Kayla, Tony, and Danielle kept track of their communications over a week. The data collected is represented in this display.

Which questions can be answered from this display?

The correct answer is wider confidence intervals.

In statistics, a confidence interval is a range of values that is likely to contain the true population parameter with a certain degree of confidence. The level of confidence is typically set at 90%, 95%, or 99%, and represents the percentage of intervals that would contain the true parameter if we were to repeat the sampling process many times.

Higher levels of confidence mean that we are more certain that the true parameter falls within the interval. However, achieving higher levels of confidence requires larger sample sizes or larger standard errors, which in turn leads to wider confidence intervals.

To understand this, consider the formula for a confidence interval:

CI = point estimate ± margin of error

The margin of error is a function of the sample size and the standard error. As the sample size increases or the standard error decreases, the margin of error decreases, which leads to a narrower confidence interval. Conversely, if we want to increase the level of confidence, we need to increase the margin of error, which requires either a larger sample size or a larger standard error, leading to wider confidence intervals.

Therefore, higher levels of confidence generally require wider confidence intervals, which means we are less precise in our estimate but more certain that the true parameter falls within the interval.

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Answer:

????????????

Step-by-step explanation:

?????????????

Step-by-step explanation:

Let the principal deposited by Ram at the age of 16 be P.

After the balance became 121/700 times of the initial principal, we have:

(121/700)P = P(1 + 10/100)^n

where n is the number of years for which the amount is compounded annually.

Simplifying the above equation, we get:

n = log(121/700)/log(1.1)

After Ram deposited 1/70 of the existing balance, his new balance became:

(121/700)P + (1/70)[(121/700)P] = (121/700)P(1 + 1/10)

After 1 year of this deposit, the new balance became:

(121/700)P*(1 + 1/10)(1 + 20/100) = (121/700)P(11/10)*(6/5) = (363/350)P

Given that this amount is Rs. 359 less than thrice of the initial principal, we get:

(363/350)P = 3P - 359

=> P = Rs. 2450

Therefore, the sum deposited by Ram at the age of 16 years was Rs. 2450.

The loan amount borrowed by Ram from the bank is Rs. 1,10,000 at a rate of 15% p.a. compounded annually for 4 years. Let the interest paid by Ram at the end of the 1st year be I1.

The amount to be paid by Ram at the end of the 1st year = 1,10,000*(1 + 15/100) = Rs. 1,26,500

Out of this, Ram pays only the interest amount, i.e., I1.

The remaining amount to be paid by Ram after the 1st year = 1,26,500 - I1

This amount is to be paid in 3 years, at a rate of 15% p.a. compounded annually.

Let the equal installments to be paid by Ram for the next 3 years be X.

Therefore, we have:

X*(1 + 15/100)^3 + X*(1 + 15/100)^2 + X*(1 + 15/100) = 1,26,500 - I1

Solving the above equation, we get:

X = Rs. 36,285.47

Therefore, the total interest paid by Ram to the bank is:

I1 + 3X - 1,10,000 = I1 + 336,285.47 - 1,10,000 = Rs. 59,856.41

To avoid borrowing an extra loan, the withdrawn amount should be equal to or greater than the amount required to start the business.

The withdrawn amount is given by:

3P - 359 = 3*2450 - 359 = Rs. 6891

Therefore, Ram should have waited for the amount in his bank account to become Rs. 6891 or more, which would take n years, where:

(121/700)2450(1 + 10/100)^n >= 6891

Solving the above equation, we get:

n >= 4.52

Therefore, Ram should have waited for at least 5 years to start the business, to avoid borrowing an extra loan.

The information inferred from the graphs includes the overall trend, frequency of responses, and distribution of data, and predicting what the graphs would look like if we polled 100 people depends on factors such as sample population, question being asked, and range of possible responses.

From the graphs created, we can infer several things such as the overall trend, frequency of responses, and distribution of the data. For instance, if the graphs depict a bell-shaped curve, we can conclude that the responses were normally distributed, and that most people fell within the average range of responses. Conversely, if the graphs show a skewed distribution, it can indicate that a particular response was more common among the participants. We can also deduce from the graphs which response was the most popular and which was the least popular.

As for predicting what the graphs would look like if we polled 100 people, it would depend on various factors such as the sample population, the question being asked, and the range of possible responses. However, we can assume that with a larger sample size, the graphs would provide a more accurate representation of the population as a whole, and the distribution of responses would be more normal.

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The two types of errors you might make in conducting hypothesis test is type I error occurs when we conclude that the new marketing campaign does not increase sales when it actually does, This would lead to missed opportunities for increased sales and revenue for the company. Type II error occurs when we conclude that the new marketing campaign does not increase sales when it actually does, This would lead to missed opportunities for increased sales and revenue for the company. So, the correct option is A) and D).

Type I error is also known as a "false positive" error, where we reject the null hypothesis (i.e., conclude that the marketing campaign does increase sales) when it is actually false (i.e., the campaign does not have an effect on sales). This would lead to missed opportunities for increased sales and revenue for the company.

Type II error is also known as a "false negative" error, where we fail to reject the null hypothesis (i.e., conclude that the marketing campaign does not increase sales) when it is actually false (i.e., the campaign does have an effect on sales). This would also lead to missed opportunities for increased sales and revenue for the company.

Therefore, it is important to choose an appropriate significance level (alpha) and calculate the power of the test to minimize the risk of both types of errors in a hypothesis test. The correct answers are (A) and (D).

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--The given question is incomplete, the complete question is given

"Suppose you are conducting a hypothesis test to determine if a new marketing campaign increases sales for a particular product. What are the two types of errors you might make in conducting this test? (select all that apply)

a) A type I error occurs when we conclude that the new marketing campaign does not increase sales when it actually does. This would lead to missed opportunities for increased sales and revenue for the company.

b) A type II error occurs when we conclude that the new marketing campaign increases sales when it actually does not. This would lead to wasted resources on the marketing campaign and potentially lost sales if the company shifts resources away from a successful strategy.

c) A type I error occurs when we conclude that the new marketing campaign increases sales when it actually does not. This would lead to wasted resources on the marketing campaign and potentially lost sales if the company shifts resources away from a successful strategy.

d) A type II error occurs when we conclude that the new marketing campaign does not increase sales when it actually does. This would lead to missed opportunities for increased sales and revenue for the company."--

The total width of the maze is 51.5+2=53.5 yards and the total length of the maze is 105 yards.

Equations are mathematical statements with two algebraic expressions on either side of an equals (=) sign. It illustrates the equality between the expressions written on the left and right sides. To determine the value of a variable representing an unknown quantity, equations can be solved. A statement is not an equation if there is no "equal to" symbol in it. It will be regarded as an expression.

The total area of the maze and walkway is equal to the area of the maze plus the area of the walkway. The area of the walkway is equal to the length of the maze plus twice its width since the walkway is one yard wide on both sides of the maze.

Let x be the width of the maze. So, (x+2) is the width of the maze and walkway, and (2x+2) is the length of the maze and walkway.

length×width=area

(2x+2)(x+2)=5616

2x²+6x+4=5616

2x²+6x=5612

Write in the form x2+bx=d by dividing both sides by 2:

x²+3x=2806

Complete the square by adding (23)2=2.25 to each side:

x²+3x+2.25=2806+2.25

(x+1.5)²=2808.25

Take the square root of both sides:

x+ 1.5 = ±√2808.25

x = −1.5±√2808.25

x ≈ −54.5 or x ≈ 51.5

A negative solution does not make sense so we take x≈51.5.

Hence, the total width of the maze is 51.5+2=53.5 yards and the total length of the maze is

2(51.5)+2

= 105 yards.

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Answer:

The lateral surface area of the given square-based pyramid is 74 in².

Step-by-step explanation:

The lateral surface area of a three-dimensional object is the total surface area excluding the area of the base(s), so the total area of the "sides".

Therefore, the lateral surface area of a square-based pyramid is the sum of the area of the four triangular sides.

The area of a triangle is half the product of its base and height.

Therefore, the formula for the lateral surface area (L.S.A.) of a square-based pyramid is:

[tex]\boxed{\begin{minipage}{6 cm}\underline{L.S.A. of a square-based pyramid}\\\\$\sf L.S.A.= 2 \cdot s \cdot \ell$\\\\where:\\ \phantom{ww}$\bullet$ $s$ is the base length. \\ \phantom{ww}$\bullet$ $\ell$ is the slant height.\\\end{minipage}}[/tex]

From inspection of the given diagram:

Substitute the given values into the formula to calculate the L.S.A. of the given square-based pyramid:

[tex]\begin{aligned}\implies \sf L.S.A.&=\sf 2 \cdot 5 \; in \cdot 7.4 \; in\\&=\sf 10 \; in \cdot 7.4 \; in\\&=\sf 74 \; in^2\end{aligned}[/tex]

Therefore, the lateral surface area is 74 in².

The lateral area of the pyramid is approximately 74square inches.

What defines a pyramid's horizontal side?

The triangular faces of the pyramid that converge at the apex are known as the lateral faces. These are the four triangles that encircle the square base in the provided net. As a result, the entire area of the four triangles makes up the pyramid's lateral surface area.

According to the given information:To find the lateral area of a pyramid, we need to find the sum of the areas of all the triangular faces except for the base. The lateral area formula for a pyramid is given by:Lateral area = 0.5 × Perimeter of base × Slant height where the slant height is the height of each triangular face.

First, let's find the perimeter of the base. Since the base is a square with a side length of 5.1 inches, its perimeter is:

Perimeter of base = 4 × Side length = 4 × 5.1 in = 20.4

In Next,

let's find the slant height. We can use the Pythagorean theorem to find the slant height of each triangular face.

The slant height is the hypotenuse of a right triangle with one leg equal to half the base length (since the base is a square) and the other leg equal to the height of the pyramid.

Half the base length = 5.1 in / 2 = 2.55

In Using the Pythagorean theorem, we get:Slant height = √(2.55² + 7²) ≈ 7.45 In Finally,

we can use the lateral area formula to find the lateral area of the pyramid:

Lateral area = 0.5 × Perimeter of base × Slant height= 0.5 × 20.4 in × 7.45 in≈ 76.02 in²

Therefore, the lateral area of the pyramid is approximately 76.02 square inches. Which is near to 74

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if you could fly like a bird, your trip would be approximately 3.9 miles shorter than the route you currently have to take (9 - 5.1 = 3.9).

If you were able to fly like a bird, you could take a direct path from your house to the grocery store, which would be shorter than the route you currently have to take.

By looking at the given directions, we can see that you have to travel a total of 1+1+5+2=9 blocks, with each block being a mile long, so your total distance traveled is 9 miles.

If you could fly like a bird, you could take a straight-line path from your house to the grocery store, which would be the hypotenuse of a right triangle with sides of 1 block and 5 blocks. Using the Pythagorean theorem, we can calculate that the distance of the hypotenuse would be the square root of[tex](1^2 + 5^2)[/tex] = square root of 26, which is approximately 5.

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After dividing the given expression 4×10²/4×10¹, we have the resultant answer which is 10.

An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles.

Algebraic expressions contain both numbers and variables in terms, separated by operators, whereas mathematical expressions only contain numbers and operators.

So, we have the expression:

= 4×10²/4×10¹

Now, solve as follows:

= 4×10²/4×10¹

= 4×100/4×10

= 400/40

= 10/1

= 10

Therefore, after dividing the given expression 4×10²/4×10¹, we have the resultant answer which is 10.

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the man has #112,500 left as disposable income after paying taxes.

the actual amount paid on the article after the 15% discount is N600.00.

To calculate the disposable income of the man, we first need to determine his taxable income.

Taxable income = Annual income - Tax-free pay

Taxable income = #140,000 - #30,000 = #110,000

Next, we need to calculate the tax he has to pay on his taxable income.

Tax = Tax rate x Taxable income

Note that the tax rate is given in kobo, so we need to convert it to naira.

1 naira = 100 kobo, so 25 kobo = 0.25 naira

Tax = 0.25 x #110,000 = #27,500

Finally, we can calculate the disposable income by subtracting the tax from the annual income.

Disposable income = Annual income - Tax

Disposable income = #140,000 - #27,500 = #112,500

Therefore, the man has #112,500 left as disposable income after paying taxes.

4.If the discount allowed on an article is 15%, it means that the customer will pay only 85% of the original price of the article after the discount is applied.

Let the original price of the article be P.

Then, the amount paid by the customer after the 15% discount is 85% of the original price:

Amount paid = 0.85P

We know that the customer received N600.00.

So, we can set up an equation:

0.85P = N600.00

Solving for P, we get:

P = N600.00 / 0.85

P = N705.88 (rounded to two decimal places)

Therefore, the actual amount paid on the article after the 15% discount is N600.00.

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The answer is  3u•v = 3(-6) = -18. So, the result is a Scalar

In physics, a scalar quantity is a physical quantity that has only magnitude, but no direction. Examples of scalar quantities include mass, time, temperature, energy, and distance. Scalar quantities are represented by a single numerical value, and they can be added, subtracted, multiplied, and divided like regular numbers.

On the other hand, a vector quantity is a physical quantity that has both magnitude and direction. Examples of vector quantities include displacement, velocity, acceleration, force, and momentum. Vector quantities are represented by a magnitude (the size or amount of the quantity) and a direction (the orientation of the quantity in space), and they are usually written as boldface letters or with an arrow above them. Unlike scalar quantities, vector quantities cannot be added, subtracted, multiplied, or divided like regular numbers. Instead, they must be combined using vector operations such as vector addition, dot product, and cross product.

The dot product of two vectors u and v is defined as:

u•v = (u1 * v1) + (u2 * v2)

where u1 and u2 are the components of vector u, and v1 and v2 are the components of vector v.

Using this formula, we can calculate:

u•v = (7 * -2) + (2 * 4) = -14 + 8 = -6

So, 3u•v = 3(-6) = -18.

Therefore, the result is a scalar.

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Gabrielle and Leslie are approximately 9.2 meters apart, rounded to the nearest tenth, given that Mrs. Kelley is standing 6 meters in front of Gabrielle and Leslie and Mrs. Kelley and Leslie are 7 meters apart.

Where M represents Mrs. Kelley, G represents Gabrielle, and L represents Leslie.

We can see that a right triangle is formed by Gabrielle, Mrs. Kelley, and the distance between Gabrielle and Leslie. We can use the Pythagorean theorem to find the missing side:

[tex]c^2 = a^2 + b^2[/tex]

where c is the distance between Gabrielle and Leslie, a is 6 meters (the distance between Mrs. Kelley and Gabrielle), and b is the distance between Mrs. Kelley and Leslie (which is equal to 7 meters).

Substituting the values, we get:

[tex]c^2 = 6^2 + 7^2[/tex]

[tex]c^2[/tex] = 36 + 49

[tex]c^2[/tex] = 85

Taking the square root of both sides, we get:

c ≈ 9.2

Therefore, Gabrielle and Leslie are approximately 9.2 meters apart, rounded to the nearest tenth.

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Answer: To solve the quadratic equation, we can start by simplifying the left side of the equation:

9(x^2 −10)−3 = 7

9x^2 - 90 - 3 = 7

9x^2 - 93 = 7

Next, we can isolate the x^2 term by adding 93 to both sides:

9x^2 = 100

Finally, we can solve for x by taking the square root of both sides and considering both the positive and negative square roots:

x = ±√(100/9)

x = ±(10/3)

Therefore, the solutions to the quadratic equation are:

x = 10/3 or x = -10/3.

Step-by-step explanation:

Answer: x=10/3

Step-by-step explanation:

9(x^2-10)-3=7

=>9(x^2-10)=7+3

=>x^2-10=10/9

=>x^2=10/9+10

=>x^2=100/9

=>x=10/3

Answer:

$750

Step-by-step explanation:

.8x = $600, so x = $750

Because the distances XY and X'Y' are different, we can conclude that the rotation is incorrect (it shouldn't affect the lengths of the sides).

Let's look at the original triangle, we can see that the distance between the vertices X (2, 5) and Y (10, 9) is:

Distance = √( (2 - 10)² + (5 - 9)²)

Distance = 8.94

The distance between the corresponding vertices in the second triangle X' (0, 1) and Y' (3, 9) is:

D = √( (0 - 3)² + (1 - 9)²) = 8.54

So that distance changed, which means that the shape of the triangle is now different, so the rotation is not done correctly.

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Thus, the theoretical probability (winning the game) when given that when orange marble or yellow marble appears is 1/4.

The concept of probability can be applied to either individual events or relationships between them, including intersection and union, in the field of accounting. Any event's probability value could be between 0 and 1 (or 0% and 100%).

Given data:

probability = favourable outcome  / total outcome

probability (winning the game) --> only when orange marble or yellow marble appears.

Thus,

Total favourable outcome = 1 yellow +  2 orange = 3.

probability (winning the game) = 3/12 = 1/4

Thus, the  theoretical probability (winning the game) when given that when orange marble or yellow marble appears is 1/4.

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Correct question:

A bag has 12 different-colored marbles: 4 green, 5 blue, 1 yellow, and 2 orange. You draw a marble from the bag. If you get an orange marble or yellow marble, you win a prize.

What is the theoretical probability of winning the game?

Answer:a) r >= -20
             b) -10 <= x, or x >=-10
             c) q<= 5

Step-by-step explanation:
a)-11r >=220
 -11r/-11 >= 220/-11
  r>=-20

b) -8<=-7+x/10
   -8+7<=x/10
   -1 <=x/10
    -10 <=x

c) -1-6(5q-2)>=-139
  -1 -30q +12 >= -139
   -30q+12-1>=-139
   -30q+11>=-139
   -30q >= -139-11
   -30q>= -151
    -30q/-30 >= -151/-31
    q <= 5,

u will notice the sign has changed from >= to <= it is coz when the negative signs erase themselves.
   

The constant of variation and write a direct variation equation relating s and t direct variation s= (1/2)t and t=32.

Direct Variation is that in which there is a  relationship between two variables in which one is a constant multiple of the other. For example, when one variable changes the other also changes, then they are said to be in proportion. If y is directly proportional to x then the equation is of the form y = k x (where k is a constant).

Given that,

s varies directly as t

i.e. s∝ t

   ⇒s= kt [ where k is the variational constant] -------------(1)

Given that s=8 when t= 16

putting these values in equation (1) we get,

8= k× 16

k= 8/16

k= 1/2

so, s= (1/2)t

If s= 16 then t= 16×2=32

Hence, s= (1/2)t and t=32

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Answer:

Mode: 5
Mean: 6
Median: 5

Step-by-step explanation:

- The mean is the sum of the data divided by the number of data points.

- The mode is the data point that appears most frequently.

- The median is the middle value in the sorted data set.

Answer:

54

Step-by-step explanation:

║∦⇵π⇅⇵⇅⇵⇅⇵⇵⇵⇵⇵⇵⇄⇆⇄⇆

Answer:

[tex]y=-2x+9[/tex]

Step-by-step explanation:

Perpendicular lines have slopes that are negative reciprocals of each other, so the line we need to find has a slope of [tex]-2[/tex].

Using point-slope form,

[tex]y+7=-2(x-8) \\ \\ y+7=-2x+16 \\ \\ y=-2x+9[/tex]

Using the graph,

The 1st statement is true as we can see from the graph the runner starts at origin 0 and till it reaches the point (12,1) it grows steadily as the graph is steep and increasing as we move forward.

This 2nd statement is also true as we can see in the graph that the 2nd runner's graph is a lot steeper than the 1st runner & stops at a point (20,2).

An organised representation of the data is all that the graph is. It facilitates our understanding of the facts. Data is the term used to describe the numerical information obtained through observation.

In the first statement:

It says that the runner begins the race & runs at a steady pace for 1 minute.

This statement is true as we can see from the graph the runner starts at origin 0 and till it reaches the point (12,1) it grows steadily as the graph is steep and increasing as we move forward.

Now in the 2nd statement it says, another runner runs from point (12,1) and runs at a steady but faster pace for 1 minute.

This statement is also true as we can see in the graph that the 2nd runner's graph is a lot steeper than the 1st runner.

So, the 2nd runner is going faster than the 1st and we can see that the 2nd runner stops at a point (20,2).

That shows the 2nd runner also ran for 1 minute.

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Answer:

Step-by-step explanation:

i would say 45cm

Step-by-step explanation:

The volume of a cone is given by the formula (1/3)πr^2h, where r is the radius of the base and h is the height. So, the volume of the cone is:

V_cone = (1/3)π(2 ft)^2(3 ft)

V_cone = (4/3)π ft^3

The volume of a cylinder is given by the formula πr^2h, where r is the radius of the base and h is the height. So, the volume of the cylinder is:

V_cylinder = π(2 ft)^2(10 ft)

V_cylinder = 40π ft^3

To find the total volume, we add the volumes of the cone and cylinder:

V_total = V_cone + V_cylinder

V_total = (4/3)π + 40π

V_total = 133.2 ft^3

Rounding off the answer to the nearest tenth of a cubic foot, the total volume is approximately 133.2 ft^3.

Answer: 138.23 ft³

Step-by-step explanation:

V = V of Cylinder + V of Cone

V of Cylinder = πr^2h

V of Cylinder = (2)^2 x 10 x π

V of Cylinder = 40π

V of Cone = πr^2(h/3)

V of Cone = π x (2)^2 x 3/3

V of Cone = π x 4 x 1

V of Cone = 4π

Total V = 40π + 4π

Total V = 138.23 ft³

There are 1008 different combinations of three chocolates that they can choose.

What is a combination?

Combination is a way of selecting objects from a larger set, where the order of the selected objects does not matter.

There are three separate choices to be made, so we need to multiply the number of choices for each step.

First, the daughter chooses one of the 8 milk chocolates. She has 8 options.

Next, the son chooses one of the 9 plain chocolates. He has 9 options.

Finally, Mrs. Sweet chooses one of the remaining 14 chocolates (8 milk and 9 plain chocolates have already been chosen). She has 14 options.

To find the total number of combinations, we multiply the number of choices for each step:

8 x 9 x 14 = 1008

Therefore, there are 1008 different combinations of three chocolates that they can choose.

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The graph of the quadratic function is plotted, in which Vertex (h,k) = (-4,3).

In mathematics, a quadratic equation is one with degree 2, which means that its maximum exponent is 2. A quadratic has the formula y = ax2 + bx + c, where a, b, and c are all numbers and a cannot be zero.

Given quadratic function:

y = (x+4)²+3

Now, the equation in vertex form is:

y = a(x - h)² + k

(h,k) are the vertex.

On comparing:

a = 1, h = -4, and k = 3

Vertex (h,k) = (-4,3)

Decide on a number of points along the parabola, making sure to get points on each side of the parabola and replacing x with both positive and negative numbers.

The ideal mirror image is: y = (x+4)²+3

x = 0, y = 19

x = 1, y = 28

x = 2, y = 39

x = -4, y = 3

Plot the vertex together with the points you choose. With the vertex serving as the minimum point, draw a parabola (curve) it through points.

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Complete question:

Given quadratic equation y=(x+4)²+3. Plot the graph.

Answer:

To solve this problem, we need to set the equation of the line and the equation of the given curve equal to each other and then solve for x. Once we find the value of x, we can substitute it back into either equation to find the corresponding y-coordinate. Here are the steps:

1. Set the equations equal to each other:

4 - 2x = 1

2. Solve for x:

4 - 2x - 1 = 0

3 - 2x = 0

3 = 2x

x = 3/2

2. Substitute x back into either equation to find y:

y = 4 - 2(3/2)

y = 1

Therefore, the point where the graph of y=4-2x crosses the line y=1 is at (3/2, 1).

To find when the daily ticket sales will peak, we need to find the maximum value of the quadratic equation [tex]N &= -0.3x^2+9x+13[/tex].

How to calculate no. of ticket solds and peak ticket sales?

Part 1:

The maximum value of a quadratic equation of the form [tex]y &= ax^2 + bx + c[/tex] occurs at [tex]x = \frac{-b}{2a}[/tex]. So in our case, the peak day will be at:

[tex]x &= \frac{-9}{2(-0.3)} = 15[/tex]

Therefore, the peak will be reached 15 days after the concert was first announced.

Part 2:

To find the number of tickets sold on that day, we substitute x=15 into the equation [tex]N &= -0.3x^2+9x+13[/tex]:

[tex]N &= -0.3(15)^2 + 9(15) + 13 = 67[/tex]

So the number of tickets sold on the day of the peak will be 67.

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Answer:

Solving for the unknown variable:

cosθ = sinθ

Dividing both sides by cosθ, we get:

tanθ = 1

Taking the inverse tangent of both sides, we get:

θ = π/4 + nπ, where n is an integer.

Therefore, the exact general solution is:

θ = π/4 + nπ, where n is an integer.

1 + sinθ = 2cos^2 (θ)

Using the identity cos^2 (θ) + sin^2 (θ) = 1, we can rewrite the equation as:

1 + sinθ = 2(1 - sin^2 (θ))

Expanding the right-hand side, we get:

1 + sinθ = 2 - 2sin^2 (θ)

Rearranging, we get:

2sin^2 (θ) + sinθ - 1 = 0

Using the quadratic formula, we get:

sinθ = (-1 ± √5)/4

Taking the inverse sine of both solutions, we get:

θ = arcsin [(-1 ± √5)/4] + 2nπ or π - arcsin [(-1 ± √5)/4] + 2nπ, where n is an integer.

Therefore, the exact general solution is:

θ = arcsin [(-1 ± √5)/4] + 2nπ or π - arcsin [(-1 ± √5)/4] + 2nπ, where n is an integer.

sin3θ = -1

Taking the inverse sine of both sides, we get:

3θ = (-1)^n (π/2) + 2nπ, where n is an integer.

Dividing both sides by 3, we get:

θ = (-1)^n (π/6) + (2nπ)/3, where n is an integer.

Therefore, the exact general solution is:

θ = (-1)^n (π/6) + (2nπ)/3, where n is an integer.

The area of a square can be calculated by squaring the length of one of its sides. So, we can find the length of the side of the playroom by taking the square root of 130.4:

√130.4 ≈ 11.4

Therefore, the measurement closest to the side length of the playroom in feet is 11 ft. Answer: 11 ft.