = 8x² + 48x

The benefit of internal data is that internal data is more reliable and easier to collect.

Internal data are unique facts and figures that originate directly from the systems of the business in the issue. Internal data is rarely accessible to or studyable by outside parties without the express consent of the corporate entity.

The success of the company's present processes can be seen through internal data. Internal data, which is gathered from sources like website KPIs and customer surveys, is a priceless resource for assessing company policies, goods and branding, and worker efficiency.

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Answer:

x = 71

Step-by-step explanation:

180 degrees is how many degrees were in a triangle.

Add up the angles then subtract by 180 and there is your answer.

Hence, the distance between Hawkes Meadow and Brunswick Street is roughly 11.1 kilometres (km).

Distance is a unit used to express the distances between two objects or positions. This is a scalar value that is typically stated in length units like feet, metres, kilometres, or miles. Distance between two things is a physical concept that specifies their separation; it is frequently used to estimate how much space there is between them. The shortest distance, also referred to as "as the crow flies," can be calculated either along a horizontal path or a curved path. Displacement, or the shift in an object's position from its original location to its end position, is a notion in physics that is closely connected to distance.

given

Using the data from the timetable and the calculation, we can determine how many kilometres the bus travels between Hawkes Meadow and Brunswick Street:

Distance is determined by multiplying time and speed.

According to the timetable, the bus travels from Hawkes Meadow to Othello Avenue in 15 minutes (from 14:25 to 14:32), and from Othello Avenue to Brunswick Street in 8 more minutes (i.e., from 14:32 to 14:40). Hence, it takes 15 + 8 = 23 minutes to get from Hawkes Meadow to Brunswick Street.

Also, we are informed that the bus travels at an average speed of 29 km/h between Hawkes Meadow and Brunswick Street.

To utilise the calculation, we must first divide the time in minutes by 60 to convert it to hours:

23, divided by 60 hours, or 0.3833 hours (rounded to 4 decimal places)

Hence, we can compute the distance using the following formula:

11.12 kilometres is equal to 29 km/h times 0.3833 hours (rounded to 2 decimal places)

Hence, the distance between Hawkes Meadow and Brunswick Street is roughly 11.1 kilometres (km).

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Answer: Without the graphs posted in your question, unfortunately it is not possible for me to provide you an answer. Please include the graphs given in your question so that I can provide an accurate response.

Answer:

10π cm

Step-by-step explanation:

C = πd

First, we find the hypotenuse of the triangle which is the diameter of the circle.

a² + b² = c²

6² + 8² = c²

c² = 100

c = 10

d = 10 cm

C = πd

C = 10π cm

The choice 1 and choice 4 indicate the correct advantage of using random forest versus a single decision tree.

Random forest refers to the machine learning algorithm using several decision trees formed by random data subset and it's features. While single decision tree as indicated by name is based on one decision tree.

Decision tree finds the application in regression and construction tasks, where the outcome is generated through decision made from tree like model.

The decision tree is easily readable and interpretable by humans. The disadvantage of decision tree includes sensitivity to minor data changes.

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The amount of time it takes to produce 2700 watts of power depends on the rate at which the power is produced. If we know the rate at which the power is produced, we can use the formula:

time = energy / power

where energy is the amount of energy produced (in watt-hours), power is the rate of power production (in watts), and time is the amount of time it takes (in hours).

If the rate at which the power is produced is constant at 2700 watts, then the amount of time it takes to produce 2700 watts of power is:

time = energy / power

time = 1 watt-hour / 2700 watts

time = 0.00037 hours

So, it would take approximately 0.00037 hours, or about 2.2 seconds, to produce 2700 watts of power at a constant rate of 2700 watts.

Step-by-step explanation:

Not sure why you would want this...here is the result using a big number calculator on the internet:

2^2000 =

114,813,069,527,425,452,423,283,320,117,768,198,402,231,770,208,869,520,047,764,273,682,576,626,139,237,031,385,665,948,631,650,626,991,844,596,463,898,746,277,344,711,896,086,305,533,142,593,135,616,665,318,539,129,989,145,312,280,000,688,779,148,240,044,871,428,926,990,063,486,244,781,615,463,646,388,363,947,317,026,040,466,353,970,904,996,558,162,398,808,944,629,605,623,311,649,536,164,221,970,332,681,344,168,908,984,458,505,602,379,484,807,914,058,900,934,776,500,429,002,716,706,625,830,522,008,132,236,281,291,761,267,883,317,206,598,995,396,418,127,021,779,858,404,042,159,853,183,251,540,889,433,902,091,920,554,957,783,589,672,039,160,081,957,216,630,582,755,380,425,583,726,015,528,348,786,419,432,054,508,915,275,783,882,625,175,435,528,800,822,842,770,817,965,453,762,184,851,149,029,376

Answer:x-3y=-12

Step-by-step explanation:just trust me

If Tayugineans drink an average of 1.64 cups of coffee per day, then 0.38 percent will drink less than 1 cup-of-coffee per day.

The average number of cups of coffee drink is = 1.64 cups,

The variance is = 0.0576, So, the standard-deviation is = √0.0576 = 0.24,

Let us assume that the variable is normally-distributed with a standard deviation of 0.24 cup.  

We have to find the persons who drink less than 1 cup of coffee,

It is calculated as : P(x < 1) ,

The corresponding z-value is = z = (1 - 1.64)/0.24 = -2.6667 ,

So, P(z < -2.67) = 0.0038 = 0.38%.

Therefore, 0.38% drink less than 1 cup of coffee per day.

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If the sum of the height and the perimeter of the square base is 14 in, the maximum possible volume is 54 in³.

Let x represent the square box's rectangular base's side length.

Suppose h is the height.

Volume V should be used.

Considering that the square base's height plus perimeter equals 14 inches.

h + 4x = 18

Subtract 4x on both side, we get

h = 18 - 4x..........(1)

The volume is given as:

V = Area × Height

V = x² × h

Substitute the value of h

V = x² × (18 - 4x)

V = 18x² - 4x³..........(2)

Differentiate the equation 2 with respect to x

dV/dx = 36x - 12x²..........(3)

For the critical numbers,

dV/dx = 0

So, 36x - 12x² = 0

x(36 - 12x) = 0

x(-12x + 36) = 0

Equating equal to 0

x = 0                      -12x + 36 = 0

x = 0                      -12x = -36

x = 0                       x = 3

Differentiate the equation 3 with respect to x

d²V/dx² = 36 - 24x

At x = 3 the value of d²V/dx² is

d²V/dx²  = 36 - 24 × 3

d²V/dx² = 36 - 72

d²V/dx² = -36 < 0

x = 3 corresponds to the maximum of the volume V according to the second derivative test.

Now from the equation 1;

[tex]V_{\text{max}}[/tex] = 18(3)² - 4(3)³

[tex]V_{\text{max}}[/tex] = 18(9) - 4(27)

[tex]V_{\text{max}}[/tex] = 162 - 108

[tex]V_{\text{max}}[/tex] = 54 in³

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Answer:

If 25 is 40% of a value, what is that value?

A. 85

B. 12.5

D. 60

Step-by-step explanation:

You're welcome.

Answer:

C) 62.5

Step-by-step explanation:

Write the expression:

40% of x = 25

Then calculate:

4x/10 = 25

4x = 250

x = 62.5

we have derived the formula T = √(8R/2.5) for the time taken for a rope of length R to be rotated about a central point, made R the subject of the formula, and used the formula to calculate the length of a rope that takes 16 seconds for a single revolution and the time taken for a rope of length 18 metres to complete a revolution.

How to solve questions?

a) To make R the subject of the formula, we need to isolate it on one side of the equation. We start by rearranging the formula:

T = √(8R/2.5)

Squaring both sides of the equation, we get:

T²= 8R/2.5

Multiplying both sides by 2.5, we get:

2.5T²= 8R

Finally, dividing both sides by 8, we get:

R = 2.5T²/8

Therefore, R is given by the formula R = 2.5T²/8.

b) To find the length of rope if it takes 16 seconds for a single revolution, we can substitute T = 16 into the formula we derived in part (a):

R = 2.5T²/8 = 2.5(16)²/8 = 100

Therefore, the length of the rope is 100 metres.

c) To find the time taken for a rope of length 18 metres to complete a revolution, we can again use the formula we derived in part (a), but this time we substitute R = 18:

T = √(8R/2.5) = √(8*18/2.5) = √(115.2) ≈ 10.7 seconds

Therefore, the time taken for a rope of length 18 metres to complete a revolution is approximately 10.7 seconds, rounded to one decimal place.

In summary, we have derived the formula T = √(8R/2.5) for the time taken for a rope of length R to be rotated about a central point, made R the subject of the formula, and used the formula to calculate the length of a rope that takes 16 seconds for a single revolution and the time taken for a rope of length 18 metres to complete a revolution.

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Let's start by using algebra to solve for the number of boxes in small and large vans.

Let x be the number of small vans and y be the number of large vans. Based on the ratio given, we know that:

- Number of boxes in small vans = 3/8 * Total number of boxes

- Number of boxes in large vans = 5/8 * Total number of boxes

Using the information provided, we can write an equation:

5x + 25y = 400

Simplifying, we get:

x + 5y = 80

Now let's use the inequality provided to determine if less than 30% of the vans filled with boxes are large vans. We know that:

- Total number of vans filled with boxes = x + y

- Less than 30% of these vans are large vans, so y/(x+y) < 0.3

Substituting x + 5y = 80, we can simplify the inequality:

y/(x+y) < 0.3

y/80 < 0.3

y < 24

So if y (the number of large vans) is less than 24, then less than 30% of the vans filled with boxes are large vans.

Now we can solve for x and y using substitution. From x + 5y = 80, we get x = 80 - 5y. Substituting into 5x + 25y = 400, we get:

5(80 - 5y) + 25y = 400

400 - 20y = 400

20y = 0

y = 0

This means that y cannot be less than 24, as y = 0 would result in no large vans at all. Therefore, Hashem's statement is incorrect, and the number of large vans must be equal to or greater than 24.