(1 point) Use the linear approximation to estimate (-2.02)2(2.02)3 = Compare with the value given by a calculator and compute the percentage error: Error = %

Answer 1 inch per hour

Step-by-step explanation:

Answer:

1 inch per hour

Step-by-step explanation:

Let x represent inches burned per hour

The equation to finding the rate of the candle burning:

3x=3

For those who dont know, whenever you have a variable with a term, its basically multiplying, and we want to do the opposite to get x by itself, so we divide

x=3/3

Which can be simplified as:

x=1

Where x is the inches burned per hour, which equal 1

The correct answer is 904.78 cm2

As y varies inversely as x, the value of y when x = 3 is 45.

Inverse proportionality is expressed as:

y ∝ 1/x

Hence:

y = k/x

Where k is the constant of proportionality.

First, we determine the constant of proportionality.

Using the information given in the problem.

When x = 5, y = 27

Substituting these values into the formula, we get:

y = k/x

27 = k/5

k = 135

Now that we have found the value of k, we can use the formula to find y when x = 3. Substituting x = 3 and k = 135, we get:

y = k/x

y = 135/3

y = 45

Therefore, the value of y is 45.

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The equation of the tangent line at (6, f(6)) is y = 6x - 48.

a. The slope of the secant line joining (2, f(2)) and (7, f(7)) is:

slope = (f(7) - f(2)) / (7 - 2)

We can find f(7) and f(2) by plugging in x = 7 and x = 2 into the expression for f(x):

f(7) = 7² - 6(7) = 7

f(2) = 2² - 6(2) = -8

Substituting these values into the slope formula, we get:

slope = (7 - (-8)) / (7 - 2) = 3

Therefore, the slope of the secant line joining (2, f(2)) and (7, f(7)) is 3.

b. The slope of the secant line joining (6, f(6)) and (6 + h, f(6 + h)) is:

slope = (f(6 + h) - f(6)) / ((6 + h) - 6) = (f(6 + h) - f(6)) / h

We can find f(6) and f(6 + h) by plugging in x = 6 and x = 6 + h into the expression for f(x):

f(6) = 6² - 6(6) = -12

f(6 + h) = (6 + h)² - 6(6 + h) = h² - 6h + 36 - 36 - 6h = h² - 12h

Substituting these values into the slope formula, we get:

slope = (h² - 12h - (-12)) / h = h - 12

Therefore, the slope of the secant line joining (6, f(6)) and (6 + h, f(6 + h)) is h - 12.

c. The slope of the tangent line at (6, f(6)) is the derivative of f(x) at x = 6:

f'(x) = 2x - 6

f'(6) = 2(6) - 6 = 6

Therefore, the slope of the tangent line at (6, f(6)) is 6.

d. To find the equation of the tangent line at (6, f(6)), we use the point-slope form of a line:

y - f(6) = f'(6)(x - 6)

Substituting f(6) and f'(6) into this equation, we get:

y - (-12) = 6(x - 6)

Simplifying, we get:

y = 6x - 48

Therefore, the equation of the tangent line at (6, f(6)) is y = 6x - 48.

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We can see here that the equation of the blue line is: y = -x -1.

Finding the value(s) of the variable(s) that make the equation true is the aim of an equation. These numbers are referred to be the equation's roots or solutions.

Finding an equation's answers may require algebraic manipulation, substitution, factoring, or other techniques, depending on how difficult the problem is.

The two points can be seen as thus:

(-1, 0) (0, -1)

Slope = -1.

Suppose, y = -x + a       (1, 0)

0 = - (-1) + a

a = -1

Thus, the equation is y = -x -1.

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Answer:

The radius of the circle is 5/√2 = (5√2)/2 inches.

Area of circle = π((5√2)/2)^2

= 25π/2 square inches

Area of triangle = (1/2)(5√2)((5√2)/2)

= 25/2 square inches

Area of shaded region

= (25/2)(π - 1) = 26.77 square inches

The function used to add together the last employee's sales for the three months on the Employee Sales Summary sheet is: =SUM('Employee Sales January:Employee Sales March'!E5)

SUM(E16): This function adds up the values in cells E16 from the current sheet. If the last employee's sales for the three months are stored in cells E16, E17, and E18, then this function would correctly calculate the total.

E16+E16+E16: This expression adds up the value in cell E16 three times. If the last employee's sales for the three months are stored in cells E16, E17, and E18, then this expression would not calculate the total correctly.

SUM('Employee Sales October:Employee Sales December'!E16): This function adds up the values in cell E16 from all sheets between Employee Sales October and Employee Sales December (inclusive). If the last employee's sales for the three months are stored in cells E16, E17, and E18 on different sheets, then this function could be used to calculate the total.

SUM('Employee Sales January:Employee Sales March'!E5): This function adds up the values in cell E5 from all sheets between Employee Sales January and Employee Sales March (inclusive).

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Full Question: On the Employee Sales Summary sheet, the function used to add together the last employee's sales for the three months is ___________________.

Group of answer choices

The inverse of the function y= (2/3)x^5-10 is y = [3/2(x + 10)]^1/5

From the question, we have the following parameters that can be used in our computation:

y= (2/3)x^5-10

Swap the ocurrence of x and y

so, we have the following representation

x = (2/3)y^5-10

Next, we have

(2/3)y^5 = x + 10

This gives

y^5 = 3/2(x + 10)

Take the fifth root of both sides

y = [3/2(x + 10)]^1/5

Hence, the inverse function is y = [3/2(x + 10)]^1/5

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Chelsey is correct that we use the same slope as the radius to write the equation of the tangent line, even though the slope of the radius is undefined at the points of tangency.

The perimeter will be 20 ✓2 units.

The area will be 50 units²

Chelsey is correct, and the reason is that a tangent line to a circle at a given point is always perpendicular to the radius of the circle at that point. This means that the slope of the tangent line and the slope of the radius at the point of tangency are negative reciprocals of each other.

Chelsey is correct that we use the same slope as the radius to write the equation of the tangent line.

The perimeter will be:

= 4 × BC

= 20 ✓2

The area will be:

= BC²

= (5✓2)²

= 50

It should be noted that BCDE is a square as EBC is 90°.

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Answer:

a) 18.84 cm

b) 28.26 cm²

Step-by-step explanation:

We Know

Each pancake is a circle with a diameter of 6 centimeters.

a)  Calculate the circumference of each pancake.

Circumference of circle = d · π

d = 6 cm

We Take

6 · 3.14 = 18.84 cm

So, the circumference of each pancake is 18.84 cm.

b) Calculate the area of each pancake.

Area of circle = r² · π

r = 1/2 · d

r = 1/2 · 6 = 3 cm

We Take

3² · 3.14 = 28.26 cm²

So, the area of each pancake is 28.26 cm².

The volume of the unoccupied space inside the cube is the volume of the cube minus the volume of the ball, which is approximately 166.

1. D. Cone. When a tree is cut down, its trunk typically has a roughly cylindrical shape with a tapered end, which closely resembles a cone.

2. B. 166. The diameter of the ball is 11 cm, which is also the length of the diagonal of the cube. Let's call the side length of the cube "s". Then, we can use the Pythagorean theorem to find s:

s^2 + s^2 + s^2 = 11^2

3s^2 = 121

s^2 = 121/3

The volume of the cube is s^3, which is approximately 166. The volume of the ball is (4/3)πr^3, where r is the radius of the ball. Since the ball fits tightly inside the cube, its diameter is equal to the side length of the cube, which is s√3. Thus, r = (s√3)/2 - 5.5.

The volume of the unoccupied space inside the cube is the volume of the cube minus the volume of the ball, which is approximately 166.

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Answer:

16 peaches

Step-by-step explanation:

Let's break this down:

Total: 144 peaches

Number of boxes she has to fill evenly: 9

Question: How many peaches are able to fit into each box evenly?

144 peaches/9 boxes = 16 peaches

So, Ariana should pack 16 peaches into each box

Hope this helps :)

3) The expression In(4x^5) - In(x^3) - In 4 can be simplified using the properties of logarithms. We know that ln(a) - ln(b) = ln(a/b) and ln(a^n) = n ln(a), so we can write:In(4x^5) - In(x^3) - In 4 = In[(4x^5)/(x^3)] - In 4= In(4x^2) - In 4= In(4x^2/4)= In(x^2)Thus, the simplified expression is In(x^2).4) To solve this problem, we need to use the formula for compound interest:A = P(1 + r/n)^(nt)where A is the final amount, P is the initial amount, r is the interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years.We want to find t when A = 3P and r = 0.02 (since the price of beef has inflated by 2%). We are told that interest is compounded biannually, so n = 2. Plugging in these values and solving for t, we get:3P = P(1 + 0.02/2)^(2t)3 = (1.01)^2tln(3) = ln(1.01^2t)ln(3) = 2t ln(1.01)t = ln(3) / (2 ln(1.01))Using a calculator, we find t ≈ 34.64 years. Therefore, it will take about 34.64 years for the price of beef to triple at a 2% biannual inflation rate.

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It will take approximately 110 years for the price of beef to triple when inflating 2% compounded biannually.

3) To simplify the expression In(4x^5) - In(x^3) - In(4), we will use the properties of logarithms:

- In(a) - In(b) = In(a/b)
- In(a^b) = b * In(a)

So, we can rewrite the expression as:

In(4x^5 / (x^3 * 4))

Now, we can simplify the expression inside the natural logarithm:

(4x^5) / (4x^3) = x^(5-3) = x^2

Thus, the simplified expression is:

In(x^2)

4) To find how long it will take for the price of beef to triple when inflating 2% compounded biannually, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

where A is the final amount, P is the initial amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. In this case, we want the final amount to be triple the initial amount:

3P = P(1 + 0.02/2)^(2t)

To solve for t, we can divide both sides by P:

3 = (1 + 0.01)^(2t)

Now, take the natural logarithm of both sides and use the properties of logarithms:

ln(3) = ln((1 + 0.01)^(2t))
ln(3) = 2t * ln(1 + 0.01)

Finally, isolate t:

t = ln(3) / (2 * ln(1 + 0.01))

t ≈ 109.96

It will take approximately 110 years for the price of beef to triple when inflating 2% compounded biannually.

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b. Nonresponse bias creates an impact on the confidence interval, but is not accounted for by the margin of error.

Given that, the dean of students at a large college is interested in learning about the opinions of students regarding the percentage of first-year students who should be given parking privileges in the main lot. He sends out an email survey to all students about this issue, but receives very few responses from sophomores, juniors, and seniors. Based on the responses he receives, he constructs a 90% confidence interval for the true proportion of students who believe first-year students should be given parking privileges in the main lot to be (0.71, 0.79).

Response bias refers to a systematic pattern of incorrect responses in a survey, which can be caused by factors such as question wording, social desirability bias, or interviewer bias.

Nonresponse bias, on the other hand, occurs when individuals who do not respond to a survey are systematically different from those who do respond, leading to a biased estimate of the population parameter.

Sampling variation refers to the fact that different samples from the same population can yield different estimates of the population parameter due to random variation.

Under coverage bias occurs when some members of the population are systematically excluded from the sample, leading to a biased estimate of the population parameter.

In this scenario, the fact that very few sophomores, juniors, and seniors responded to the survey could potentially introduce nonresponse bias, since those who did respond may not be representative of the entire population of students.

However, the confidence interval itself does not account for nonresponse bias or any other sources of bias. Instead, it reflects the range of values that is likely to contain the true proportion of students who believe first-year students should be given parking privileges in the main lot, based on the data that was collected.

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The profit is positive, the firm makes a profit of $21,243 at the optimal level of production.

Cost of producing 40 units

We know that the total cost of producing 30 units is $6000. Let's denote the total cost function by C(x), where x is the number of units produced. Then, we have:

C(30) = $6000

The marginal cost function is given as MC = 8x + 70. Integrating this function, we get the total cost function as:

C(x) = [tex]4x^2[/tex] + 70x + C

To find the value of the constant C, we use the fact that C(30) = $6000:

4[tex](30)^2[/tex] + 70(30) + C = $6000

Solving for C, we get:

C = $300

Therefore, the total cost function is:

C(x) = [tex]4x^2[/tex] + 70x + $300

To find the cost of producing 40 units, we evaluate C(40):

C(40) = [tex]4(40)^2[/tex] + 70(40) + $300

C(40) = $7000

Therefore, the cost of producing 40 units is $7000.

Optimal level of production:

The optimal level of production is the value of x that maximizes the profit function. To find this value, we need to set the marginal cost equal to the marginal revenue:

MC = MR

8x + 70 = -6x + 55

Solving for x, we get:

x = 5/7

Since the optimal level of production should be a whole number, we round x up to 1 unit.

Therefore, the optimal level of production is 1 unit.

Profit function:

The profit function is given as:

P(x) = R(x) - C(x)

where R(x) is the revenue function and C(x) is the cost function.

The marginal revenue function is given as MR = -6x + 55. Integrating this function, we get the revenue function as:

R(x) = -[tex]3x^2[/tex] + 55x + D

To find the value of the constant D, we use the fact that the revenue at x = 80 is $14,920:

[tex]-3(80)^2[/tex] + 55(80) + D = $14,920

Solving for D, we get:

D = $21,520

Therefore, the revenue function is:

R(x) = -[tex]3x^2[/tex] + 55x + $21,520

Substituting the cost function and revenue function in the profit function, we get:

P(x) = ([tex]-3x^2[/tex] + 55x + $21,520) - (4x^2 + 25x + $300)

Simplifying, we get:

P(x) = -[tex]7x^2[/tex] + 30x + $21,220

Therefore, the profit function is P(x) = [tex]-7x^2[/tex] + 30x + $21,220.

Profit or loss at the optimal level:

To find the profit or loss at the optimal level, we evaluate the profit function at x = 1:

P(1) = [tex]-7(1)^2[/tex] + 30(1) + $21,220

P(1) = $21,243

Since the profit is positive, the firm makes a profit of $21,243 at the optimal level of production.

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Step-by-step explanation:

To estimate the difference between 5 1/2 and 3 5/9, we can first round the fractions to the nearest whole number or simpler fractions. In this case, we can round 1/2 to 1/2 and 5/9 to 1/2 as well. Now, we have:

5 1/2 − 3 1/2

Subtracting the whole numbers, we get:

5−3=2

Subtracting the fractions, we get:

1/2 − 1/2 = 0

So, the estimated difference is 2, which falls between 2 and 2 1/2. Therefore, Elise chose option D as the correct answer.

The surface area of the cylinder is 96cm²

From the question, we have the following parameters that can be used in our computation:

Using the above as a guide, we have the following:

Surface area of the cylinder = The area around the cylinder + 2 * The area of the top

Substitute the known values in the above equation, so, we have the following representation

Surface area of the cylinder = 64 + 2 * 16

Evaluate

Surface area of the cylinder = 96

Hence, the surface area of the cylinder is 96cm²

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Monica's claim is incorrect. Even though her interest rate is higher, she will earn less interest, $10.20, after one year than Paul because her initial deposit is lower.

Paul will earn more interest of $13.20 because he deposited more money, even though his interest rate is slightly lower.

To determine who will earn more interest in one year, we shall calculate the interest earned by each person using the simple interest formula.

The formula for simple interest is:

I = P * r * t

where:

I = the interest earned

P = the principal (the amount deposited)

r = the interest rate (as a decimal)

t =s the time (in years)

For Monica, we are given:

P = $300

r = 0.034 (the interest rate is 3.4%)

t = 1 (the interest earned in one year)

Plugging these values, we have:

I = 300 * 0.034 * 1 = $10.20

So, Monica will earn $10.20 in interest after one year.

For Paul, we are given:

P = $400

r = 0.033 (interest rate = 3.3%)

t = 1 (interest earned in one year)

Plugging the values, we get:

I = 400 * 0.033 * 1 = $13.20

So, Paul will earn $13.20 in interest after one year.

Therefore, Monica's claim is incorrect. Monica will earn $10.20 in interest after one year while Paul will earn $13.20 in interest after one year.

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To find the derivative of h(x) = 8x, we use the power rule, which states that the derivative of x^n is nx^(n-1). Applying this rule to h(x), we get h'(x) = 8.

To find the value of h'(4), we simply plug in x = 4 into our derivative expression: h'(4) = 8.

Therefore, the derivative of h(x) = 8x at x = 4 is h'(4) = 8.
To find the derivative of the function h(x) = 8x at x = 4, you can use the power rule for differentiation. The power rule states that if h(x) = x^n, then h'(x) = n * x^(n-1).

For h(x) = 8x, n = 1, so:

h'(x) = 1 * 8x^(1-1) = 8

Now, to find h'(4), just plug in x = 4:

h'(4) = 8

So, h'(4) = 8.

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The length of AC in triangle ABC is 24 inches.

In triangle ABC, let point D be on side AC such that AB = BD = DC = 12 inches. We are given that the measure of angle BDC is twice the measure of angle ABD.

To find the length of AC, we can use the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. In this case, we can apply the Law of Cosines to triangle ABD to find the length of AD:

AD^2 = AB^2 + BD^2 - 2 * AB * BD * cos(ABD)

Since AB = BD = 12 inches, we have:

AD^2 = 12^2 + 12^2 - 2 * 12 * 12 * cos(ABD)

AD^2 = 288 - 288 * cos(ABD)

Now, let's consider triangle BDC. We are given that the measure of angle BDC is twice the measure of angle ABD. Let's denote the measure of angle ABD as x. Therefore, the measure of angle BDC is 2x.

Since the sum of angles in a triangle is 180 degrees, we can write:

x + 2x + angle BCD = 180

3x + angle BCD = 180

angle BCD = 180 - 3x

Now, let's apply the Law of Cosines to triangle BDC to find the length of BC:

BC^2 = BD^2 + CD^2 - 2 * BD * CD * cos(BDC)

BC^2 = 12^2 + 12^2 - 2 * 12 * 12 * cos(2x)

BC^2 = 288 - 288 * cos(2x)

Since AD = DC, we have AD = 12 inches. Now we can write the equation for the total length AC:

AC = AD + DC

AC = 12 + 12

AC = 24 inches

Therefore, the length of AC in triangle ABC is 24 inches.

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If the interest is compounded daily, the interest rate is 4.5%.

To determine the interest rate, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the final amount, $790

P = the principal, $350

r = the interest rate

n = the number of times the interest is compounded per year, in this case daily (n = 365)

t = the time period in years, 18

Substituting the values :

790 = 350(1 + r/365)³⁶⁵ˣ¹⁸

790 = 350(1 + r/365)⁶⁵⁷⁰

790/350 = (1 + r/365)⁶⁵⁷⁰

ln(790/350) = 6570 * ln (1 + r/365)

Using the property of logarithms that ln(1 + x) ~ x for small values of x, we can approximate the right-hand side as:

[ln(790/350)]/6570 = r/365

r = 0.045

r = 4.5%

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If you charge more than your limit on your credit card, you will likely face several consequences, including:

1. Over-limit fees: Many credit card companies charge a fee if you exceed your credit limit. This fee can vary depending on the issuer, but it's typically around $25 to $35.

2. Increased interest rates: If you go over your credit limit, your credit card company may increase your interest rate, making it more expensive to carry a balance on your card.

3. Lower credit score: Exceeding your credit limit can negatively impact your credit score, as it's an indication of risky financial behavior.

4. Reduced credit availability: Your credit card company may reduce your credit limit if you consistently go over the limit, making it harder for you to access credit in the future.

5. Declined transactions: If you're significantly over your limit, your credit card company may start declining transactions to prevent further over-limit spending.

To avoid these consequences, it's important to monitor your spending and ensure you stay within your credit limit. If you find yourself consistently approaching your limit, consider requesting a credit limit increase or using other methods to manage your spending.

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The value of the speed of the satellite in meters per second is,

⇒ Speed = 7.5 x 10³ m /sec

We have to given that;

A satellite orbiting Earth travels 4.95×10⁷ meters for each orbit.

And, It takes 6,600 seconds to make one orbit.

Hence, the speed of the satellite in meters per second is,

⇒ Speed = Distance / Time

⇒ Speed = 4.95×10⁷/ 6,600

⇒ Speed = 0.00075 x 10⁷

⇒ Speed = 7.5 x 10⁻⁴ x 10⁷

⇒ Speed = 7.5 x 10³ m /sec

Thus, The value of the speed of the satellite in meters per second is,

⇒ Speed = 7.5 x 10³ m /sec

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The area k of the triangle a = 3, c = 2, b = 135 degrees is approximately 1.06125 square units.

The area k of the triangle, we can use the formula:
k = (1/2) * b * c * sin(A)
where A is the angle opposite side a.
Find A, we can use the fact that the angles in a triangle add up to 180 degrees:
A + B + C = 180
Substituting in the given values, we get:
A + 135 + 180 = 360
A = 45 degrees
Now we can plug in all the values into the area formula:
k = (1/2) * 2 * 3 * sin(45)
k = 1.5 * 0.707
k = 1.06125
Therefore, the area k of the triangle is approximately 1.06125 square units.

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The sum of the arrivals and departures from these three cities is 3x + 18.1 million.

If we let x be the number of arrivals and departures at Frankfurt airport, then the number of arrivals and departures at London airport is x + 12.9 million

The number of arrivals and departures at Paris airport is x + 5.2 million.

The sum of the arrivals and departures from these three cities is:

x + (x + 12.9 million) + (x + 5.2 million)

Simplifying this expression, we can combine like terms:

3x + 18.1 million

Therefore, the sum of the arrivals and departures from these three cities is 3x + 18.1 million.

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The sum of all the angles in a quadrilateral is 360° ,So The angle measure indicated is 137°.

What is radius?

In classical geometry, the radius of a circle or sphere is any line segment that links the object's centre to its edge; in more modern usage, the term also refers to the length of such line segments. The Latin term "radius," which may also be used to describe a chariot wheel spoke, is where the word "radius" first appeared.

The length of tangents drawn from an external point is known to be constant. The circle's radius across the point of contact and any other point on the circle are perpendicular to the tangent. A quadrilateral has 360° of angles total. In light of this, 137° is the indicated angle measurement.

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Correct question is

Find the angle measure indicated. Assume that lines which appear to be tangent are tangent.

If Liquid a has a density of 1. 2 g/cm³, 150 cm of Liquid a is mixed with some of Liquid b to make Liquid c whose mass is 220 g and has a density of  1.1 g/cm³, then the density of liquid B is 0.8 g/cm³.

To find the density of liquid B, you can follow these steps:
1. Calculate the mass of liquid A using its density and volume:
Liquid A has a density of 1.2 g/cm³ and a volume of 150 cm³.

Mass of A = Density of A × Volume of A = 1.2 g/cm³ × 150 cm³ = 180 g

2. Calculate the mass of liquid B using the mass of liquid C and mass of liquid A:
Liquid C has a mass of 220 g.
Mass of B = Mass of C - Mass of A = 220 g - 180 g = 40 g

3. Calculate the volume of liquid C using its mass and density:
Liquid C has a density of 1.1 g/cm³.
Volume of C = Mass of C ÷ Density of C = 220 g ÷ 1.1 g/cm³ = 200 cm³

4. Calculate the volume of liquid B using the volume of liquid C and the volume of liquid A:
Volume of B = Volume of C - Volume of A = 200 cm³ - 150 cm³ = 50 cm³

5. Calculate the density of liquid B using it's mass and volume:
Density of B = Mass of B ÷ Volume of B = 40 g ÷ 50 cm³ = 0.8 g/cm³

So, the density of liquid B is 0.8 g/cm³.

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Jackson's gross monthly income for 40 hours per week is approximately $3,201.70 and gross annual income s $38,480.

To find Jackson's gross monthly income, we first need to find his gross weekly income.

Jackson's hourly wage is $18.50, so his weekly gross income for 40 hours of work is:

40 hours/week x $18.50/hour = $740/week

Calculate annual income:

To determine the gross annual income, we need to consider how many weeks there are in a year. Assuming 52 weeks in a year:

Annual income = Weekly income * Number of weeks in a year

Annual income = $740 * 52 = $38,480

To find Jackson's gross monthly income, we can multiply his weekly gross income by the number of weeks in a month (approximately 4.33):

$740/week x 4.33 weeks/month ≈ $3,201.70/month

Therefore, Jackson's gross monthly income for 40 hours per week is approximately $3,201.70.

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The problem presents a scenario where a square patio is placed in the centre of a 10m x 10m square backyard. The side length of the patio is given by x, and the remaining area of grass is expressed as A=-x^2+100.

To find the area of grass remaining, we can substitute different values of x into the equation.

For instance, if the patio is 5m x 5m, then x = 5 and the area of grass remaining is[tex]A = -5^2 + 100 = 75[/tex] square metres. Similarly, if the patio is 8m x 8m, then x = 8 and the area of grass remaining is [tex]A = -8^2 + 100 = 36[/tex] square metres.

As we can see, the area of grass remaining decreases as the size of the patio increases.

This problem illustrates the concept of content loaded and content remaining, where the initial content is the entire area of the square backyard, and the loaded content is the area of the square patio.

The remaining content is what is left after the loaded content is subtracted from the initial content. In this case, the loaded content is the patio area, and the remaining content is the grass area.

In summary, the area of grass remaining in the backyard after a square patio is placed in the centre can be calculated using the equation [tex]A=-x^2+100,[/tex] where x is the side length of the patio in metres.

The concept of content loaded and content remaining is also illustrated in this problem, where the loaded content is the patio area and the remaining content is the grass area.

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