How Does This Topic Reinforce Or Change Your Belief About The Application Of Math To Your Daily Life? (2024)

If there is no significant difference found, then the null hypothesis can be accepted, indicating that donating to charity does not necessarily lead to greater happiness levels compared to spending on oneself.

The appropriate null hypothesis for this study is that there is no significant difference in happiness levels between those who donate to charity and those who spend money on themselves.

The alternative hypothesis, on the other hand, is that donating to charity results in greater happiness levels than spending on oneself, on average. In order to test these hypotheses, statistical methods such as a t-test or ANOVA can be used to analyze the data collected from the participants.

If the results of the statistical analysis indicate a significant difference in happiness levels between the two groups, then the alternative hypothesis can be accepted, providing convincing evidence that donating to charity can lead to greater happiness levels.

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The height of an object dropped off a 350 foot cliff after 1.5 seconds is 314 feet

We have,

An equation is an expression that shows the relationship between two or more numbers and variables.

Let h(t) represent the height of an object after t seconds, hence:

h(t) = -16t² + 350

Hence:

At time t = 1.5:

h(1.5) = -16(1.5)² + 350 = 314 feet

The height of an object dropped off a 350 foot cliff after 1.5 seconds is 314 feet

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The length of the hypotenuse AC is 5 units.

The correct answer is: A) 5 units.

In the given right triangle ABC, we are given the lengths of the two legs, AB and BC, as 3 units and 4 units, respectively.

We need to determine the length of the hypotenuse, AC.

According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, it can be expressed as :

[tex]AC^2 = AB^2 + BC^2[/tex]

Substituting the given lengths, we have:

[tex]AC^2 = (3 units)^2 + (4 units)^2[/tex]

[tex]AC^2 = 9 units^2 + 16 units^2[/tex]

[tex]AC^2 = 25 units^2[/tex]

To find the length of AC, we take the square root of both sides:

AC = √(25 [tex]units^2[/tex])

AC = 5 units.

Therefore, the length of the hypotenuse AC is 5 units.

The correct answer is:

A) 5 units.

The Pythagorean theorem is a fundamental concept in geometry that relates the sides of a right triangle.

It is commonly used to find missing side lengths or to determine if a triangle is a right triangle.

In this case, by applying the theorem, we were able to calculate the length of the hypotenuse AC based on the given side lengths.

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The complete question may be like:

In a right triangle ABC, angle B is a right angle. The lengths of the two legs are given by AB = 3 units and BC = 4 units. The length of the hypotenuse AC can be determined using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. What is the length of the hypotenuse AC?

A) 5 units

B) 7 units

C) 9 units

D) 12 units

approximately 1.81 meters.
Step-by-step explanation:
Let's call the height of the tree "h". We can set up a proportion based on the similar triangles formed by Nicole, her shadow, the tree, and the tree's shadow:
(height of Nicole) / (length of Nicole's shadow) = (height of tree) / (length of tree's shadow)
Substituting the given values, we get:
1.55 / 25.6 = h / 30.05
Simplifying and solving for h, we get:
h = (1.55 / 25.6) * 30.05
= 1.81181640625
Rounding to the nearest hundredth, we get:
h ≈ 1.81 meters
the height of the tree is approximately 1.81 meters.

Answer:

2√6

Step-by-step explanation:

√24=√2*2*2*3 the square root is to all numbers

√24=2√6

Given the difference that exists between the the total blue area and the total green area, we can see that the blue area is larger

How to solve for the larger area

We have to solve for the blue area

2 {area of the largest face}

The largest face is 9 x 5

2[9 * 5]

= 90 cm²

Then total green area calculation

= 2{5 * 3} + 2{9 * 3}

= 84 cm²

Hence given the difference that exists between the the total blue area and the total green area, we can see that the blue area is larger

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Answer:

3/10

Step-by-step explanation:

a. The function (f o g)(x) = 3/(3 + x)

b. In interval notation, the domain of the function (f o g)(x) is (-3, ∞).

What is a function?

A function is a mathematical equation that shows the relationship between two variables.

a. Given the functions f(x) = x/x + 2 and g(x) = 6/x, we desire to find (f o g)(x). We proceed as follows

We know that (f o g)(x) is a composite function and (f o g)(x) = f(g(x))

Since f(x) = x/x + 2 and g(x) = 6/x,

So f(g(x)) = x/(x + 2)

= g(x)/[g(x) + 2]

= 6/x ÷ (6/x + 2)

= 6/x ÷ (6 + 2x)/x

= 6/x × x/(6 + 2x)

= 6/(6 + 2x)

= 6/2(3 + x)

= 3/(3 + x)

So, f(g(x)) = 3/(3 + x)

b. To find the domain of (f o g)(x) in interval notation, we proceed as follows

Since (f o g)(x) = f(g(x)) = 3/(3 + x)

Now, f(g(x)) is valid when (3 + x) > 0

So, 3 + x > 0

x > - 3

So, in interval notation, the domain of (f o g)(x) is (-3, ∞). We use the open bracket on the left side of the - 3 since -3 is not included.

So, in interval notation, the domain of (f o g)(x) is (-3, ∞).

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Answer:

Step-by-step explanation:

V = π.[tex]r^{2}[/tex].h

100π = π.[tex](5)^{2}[/tex].h

h = 4

Answer:

4

Step-by-step explanation:

The equation for the volume of a cylinder is [tex]V=\pi r^{2} h[/tex]. So if V is 100[tex]\pi[/tex], it means that [tex]\pi hr^{2}[/tex] is [tex]100\pi[/tex]. So the you plug in the information you know [tex]25h\pi =100\pi[/tex]. The pi cancels out and you end up with [tex]25h=100[/tex] meaning that h=4

The measure of DF is 82 and measure of ∠PQR is 59°.

We know that,

For the first figure,

The secant- secant angle CEG is half the difference of the measures of the intercepted arcs, that is

∠ CEG = (1/2) (CG - DF)

44° = (1/2)( 170 - DF)

Multiply both sides by 2 to clear the fraction

88° = 170 - DF

subtract 170 from both sides

- 82 = - DF

multiply both sides by - 1

DF = 82°

For the second figure,

The tangent- tangent angle PQR is half the difference of the measures of the intercepted arcs.

the intercepted arcs are PSR and RP

the sum of the arcs in a circle = 360° , so

Hence,

PSR = 360° - RP = 360° - 121° = 239°

Therefore,

∠ PQR = (1/2)( 239 - 121)° = (1/2) × 118° = 59°

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Answer: f(0) = 4

Step-by-step explanation:

First, we see that 0 is greater than -2, so we will substitute 0 for x into the x + 4 part of the piece-wise function. Then we will simplify.

f(x > -2) = x + 4

f(0) = 0 + 4

f(0) = 4

Solution:

(g o g)(x) = -((-x^2 + 5)^2) + 5

Answer:

(g o g)(x) = -x^4 + 10x^2 - 20

Based on the information, it should be noted that the perimeter that they need is 130 yds to build up the fence.

How to explain the value

Attached you can find a picture that will guide the explanation. We will assume that each line is supposed to be a fence. We will assume also that the diagonals that cross any subfield that has the same vegetable is unnecessary.

On the same fashion, the diagonal of the spinach-celery sector is 10. Recall that the field is a rectangle, so the perimeter of the outter fence is 2*(12+6)+2*(8+9) = 70 yds. Then, the total perimeter is given by

outter fence (70)+

fence spinach-watermelon (8) +

fence red peppers-watermelon(12)+

fence red peppers-carrots (15)+

fence carrots-tomatoes (9)+

fence celery-tomatoes(6) +

fence spinach-celery(10)

= 130 yds

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Answer:

sin(θ) = -12/13

Step-by-step explanation:

You want the sine of the third-quadrant angle whose cosine is -5/13.

3rd Quadrant

In the third quadrant, both the sine and the cosine are negative. The sine of the angle can be found from your knowledge of the {5, 12, 13} Pythagorean triple, or using the Pythagorean trig identity:

sin(θ) = -√(1 -cos(θ)²)

sin(θ) = -√(1 -(-5/13)²) = -√(144/169)

sin(θ) = -12/13

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The minimum amount of plastic wrap needed is 445.08 square inches

How much plastic wrap is needed?

To get minimum amount of plastic wrap needed to completely wrap 6 containers, we have to get total surface area of all 6 containers.

The surface area of one container is found using [tex]SA = 2\pi r^2 + 2\pi rh[/tex]

The diameter of each container is 4.5 inches

The radius is half of diameter = 2.25 inches.

The height of each container is 3 inches.

Substituting these values.

SA = 2 x 3.14 x 2.25^2 + 2 x 3.14 x 2.25 x 3

SA = 74.1825

SA = 74.18 square inches

The total surface area of 6 containers is:

= 6 x 74.18

= 445.08 square inches.

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Answer:

44 degrees.

Step-by-step explanation:

Interior angles in a triangle add up to 180 degrees. The little square in the top right hand corner of the right-angled triangle indicates a right angle. Right angles are always equal to 90 degrees. Do 180-90-46=44, because there are three angles in the triangle that have a cumulative value of 180 degrees, and we know the value of two of them (90 and 46), so 44 degrees must be the value of the remaining angle.

Answer:

Part A: -x^2 + 15x + 1

Part B: 4x^3 - 2x^2 - 13x - 7

Part C: No

Step-by-step explanation:

Part A:

To find the total length of the two sides, we simply add them together:

(8x^2 - 5x - 2) + (7x - x^2 + 3)

= -x^2 + 15x + 1

Therefore, the total length of the two sides is -x^2 + 15x + 1.

Part B:

To find the length of the third side, we need to use the fact that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. So we set up an inequality:

(8x^2 - 5x - 2) + (7x - x^2 + 3) > (length of third side)

Simplifying, we get:

6x^2 + 2x + 1 > (length of third side)

Now we use the fact that the perimeter of the triangle is equal to the sum of the lengths of all three sides:

(length of side 1) + (length of side 2) + (length of third side) = 4x^3 - 3x^2 + 2x - 6

Substituting in the expressions for side 1 and side 2, we get:

(8x^2 - 5x - 2) + (7x - x^2 + 3) + (length of third side) = 4x^3 - 3x^2 + 2x - 6

Simplifying, we get:

-x^2 + 15x + 1 + (length of third side) = 4x^3 - 3x^2 + 2x - 6

Now we can solve for the length of the third side:

length of third side = 4x^3 - 2x^2 - 13x - 7

Therefore, the length of the third side is 4x^3 - 2x^2 - 13x - 7.

Part C:

To see if the polynomials are closed under addition and subtraction, we need to check whether the sum or difference of any two polynomials in the form given will also be of that form.

For Part A, we found that the total length of the two sides was -x^2 + 15x + 1. To check if this polynomial is of the same form, we compare it to the general form of a polynomial in this problem:

ax^2 + bx + c

We can see that the polynomial -x^2 + 15x + 1 is indeed of this form, with a = -1, b = 15, and c = 1. Therefore, the polynomials are closed under addition.

For Part B, we found that the length of the third side was 4x^3 - 2x^2 - 13x - 7. To check if this polynomial is of the same form, we again compare it to the general form of a polynomial in this problem:

ax^2 + bx + c

We can see that the polynomial 4x^3 - 2x^2 - 13x - 7 is not of this form, since it has a degree of 3 rather than 2. Therefore, the polynomials are not closed under subtraction.

Overall, we can say that the polynomials are closed under addition, but not under subtraction.

There would be 210 Different combinations of 6 records that could be played out of the 10 available records.

The number of ways the program for this radio show segment can be arranged, we need to use the formula for permutations, which is:nPr = n! / (n - r)!

where n is the total number of items to choose from and r is the number of items to be selected.

In this case, the disc jockey can play 6 records out of a total of 10. So, we can plug in n = 10 and r = 6 into the formula and calculate the number of permutations:

10P6 = 10! / (10 - 6)!

= 10! / 4!

= (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (4 x 3 x 2 x 1)

= (10 x 9 x 8 x 7 x 6 x 5)

= 151,200

Therefore, there are 151,200 ways that the program for this radio show segment can be arranged from the 10 available records.

It's important to note that the order in which the records are played matters, so this is a case of permutations rather than combinations. If the order didn't matter and we were simply selecting 6 records out of 10, we would use the formula for combinations instead:

nCr = n! / (r! * (n - r)!)

In this case, we would have:

10C6 = 10! / (6! * (10 - 6)!)

= 10! / (6! * 4!)

= (10 x 9 x 8 x 7) / (4 x 3 x 2 x 1)

= 210

Therefore, there would be 210 different combinations of 6 records that could be played out of the 10 available records.

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The three final discount dates are 28 January, 7 February, and 17 February, and the net payment date is 9 March, based on the given invoice terms.

To determine the three final discount dates and the net payment date based on the invoice terms provided (6/10, 4/20, 1/30, n/50), we need to understand the meaning of the terms.

The format for these terms is a discount percentage followed by the number of days within which the discount can be availed. The final discount date is calculated by adding the specified number of days to the invoice date, while the net payment date is determined by adding the maximum number of days for payment.

Given that the invoice is dated 18 January, we can calculate the final discount dates and net payment date as follows:

6/10: This term offers a 6% discount if paid within 10 days. The final discount date is calculated by adding 10 days to the invoice date (18 January + 10 days), resulting in 28 January.

4/20: This term offers a 4% discount if paid within 20 days. The final discount date is calculated by adding 20 days to the invoice date (18 January + 20 days), resulting in 7 February.

1/30: This term offers a 1% discount if paid within 30 days. The final discount date is calculated by adding 30 days to the invoice date (18 January + 30 days), resulting in 17 February.

n/50: This term does not specify a discount percentage but offers a payment period of 50 days. The net payment date is calculated by adding 50 days to the invoice date (18 January + 50 days), resulting in 9 March.

The three final discount dates are 28 January, 7 February, and 17 February, and the net payment date is 9 March, based on the given invoice terms.

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Dolma received a total amount of Rs 1,29,260 at the beginning of BS 2080.

Given:

Principal amount (P) = Rs 1,20,000

Interest rate (R) = 9% per annum

Income tax rate (T) = 5% of the interest earned

First, we calculate the interest earned

Interest (I) = P x R/100

= 1,20,000 * 9/100

= 10,800

Next, we calculate the income tax

Income Tax = I x T/100

= 10,800 x 5/100

= 540

Now, Total amount = P + I - Income Tax

= 1,20,000 + 10,800 - 540

= Rs 1,29,260

Therefore, Dolma received a total amount of Rs 1,29,260 at the beginning of BS 2080.

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Answer: -7x² - 10x + 3

Step-by-step explanation: To find A + B, we need to add the like terms. Combining the terms of A and B, we get: -7x² - 10x + 3. Therefore, the sum of A and B is -7x² - 10x + 3.

The value of the cosine X is 0.724

How to determine the value

We need to know the different trigonometric identities.

These identities are written as;

sinecosinetangentcotangentsecantcosecant

From the information given, we have that;

Z=90°, XZ = 20, YX = 29, and ZY = 21.

The ratio for the cosine identity is;

cos θ = adjacent/hypotenuse

Adjacent side = XZ = 20

Hypotenuse side; YX = 29

then, we have;

cos X = 21/29

cos X = 0.724

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Answer:

{(15, 2988.5), (21, 2670.5), (27, 2352.5), (33, 2034.5)}

Step-by-step explanation:

Let's analyze both sets of ordered pairs:

Set 1: {(0, 3600), (4, 3526), (8, 3353.5), (12, 3147.5)}

The x-values in this set increase by a constant amount of 4: 0, 4, 8, 12.

Now let's examine the corresponding y-values:

3600, 3526, 3353.5, 3147.5

To determine the rate of change between the y-values, we subtract consecutive pairs:

3526 - 3600 = -74

3353.5 - 3526 = -172.5

3147.5 - 3353.5 = -206

The y-values in this set decrease, but the rate of change is not constant. The differences between consecutive y-values are not the same, indicating that the relation is not linear.

Set 2: {(15, 2988.5), (21, 2670.5), (27, 2352.5), (33, 2034.5)}

The x-values in this set also increase by a constant amount of 6: 15, 21, 27, 33.

Now let's examine the corresponding y-values:

2988.5, 2670.5, 2352.5, 2034.5

Again, we calculate the differences between consecutive y-values:

2670.5 - 2988.5 = -318

2352.5 - 2670.5 = -318

2034.5 - 2352.5 = -318

In this set, the y-values decrease by a constant rate of 318. The rate of change between consecutive y-values is the same, indicating that the relation is linear.

Therefore, the set of ordered pairs {(15, 2988.5), (21, 2670.5), (27, 2352.5), (33, 2034.5)} represents a linear relation.

The total area that Molly needs to paint is: 300 sq. inches

How to find the total surface area?

To find the total surface area of the given figure, we will add all surface areas together.

Prism: 6 surfaces

Surface area:

Front = 6 x 6 = 36

Back = 6 x 6 = 36

Top = 6 x 6 = 36

Bottom (base) = 6 x 6 = 36

Side = 6 x 6 = 36

Other side = 6 x 6 = 36

Total = 216 square inches

Pyramid: 5 surfaces

Surface area:

Bottom (base) = 6 x 6 = 36

Front triangle = (6 x 4)/2 = 12

Back = (6 x 4)/2 = 12

Side = (6 x 4)/2 = 12

Other side = (6 x 4)/2 = 12

Surface area = 84 square inches

Thus:

TOTAL surface area = 216 sq. inches + 84 sq. inches

= 300 sq. inches

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By using definition of area of Trapezoid, The area of garden is,

⇒ Area = 504 feet²

We have to given that;

A raised garden is shown below.

Since, The multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

Since, Garden is in shape of trapezoid.

And, Bases of garden are,

⇒ a = 26 feet

⇒ b = 32 feet

Height of garden is,

⇒ h = 18 feet

Since, We know that;

Area of trapezoid is,

A = (a + b) h/2

Where, a and b are base and h is height

Hence, We can formulate;

The area of garden is,

Area = (24 + 32) × 18 / 2

Area = 504 feet²

Therefore, The area of garden would be,

⇒ Area = 504 feet²

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The value of the angles in the figure are:

m∠1 = 114°

m∠3 = 114°

m∠4 = 66°

How to find m∠1, m∠3 and m∠4 in the figure?

To answer the question, we need some angle geometry knowledge:

We have:

m∠2 = 66°

Thus, we can find the angles in the figure as follow:

m∠1 = 180° - 66° (sum of angles on a straight line is 180°)

m∠1 = 114°

m∠3 = m∠1 (vertically opposite angles are equal)

m∠3 = 114°

m∠4 = m∠2 (vertically opposite angles are equal)

m∠4 = 66°

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Answer:

[tex]\textsf{7.} \quad a_n=2n+13[/tex]

[tex]\textsf{8.} \quad a_n=8n+17[/tex]

Step-by-step explanation:

To write a rule for the nth term of a arithmetic sequence, we can use the following formula:

[tex]\boxed{\begin{minipage}{8 cm}\underline{General form of an arithmetic sequence}\\\\$a_n=a+(n-1)d$\\\\where:\\\phantom{ww}$\bullet$ $a_n$ is the nth term. \\ \phantom{ww}$\bullet$ $a$ is the first term.\\\phantom{ww}$\bullet$ $d$ is the common difference between terms.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

[tex]\hrulefill[/tex]

Question 7

Given terms:

a₂ = 17a₁₁ = 35

Substitute the given values into the formula to create two equations:

[tex]\begin{aligned}a_2=a+(2-1)d&=17\\a+d&=17\end{aligned}[/tex]

[tex]\begin{aligned}a_{11}=a+(11-1)d&=35\\a+10d&=35\end{aligned}[/tex]

Rearrange the first equation to isolate d:

[tex]d=17-a[/tex]

Substitute this into the second equation and solve for a:

[tex]\begin{aligned}a+10(17-a)&=35\\a+170-10a&=35\\170-9a&=35\\-9a&=-135\\a&=15\end{aligned}[/tex]

Substitute the found value of a into the equation for d and solve for d:

[tex]d=17-15=2[/tex]

Therefore, the rule for the nth term of the given arithmetic sequence is:

[tex]\begin{aligned}a_n&=15+(n-1)2\\ &=15+2n-2\\&=2n+13\end{aligned}[/tex]

[tex]\hrulefill[/tex]

Question 8

Given terms:

a₉ = 89a₁₅ = 137

Substitute the given values into the formula to create two equations:

[tex]\begin{aligned}a_9=a+(9-1)d&=89\\a+8d&=89\end{aligned}[/tex]

[tex]\begin{aligned}a_{15}=a+(15-1)d&=137\\a+14d&=137\end{aligned}[/tex]

Rearrange the first equation to isolate a:

[tex]a=89-8d[/tex]

Substitute this into the second equation and solve for d:

[tex]\begin{aligned}a+14d&=137\\(89-8d)+14d&=137\\89+6d&=137\\6d&=48\\d&=8\end{aligned}[/tex]

Substitute the found value of d into the equation for a and solve for a:

[tex]\begin{aligned}a&=89-8(8)\\&=89-64\\&=25\end{aligned}[/tex]

Therefore, the rule for the nth term of the given arithmetic sequence is:

[tex]\begin{aligned}a_n&=25+(n-1)8\\ &=25+8n-8\\&=8n+17\end{aligned}[/tex]

a) We estimate that there are approximately 4700 rings weighing between 1.4 g and 3 g of gold. b) Because we are using the histogram to approximate the number of rings in each mass range, the value we calculated in part a) is only an estimate.

How to answer the aforementoned questions

a) To estimate the number of rings that contain between 1.4 g and 3 g of gold, we need to find the area of the bars in the histogram that correspond to this range. We can do this by multiplying the height of each bar by its width and adding up the results.

From the histogram, we can see that the height of the bar for the range 1.4 g to 2 g is 44 and its width is 50, while the height of the bar for the range 2 g to 3 g is 25 and its width is 100. So, the estimate for the number of rings that contain between 1.4 g and 3 g of gold is:

44 × 50 + 25 × 100 = 2200 + 2500 = 4700

Therefore, we estimate that there are around 4700 rings that contain between 1.4 g and 3 g of gold.

b) Because we are using the histogram to approximate the number of rings in each mass range, the value we calculated in part a) is only an estimate.

The frequency density is represented by the histogram, which is the frequency divided by the width of each bar. To calculate the true frequency, multiply the frequency density by the width of the bar. This, however, assumes that the mass of gold in each ring is distributed evenly across the range of the bar, which may not be the case.

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The probability that Isaiah will make both penalty kicks, as simulated by flipping two fair coins 20 times, is 25%.

Since Isaiah scores with a 50% success rate on penalty kicks, the probability of him scoring a goal on any given penalty kick is 0.5 or 50%.

If he flips two fair coins to simulate 20 trials, there are four possible outcomes: HH (two goals), HT (one goal and one miss), TH (one goal and one miss), and TT (two misses).

Out of the 20 trials, there are two possible ways for Isaiah to score both of his penalty kicks: HH and HH. The probability of getting HH on two coin flips is (0.5) x (0.5) = 0.25 or 25%.

Therefore, the probability that Isaiah will make both penalty kicks, as simulated by flipping two fair coins 20 times, is 25%.

Expressed as a percentage, this is:

P(two goals) = 25%

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Given statement solution:- The final sale price of the shirt is $13.50.

To calculate the final sale price of the shirt, we need to apply the discounts sequentially.

First, let's calculate the price after the initial 25% discount. The discount is calculated by multiplying the original price by (1 - discount percentage):

25% discount = 0.25

Price after the initial discount = $20 * (1 - 0.25) = $20 * 0.75 = $15.

Next, we need to calculate the additional discount of 10% after the first discount is taken. Again, we'll use the same formula:

10% discount = 0.10

Price after the additional discount = $15 * (1 - 0.10) = $15 * 0.90 = $13.50.

Therefore, the final sale price of the shirt is $13.50.

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