tice Problems
Scientists have calculated that recycling 10 pounds of paper, results in 8 fewer gallons of
water being required to produce an equivalent amount of new paper. If a school begins a
recycling program and is able to increase the amount of paper they recycle by 500 pounds
per month, calculate number of gallons of water this conserves over an entire year.

the equivalent integral with the order of integration reversed is: ∫0^1 ∫1^2 log(x) 9(y) dydx = (9/2) (2log(2) - 1)

To write an equivalent integral with the order of integration reversed, we need to integrate first with respect to y and then with respect to x. So, we have:

∫a^b ∫f(y)g(x) F(x,y) dxdy

Reversing the order of integration, we get:

∫f(y)g(x) ∫a^b F(x,y) dydx

Now, substituting the given values for f(y), g(x), and F(x,y), we get:

∫0^1 ∫1^2 log(x) 9(y) dydx

= ∫0^1 [9(y)∫1^2 log(x) dx] dy

= ∫0^1 [9(y) (xlog(x) - x) from x=1 to x=2] dy

= ∫0^1 [9(y) (2log(2) - 2 - log(1) + 1)] dy

= ∫0^1 [9(y) (2log(2) - 1)] dy

= (9/2) [(2log(2) - 1) y] from y=0 to y=1

= (9/2) (2log(2) - 1)

Therefore, the equivalent integral with the order of integration reversed is:

∫0^1 ∫1^2 log(x) 9(y) dydx = (9/2) (2log(2) - 1)

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To approximate the values for Mₙ and S for n = 6 and 12, we'll use the trapezoidal rule (T), midpoint rule (M), and Simpson's rule (S). After calculating these approximations, we'll compute the errors Eₜ, Eₘ, and Eₛ.

For n = 6:
T₆ = (Approximation using trapezoidal rule)
M₆ = (Approximation using midpoint rule)
S₆ = (Approximation using Simpson's rule)

For n = 12:
T₁₂ = (Approximation using trapezoidal rule)
M₁₂ = (Approximation using midpoint rule)
S₁₂ = (Approximation using Simpson's rule)

Errors for n = 6:
Eₜ₆ = |Actual value - T₆|
Eₘ₆ = |Actual value - M₆|
Eₛ₆ = |Actual value - S₆|

Errors for n = 12:
Eₜ₁₂ = |Actual value - T₁₂|
Eₘ₁₂ = |Actual value - M₁₂|
Eₛ₁₂ = |Actual value - S₁₂|

As n is doubled (from 6 to 12), observe the changes in the errors:
- Eₜ and Eₘ typically decrease by a factor of about 4 (since error is proportional to 1/n² for these methods)
- Eₛ typically decreases by a factor of about 16 (since error is proportional to 1/n⁴ for Simpson's rule)

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The length of MQ and QN is 10 and 15 respectively and verify that MQ: QN = 2:3

The coordinate of M = (-12,-5)

The coordinate of N = (8,10)

n = 2 , m = 3

By using the section formula coordinate of Q =( [tex]\frac{mx_{1} + nx_{2} }{m+n }[/tex] , [tex]\frac{my_{1} + ny_{2} }{m+n}[/tex])

Coordinate of Q = ([tex]\frac{(-12)3 + 8(2)}{3+2}[/tex] , [tex]\frac{10(2) + 3(-5)}{2+3}[/tex])

Coordinate of Q = ( -4, 1)

Now using the distance formula

MQ = [tex]\sqrt{ (x_{2}- x_{1} )^{2} +(y_{2} -y_{1} )^{2}[/tex]

MQ = [tex]\sqrt{(-4+12)^{2}+(1+5)^{2} }[/tex]

MQ = √100

MQ = 10

Similarly,

QN = [tex]\sqrt{(8+4)^{2}+(10-1)^{2} }[/tex]

QN =  [tex]\sqrt{225}[/tex]

QN = 15

MQ:QN = 10:15

MA :QN = 2:3

Hence it is verified that MQ: QN = 2:3

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The height of the cone is approximately 12.15 millimeters (rounded to the nearest hundredth).

To find the height of the cone, we need to use the formula for the volume of a cone:

V = (1/3)πr²h

where V is the volume, r is the radius, h is the height, and π is approximately equal to 3.14.

We are given the volume of the cone as 2,279.64 cubic millimeters. We can plug this value into the formula and solve for h:

2,279.64 = (1/3)πr²h

Multiplying both sides by 3 and dividing by πr², we get:

h = (3 × 2,279.64) / (π × r²)

Now, we need to find the radius of the cone. Unfortunately, we are not given this information directly. However, we can use the fact that the volume of a cone is also given by:

V = (1/3)πr²h

If we rearrange this formula to solve for r², we get:

r² = 3V / (πh)

Now, we can substitute the given values for V and h and simplify:

r² = 3(2,279.64) / (π × h) ≈ 2,304.32 / h

Taking the square root of both sides, we get:

r ≈ √(2,304.32 / h)

Now, we can substitute this expression for r into our earlier formula for h:

h = (3 × 2,279.64) / (π × r²) ≈ (6,838.92 / π) / (2,304.32 / h)

Simplifying, we get:

h ≈ 2,279.64 × h / (2,304.32 / h)

h² ≈ 2,279.64 × h / (2,304.32 / h)

h³ ≈ 2,279.64

Taking the cube root of both sides, we get:

h ≈ 12.15

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Answer:

T=64

Step-by-step explanation:

Multiply both sides by 4

t/4=16

t/4×4=16×4 Cancel out the 4

t=64

This will result in Circle 1 having the same center and radius as Circle 2, thus proving that the circles are similar.

To prove that Circle 1 and Circle 2 are similar, we can apply the following transformations to Circle 1:

1. Translation: Translate Circle 1 by moving its center from (-3, 5) to (7, 5). This is a horizontal translation of 10 units to the right.

2. Dilation: Dilate Circle 1 with a scale factor of 0.4, which will reduce its radius from 10 units to 4 units (the same as Circle 2).

These transformations will result in Circle 1 having the same center and radius as Circle 2, thus proving that the circles are similar.

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The real-world problem is modeled by the proportion 66/100 = x/2,500, where 66 is the sample proportion and 2,500 represents the population size.

To find the value of x, which represents the number of individuals with a specific characteristic in the population, follow these steps:

1. Cross-multiply the terms in the proportion:
  66 * 2,500 = 100 * x
2. Simplify the equation:
  165,000 = 100x
3. Divide both sides by 100 to isolate x:
  x = 1,650

Thus, 1,650 individuals in the population share the specific characteristic represented by the sample proportion. This proportion helps us understand and predict the prevalence of a certain characteristic or behavior within a larger population, based on the information gathered from a smaller sample.

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The area of the table is approximately 651 square inches.

What is Area ?

Area is a measure of the size of a two-dimensional shape or surface, such as a rectangle, circle, or triangle. It is expressed in square units, such as square inches, square feet, or square meters.

Let's use the Pythagorean theorem to find the length of the table top:

Substituting the given values, we get:

40*40 = [tex]length^{2}[/tex] + 21*21

Simplifying and solving for length, we get:

[tex]length^{2}[/tex]= 1600 - 441

[tex]length^{2}[/tex] = 961

length = 31 inches (rounded to the nearest inch)

Now that we know the length and width of the table, we can find the area by multiplying them together:

area = length x width

area = 31 x 21

area = 651 square inches (rounded to the nearest inch)

Therefore, the area of the table is approximately 651 square inches.

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The distance between Town A and Town B is 550 km.

Let's denote the distance between Town A and Town B as D.

When the two cars first passed each other, let's assume that car A traveled a distance of x km and car B traveled a distance of D - x km.

Let's also denote the initial speeds of car A and car B as 6s and 5s, respectively, where s is some constant representing the speed of the slower car.

The time it took for the two cars to pass each other can be calculated using the formula:

time = distance / speed

For car A, the time it took to travel x km was:

x / (6s)

For car B, the time it took to travel D - x km was:

(D - x) / (5s)

Since the two cars traveled the same amount of time until they passed each other, we can set these two expressions equal to each other:

x / (6s) = (D - x) / (5s)

Solving for x, we get:

x = 6Ds / (11s)

After the speeds of both cars were reduced, car A's speed was (5/6) * 6s = 5s, and car B's speed was (3/4) * 5s = (15/4)s.

Let's denote the time it took for car A to travel the remaining distance from x to D as t.

Then, the time it took for car B to travel a distance of (D - x - 54) km is also t.

Using the new speeds, we can write the equation:

[tex](D - x - 54) = (15/4)s * t[/tex]

Solving for t, we get:

[tex]t = (4/15)(D - x - 54) / s[/tex]

The distance car A traveled after the two cars passed each other is:

D - x = D - 6Ds / (11s) = (5/11)D

The time it took for car A to travel this distance is:

[tex]t + x / (6s) = (4/15)(D - x - 54) / s + 6Ds / (66s)[/tex]

Setting these two expressions equal to each other and solving for D, we get:

D = 550 km

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1- On the 58th birthday, the account would have $24,209.98, 2- If the interest is compounded monthly, then on the 58th birthday, the account would have $26,322.47.

1- The formula for calculating the compound interest is given by A = P(1 + r/n)(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years. Here, P = $10,000, r = 0.03, n = 1, t = 40 years (58 - 18).

substituting the values in the formula, we get A = $10,000(1 + 0.03/1)1*40) = $24,209.98.

2) In this case, n = 12 (monthly compounding), and t = 12*40 (total number of months in 40 years). So, the formula for calculating the compound interest becomes A = P(1 + r/n)(nt) = $10,000(1 + 0.03/12)(12*40) = $26,322.47.

Since the interest is compounded more frequently, the amount at the end of 40 years is higher than when the interest is compounded annually.

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the distance between the two planes is |x - 19|. Since we don't have any information about the value of x, we cannot compute the exact distance. We can only give the answer in terms of |x - 19|, rounded to three decimal places.

To verify that the two planes are parallel, we need to check if their normal vectors are parallel. The normal vector of the first plane is <2, 0, 0> and the normal vector of the second plane is <2, 0, -4>. We can see that these vectors are parallel because they have the same direction but different magnitudes. Therefore, the two planes are parallel.

To find the distance between the planes, we can use the formula:

distance = |ax + by + cz + d| / √(a² + b² + c²)

where a, b, and c are the coefficients of the variables x, y, and z in the equation of one of the planes, and d is the constant term.

Let's use the first plane: 2x - 42 = 4

We can rewrite this as 2x - 38 = 0, which means that a = 2, b = 0, c = 0, and d = -38.

Substituting these values into the formula, we get:

distance = |2x + 0y + 0z - 38| / √(2² + 0² + 0²)
distance = |2x - 38| / 2
distance = |x - 19|

Therefore, the distance between the two planes is |x - 19|. Since we don't have any information about the value of x, we cannot compute the exact distance. We can only give the answer in terms of |x - 19|, rounded to three decimal places.

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The probability of rolling an even number and then an odd number is 1/4.

The probability of rolling an even number on a fair number cube is 1/2, since there are three even numbers (2, 4, 6) and six possible outcomes (1, 2, 3, 4, 5, 6).

Similarly, the probability of rolling an odd number is also 1/2.

To find the probability of rolling an even number and then an odd number, we need to multiply the probabilities of each event. So:

P(even and odd) = P(even) × P(odd)

P(even and odd) = (1/2) × (1/2)

P(even and odd) = 1/4

So the probability of rolling an even number and then an odd number is 1/4.

The number of desired outcomes for rolling an even number and then an odd number is 9

Since there are three even numbers and three odd numbers, and therefore 3 × 3 = 9 possible outcomes.

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Since the area of a similar figure is proportional to the square of its linear dimensions, if one similar figure has an area that is nine times the area of another, the larger figure must have dimensions that are three times the dimensions of the smaller figure.

This is because the area is the square of the linear dimensions. So, if we increase the linear dimensions by a factor of 3, the area increases by a factor of 3^2 = 9.

Therefore, the answer is 3.

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Over the last couple of years, Arnav's height has increased by 20% so his current height is 1.8 meters.

Arnav's height initially was 1.5 meters. Over the last couple of years, his height increased by 20%. To find the new height, we can use the formula: new height = initial height × (1 + percentage increase).

In this case, the initial height is 1.5 meters and the percentage increase is 20%, which can be expressed as a decimal (0.2). Using the formula, we can calculate Arnav's new height as follows:

New height = 1.5 meters × (1 + 0.2) = 1.5 meters × 1.2 = 1.8 meters.

After the 20% increase in height over the last couple of years, Arnav's current height is 1.8 meters.

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Correspondence is a relation of connection as it establishes a link between two sets of elements, often by relating each element in one set to a specific element in the other set.

Correspondence refers to the exchange of communication or information between two or more parties. It is a relation of connection because it involves establishing a link or connection between the sender and the receiver of the message.

For example, when two people exchange letters or emails, they establish a correspondence that connects them and allows them to communicate. Similarly, in business, correspondence can refer to the exchange of official documents such as letters, memos, and reports, which establish a connection between different departments or organizations. Overall, correspondence is an important aspect of communication that helps to establish and maintain relationships between individuals and groups.

For example, in mathematics, a correspondence can be seen when matching the elements of one set to another, such as associating students with their grades. In this case, the connection is created by linking each student to their respective grade, illustrating the concept of correspondence.

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Each box has a mass of 90 grams, which is found by setting up a proportion using the ratio of cans to boxes and the known mass of each can.

To solve this problem, we need to use proportions. We know that 3 cans have the same mass as 9 identical boxes, which means that the ratio of cans to boxes is 3:9 or simplified to 1:3.

We also know that each can has a mass of 30 grams. Therefore, we can set up the proportion:

1 can / 30 grams = 1 box / x grams

where x is the mass, in grams, of each box.

To solve for x, we can cross-multiply:

1 can * x grams = 30 grams * 1 box

x grams = 30 grams / 1 can * 1 box

Since the ratio of cans to boxes is 1:3, we can substitute 3 for the number of boxes:

x grams = 30 grams / 1 can * 3 boxes

x grams = 90 grams

Therefore, the mass of each box is 90 grams.

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The required inequalities are h + l ≤ 20, 10h + 15l ≥ 240 and 20h + 25l ≥ 440

Given, for tutoring elementary students in math Susan earns $10 per hour. She earns $15 per hour for cleaning in the library.

Let h be the number of hours Susan works in one week tutoring elementary students.

Let l be the number of hours Susan works in one week cleaning the library.

Given that each week Susan cannot work more than 20 hours.

So, h + l ≤ 20 ....(1)

Susan's total earnings must be at least $240 per week.

10h + 15l ≥ 240    ...(2)

Multiplying equation (1) by 10

10h + 10l ≤ 200    ...(3)

Adding equations (2) and (3)

20h + 25l ≥ 440

Thus, the three required inequalities are h + l ≤ 20, 10h + 15l ≥ 240 and 20h + 25l ≥ 440

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The value of b = 4

The function becomes undefined when the denominator goes to 0

3x² - 48 = 0

3x² = 48

x² = 16

x = +/ 4

From the given choices, it's x = 4

Rational expressions that utilize ratios of polynomial expressions are referred to as rational expressions. These can be written in the format p(x)/q(x), where both p(x) and q(x) are polynomials with the constraint that q(x) cannot equal zero.

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When we look at [tex]f(t)=1/7 t^3-4t-12[/tex], this is a cubic equation, and solving it analytically is not straightforward.

To find the value of r such that f(r) = 0, we need to solve the equation:

[tex]1/7 r^3 - 4r - 12 = 0[/tex]

This is a cubic equation, and solving it analytically is not straightforward.

Yet, it is possible to obtain the value of r that meets the equation using numerical schemes such as Newton-Raphson or bisection. Additionally, one can take advantage of calculation tools and graphical software to calculate an estimation of r.

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The most general antiderivative of the function g(v) = 5 cos(v) - 8/√(1-v^2) is 5 sin(v) + 8 arcsin(v) + C, where C is the constant of the antiderivative.

To find the antiderivative of the given function g(v), we can use the basic antiderivative rules. The antiderivative of 5 cos(v) is 5 sin(v), as the derivative of sin(v) is cos(v) and we only need to reverse the process.

Similarly, the antiderivative of -8/√(1-v^2) can be found using the inverse trigonometric function arcsin(v), as its derivative is -1/√(1-v^2). However, we need to include a constant of integration, denoted by C, as the antiderivative is not unique.

So the most general antiderivative of g(v) is 5 sin(v) + 8 arcsin(v) + C, where C represents the constant of the antiderivative. To check the correctness of the answer, we can differentiate it and verify if it gives us the original function g(v) as the result.

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Answer:

Step-by-step explanation:

cat

The dimensions of the given rectangle is 2 1/2 meters (width) and 3 1/3 meters (length).

First, let's define the variables for the rectangle: let the width be w meters, and the length be (4/3)w meters since the length is 4/3 times the width. The area of a rectangle is calculated by multiplying its length and width. In this case, the given area is 8 1/3 square meters.

Now, we can write an equation using the area and dimensions:

Area = Length × Width
8 1/3 = (4/3)w × w

First, convert the mixed number 8 1/3 to an improper fraction, which is 25/3. Then, we can solve for w:

25/3 = (4/3)w²

To find w², multiply both sides by 3/4:

w² = (25/3) × (3/4)
w² = 25/4

Now, take the square root of both sides:

w = √(25/4)
w = 5/2

So, the width is 5/2 meters, or 2 1/2 meters. To find the length, multiply the width by 4/3:

Length = (4/3)(5/2) = 20/6 = 10/3

The length is 10/3 meters, or 3 1/3 meters. Therefore, the dimensions of the rectangle are 2 1/2 meters (width) and 3 1/3 meters (length).

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Two events are independent if the occurrence of one event does not affect the occurrence of the other event. In other words, the probability of one event happening is not affected by whether or not the other event happens.

A simple example would be flipping a coin and rolling a die. The outcome of the coin flip does not affect the outcome of the die roll, so these events are independent.

On the other hand, two events are mutually exclusive if they cannot happen at the same time. If one event happens, the other event cannot happen. For instance, when rolling a die, the events of getting a 1 or a 2 are mutually exclusive because it is impossible to roll both numbers at the same time.

To summarize, two events are independent if the probability of one event happening is not affected by the occurrence of the other event, while two events are mutually exclusive if they cannot happen at the same time.

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The situation provided is exponential growth and the equation is

p = 225 * 1.133ˣ

This scenario represents a significant improvement as the number of students increases each year.

The basic approach to incremental growth is:

[tex]P = P_{0} * (1 + r)^t[/tex]

where:

P. = initial population

r = increase in decimal form

t = time in years

Here

P₀ = 225 (initial population in 2021).

r = 13.3% = 0.133 (growth rate as decimal) .

t = x (time in years from the opening of the school in 2021)

Substituting these values ​​in the formula we get:

[tex]p = 225 * (1 + 0.133)^x[/tex]

Simplifying further, we get:

p = 225 * 1.133ˣ

This is the equation that predicts school population x years after the school opens in 2021

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The total or actual time he needs to practice is 140 min whereas he practiced for 98 min during the school week.

We need to find the total time he must practice for a week. To find the total time we assume that the total time is x min.

Given Data:

Dmitri practices time during the school week = 98 min

Dmitri practices amount of time = 70% of his total time

Total time = x

Then the equation is given as

70% × (x) = 98

0.70 × (x) = 98

x = 98 / 0.70

x = 140

Therefore, The total time of the practices is 140 min

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The coordinates of the triangle after the dilation are given as follows:

a(0, 12), b(-6, 15) and c(9, 21).

The perimeter of the triangle increases, as the side lengths are multiplied by 3, hence the perimeter is also multiplied by 3.

A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The scale factor for this problem is given as follows:

k = 3.

The scale factor is greater than 1, meaning that the figure is an enlargement, and thus the perimeter increases.

The original vertices of the triangle are given as follows:

a(0, 4), b(−2, 5), and c(3, 7)

Hence the vertices of the dilated triangle are given as follows:

a(0, 12), b(-6, 15) and c(9, 21).

(each coordinate of each vertex is multiplied by the scale factor of 3).

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Caleb has 30 quarters.

What is arithmetic?

Mathematical arithmetic is the study of the properties of the standard operations on numbers, such as addition, subtraction, multiplication, division, exponentiation, and root extraction.

Here, we have

Given: Caleb has 51 coins (nickels, dimes, and quarters) in a jar, totaling $9. He has three more nickels than dimes.

We have to find out how many quarters Caleb has.

Let x be nickel,

y be dimes and

z be quarters

x + y + z = 51.....(1)

1 quartes = 25 cents

1 dimes = 10 cents

1 nickel = 5 cents

Now, the total dollar is $9,

5x/100 + 10y/100 + 25z/100 = 9

5x + 10y + 25z = 900

x + 2y + 5z = 180....(2)

and

y + 3 = x...(3)

Solving equation(1) and (2), we get

From (1)

x + x-3 + z = 51

2x + z = 54....(4)

From (2)

x + 2(x -3) + 5z = 180

3x + 5z = 186...(5)

Now, by solving equations (4) and (5), we get

x = 12

z = 30

Now,

y + 3 = x

y + 3 = 12

y = 9

Hence, Caleb has 30 quarters.

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0.626 liters of water is needed to fill the container to its brim.

The volume of the rectangular container can be found by multiplying the area of the base (length x width) by the height:

Volume of rectangular container = length x width x height

Since the base is a square with a side of 12 cm, the area of the base is:

Area of base = 12 cm x 12 cm = 144 cm^2

Converting the height to cm, we have:

Height = 16.5 cm

So the volume of the container is:

Volume = 144 cm^2 x 16.5 cm = 2376 cm^3

To convert the volume from cubic centimeters to liters, we divide by 1000:

Volume = 2376 cm^3 ÷ 1000 = 2.376 liters

Since there is already 1.75 liters of water in the container, the amount of water needed to fill the container to its brim is:

Amount of water needed = 2.376 liters - 1.75 liters = 0.626 liters

Therefore, 0.626 liters of water is needed to fill the container to its brim.

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The weight of Harry Skyes on Mercury is 32.8 kg, under the condition that Mercury's gravity = 3. 70 m/s².

The weight of astronaut Harry Skyes on Mercury can be evaluated using the formula:

Weight on Mercury = (Weight on Earth / 9.81 m/s²) × 3.7 m/s²

Given that Harry Skyes has a mass of approximately 85.0 kg, his weight on Mercury would be:

Weight on Mercury = (85.0 kg / 9.81 m/s²) × 3.7 m/s²

Weight on Mercury = 32.8 kg

Gravity affects weight severely and causes its change . Objects have mass, which is specified as how much matter an object contains. Weight is known as the pull of gravity on mass. The relation between weight and gravitational pull is such that, when on another celestial body, the difference in gravity would alter a person’s weight.

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[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ P(\stackrel{x_1}{x}~,~\stackrel{y_1}{y})\qquad R(\stackrel{x_2}{-2}~,~\stackrel{y_2}{-4}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ -2 +x}{2}~~~ ,~~~ \cfrac{ -4 +y}{2} \right) ~~ = ~~\stackrel{\textit{\LARGE Q} }{(-5~~,~~1)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ -2 +x }{2}=-5\implies -2+x=-10\implies \boxed{x=-8} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ -4 +y }{2}=1\implies -4+y=2\implies \boxed{y=6}[/tex]

Answer:

The answer is (-8,6)