HELPPPPPPP TRIGONOMETRY

The size of angle BAC is approximately 19.88 degrees where ABC is a right angle triangle.

A triangle is a polygon with three sides and three angles. It is a simple closed shape, and one of the basic shapes in geometry.

To calculate the size of angle BAC, we can use the trigonometric ratio of the opposite side to the hypotenuse, which is sine:

sin(BAC) = opposite/hypotenuse

sin(BAC) = BC/AC

sin(BAC) = 7.9/23.1

Now, we can use a calculator to find the inverse sine of this value:

BAC = sin^(-1)(7.9/23.1)

BAC ≈ 19.88 degrees

Therefore, the size of angle BAC is approximately 19.88 degrees.

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The complete question is ABC is a right angle triangle BC=7.9, AC=23.1 and right angle at B. Calculate the size of BAC.​

Answer:

=6.96969e+29

Step-by-step explanation:

The correct answer is sixteen (16).

To calculate the total purchase price for Martin Pincher, we need to calculate the total amount of taxes he must pay and then add the purchase price of the items.

Let's start by calculating the taxes.

State Tax:

State Tax = 0.05 x (28.61+23.27+7.96)

State Tax = 0.05 x 59.84

State Tax = 2.99

County Tax:

County Tax = 0.005 x (28.61+23.27+7.96)

County Tax = 0.005 x 59.84

County Tax = 0.30

City Tax:

City Tax = 0.025 x (28.61+23.27+7.96)

City Tax = 0.025 x 59.84

City Tax = 1.50

Total taxes = 2.99 + 0.30 + 1.50

Total taxes = 4.79

Now let's calculate the total purchase price.

Total purchase price = 28.61+23.27+7.96 + 4.79

Total purchase price = 64.63

Answer:

  $64.63

Step-by-step explanation:

You want to know the total with tax of purchases of $28.61, 23.27, and 7.96 when they are subject to taxes of 5%, 0.5% and 2.5%.

The sum of the individual prices is subject to tax that is the sum of the individual tax rates. The tax rate sum is ...

  5% +0.5% +2.5% = 8%

Then the multiplier of the price is ...

  total = subtotal × (1 +8%) = 1.08×subtotal

A calculator can do all the computations at once, so we don't actually need to know the subtotals in order to find the total sale amount.

  (28.61 +23.27 +7.96) × 1.08 = 64.6272

The tax is generally rounded to the nearest cent, so ...

  the total purchase price is $64.63.

__

Additional comment

In the days before smart cash registers, the amount of sales tax was found by looking in a table supplied by the tax authority. It was always the same for any given purchase amount.

These days, the tax may be computed on the fly in such a way that the total tax collected is as near as possible to the total tax that needs to be collected.

That means a hundred instances of this purchase might have a total price of $64.63 for 72 of them, and $64.62 for the remaining 28 of them. That way, the total of the 100 sales would be $6462.72 and exactly the right amount of tax will have been collected. (At least one pizza chain collects tax this way.)

Answer:

Compound shapes can also be called composite shapes. To find the area of compound shapes we must divide the compound shape into basic shapes and find the area of each of the basic shapes and add them together.

Answer:

Step-by-step explanation:

To determine the amount that Buggiero Brothers Oil must deposit today in order to have enough money to pay the R85 000 liability four years from today, we can use the present value formula:

Present Value (PV) = Future Value (FV) / (1 + r)^n

where:

- FV is the future value, which is R85 000

- r is the annual interest rate, which is 6%

- n is the number of years until the liability needs to be paid, which is 4

To calculate the future value of the annual deposits of R16 000, we can use the future value of an annuity formula:

FV = PMT x [(1 + r)^n - 1] / r

where:

- PMT is the annuity payment, which is R16 000

- r is the annual interest rate, which is 6%

- n is the number of years of the annuity, which is 3 (since the first payment is made one year from today)

Plugging in the numbers:

FV = R16 000 x [(1 + 0.06)^3 - 1] / 0.06

FV = R61 719.56

Therefore, the total future value that Buggiero Brothers Oil will have in four years, including the initial deposit and the annual payments, will be:

FV = R85 000 + R61 719.56

FV = R146 719.56

Plugging in the numbers to the present value formula:

PV = R146 719.56 / (1 + 0.06)^4

PV = R116 442.12

Thus, Buggiero Brothers Oil needs to deposit R116 442.12 today to have enough money to pay the R85 000 liability four years from today, assuming a 6% rate of return.

the equation of the line passing through the points (Two-fifths, 19 Over 20) and (one-third, 11 Over 12) in slope-intercept form is y = (-1/16)x + 3/5.

The correct option is: y = (-1/16)x + 3/5.

What is slope?

In mathematics, slope refers to the steepness or incline of a line on a graph. It is a measure of how much the dependent variable changes for every unit change in the independent variable.

To find the equation of the line passing through the given points in slope-intercept form, we need to first determine the slope of the line.

We can use the slope formula:

slope = (y2 - y1)/(x2 - x1)

Let's label the first point as (x1, y1) = (Two-fifths, 19 Over 20) and the second point as (x2, y2) = (one-third, 11 Over 12).

So,

x1 = Two-fifths, y1 = StartFraction 19 Over 20 EndFraction

x2 = one-third, y2 = StartFraction 11 Over 12 EndFraction

slope = (StartFraction 11 Over 12 EndFraction - StartFraction 19 Over 20 EndFraction)/(one-third - Two-fifths)

slope = (-1/240)/(1/15)

slope = -1/16

Now, we can use the point-slope form of a line to find the equation in slope-intercept form, where (x1, y1) is any point on the line and m is the slope:

y - y1 = m(x - x1)

Let's choose the first point, (x1, y1) = (Two-fifths, StartFraction 19 Over 20 EndFraction):

y - StartFraction 19 Over 20 EndFraction = (-1/16)(x - Two-fifths)

Simplifying:

y - StartFraction 19 Over 20 EndFraction = (-1/16)x + 1/8

y = (-1/16)x + 1/8 + StartFraction 19 Over 20 EndFraction

y = (-1/16)x + (10/80 + 38/80)

y = (-1/16)x + 48/80

y = (-1/16)x + 3/5

So, the equation of the line passing through the points (Two-fifths, StartFraction 19 Over 20 EndFraction) and (one-third, StartFraction 11 Over 12 EndFraction) in slope-intercept form is y = (-1/16)x + 3/5.

Therefore, the correct option is: y = (-1/16)x + 3/5.

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According to the information, the values that coincide are: 2.2 meters with 2200 millimeters, 14 decimeters with 0.014 hectometers, 0.14 meters with 0.014 decameters, 0.022 decameters with 22 centimeters.

To find the numbers that match we must take into account the relationship between the values:

According to the above, we can infer that the values that match are:

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the profit function is positive, producing 164 units of the new product line would yield a profit of $600.

We should proceed with the product line since it is profitable and we can reasonably expect to meet our break-even quantity.
To find the break-even quantity, we need to set the cost function equal to the revenue function and solve for x:
C(x) = R(x)
55x + 3500 = 80x
25x = 3500
x = 140
Therefore, the break-even quantity is 140 units.
To find the profit function, we subtract the cost function from the revenue function:
P(x) = R(x) - C(x)
P(x) = 80x - (55x + 3500)
P(x) = 25x - 3500
Now we can evaluate the profit function at different values of x to see if producing these products is profitable or not.
If we plug in x = 164, the expected sales quantity, we get:
P(164) = 25(164) - 3500
P(164) = 4100 - 3500
P(164) = 600

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The product of (15.4 × 102) · (2.8 × 10–4) in scientific notation is written as 4.312 x 10⁻¹. Thus, option C is was correct.

When a number between 1 and 10 is multiplied by a power of 10, the result is represented in scientific notation. For instance, the scientific notation for 650,000,000 is 6.5 108.

⇒ (15.4 × 10²) · (2.8 × 10⁻⁴)

= 15.4  × 2.8 × 10² × 10⁻⁴

= 43.12 × 10² × 10⁻⁴

= 4.312 × 10¹ × 10² × 10⁻⁴

= 4.312 × 10¹⁺²⁻⁴

= 4.312 × 10⁻¹

Thus, The product of (15.4 × 102) · (2.8 × 10–4) in scientific notation is written as 4.312 x 10⁻¹. Thus, option C is was correct.

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The exact value of tan(2θ) in simplest radical form is 58√(793) / 48 which has been calculated through Pythagorean theorem.

The Pythagorean Theorem can be used to find the correct angled triangle's missing length. The triangle contains three sides: the hypotenuse, this same opposite, which will always be the longest, and the adjacent side, which really doesn't touch the hypotenuse. The Pythagorean equation is: a² + b² = c².

We know that sin(θ) = 29/4 and that θ is in Quadrant I, which means that all three trigonometric functions (sine, cosine, and tangent) are positive in this quadrant.

Using the identity:

tan(2θ) = 2tan(θ) / (1 - tan²(θ))

We can find the value of tan(2θ) by first finding tan(θ) and then using it to calculate tan(2θ).

To find tan(θ), we can use the Pythagorean identity:

sin²(θ) + cos²(θ) = 1

cos²(θ) = 1 - sin²(θ)

cos(θ) = ± √(1 - sin²(θ))

Since θ is in Quadrant I, we know that cos(θ) is positive, so we take the positive square root:

cos(θ) = √(1 - (29/4)²) = √(793) / 4

Now we can find tan(θ) as:

tan(θ) = sin(θ) / cos(θ) = (29/4) / (√(793) / 4) = 29 / √(793)

Substituting this into the formula for tan(2θ), we get:

tan(2θ) = 2tan(θ) / (1 - tan²(θ))

tan(2θ) = 2(29 / √(793)) / (1 - (29 / √(793))²)

tan(2θ) = 2(29 / √(793)) / (1 - 841/793)

tan(2θ) = 58√(793) / 48

Therefore, the exact value of tan(2θ) in simplest radical form is 58√(793) / 48.

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Given:-

To find:-

Answer:-

In the given right angled triangle, we may use the trigonometric ratios. We can see that the measure of the longest side is "x" which is hypotenuse and it needs to be find out. The perpendicular in this case is "2" .

We may use the ratio of sine here as , we know that in any right angled triangle,

[tex]\implies\sin\theta =\dfrac{p}{h} \\[/tex]

And here , p = 2 and h = x , so on substituting the respective values, we have;

[tex]\implies \sin\theta = \dfrac{2}{x} \\[/tex]

Again here angle is 60° . So , we have;

[tex]\implies \sin60^o =\dfrac{2}{x} \\[/tex]

The measure of sin45° is √3/2 , so on substituting this we have;

[tex]\implies \dfrac{\sqrt3}{2}=\dfrac{2}{x} \\[/tex]

[tex]\implies x =\dfrac{2\cdot 2}{\sqrt3}\\[/tex]

Value of √3 is approximately 1.732 . So we have;

[tex]\implies x =\dfrac{4}{1.732} \\[/tex]

[tex]\implies \underline{\underline{\red{\quad x = 2.31\quad }}}\\[/tex]

Hence the value of x is 2.31 .

Answer:

The length of side x to the nearest tenth is 2.3.

Step-by-step explanation:

From inspection of the given right triangle, we can see that the interior angles are 30°, 60° and 90°. Therefore, this triangle is a 30-60-90 triangle.

A 30-60-90 triangle is a special right triangle where the measures of its sides are in the ratio  1 : √3 : 2.  Therefore, the formula for the ratio of the sides is b: b√3 : 2b where:

We have been given the side opposite the 60° angle, so:

[tex]\implies b\sqrt{3}=2[/tex]

Solve for b by dividing both sides of the equation by √3:

[tex]\implies b=\dfrac{2}{\sqrt{3}}[/tex]

The side labelled "x" is the hypotenuse, so:

[tex]\implies x=2b[/tex]

Substitute the found value of b into the equation for x:

[tex]\implies x=2 \cdot \dfrac{2}{\sqrt{3}}[/tex]

[tex]\implies x=\dfrac{4}{\sqrt{3}}[/tex]

[tex]\implies x=2.30940107...[/tex]

[tex]\implies x=2.3\; \sf (nearest\;tenth)[/tex]

Therefore, the length of side x to the nearest tenth is 2.3.

Answer: To find the least positive value of x that satisfies the given congruence, we can use the trial and error method or algebraic manipulation.

Using the trial and error method, we can start by plugging in values of x and checking if the congruence is satisfied.

For x = 1, (x + 3) = 4, so 89 ≡ 0 (mod 4), which is not a multiple of 3.

For x = 2, (x + 3) = 5, so 89 ≡ 1 (mod 4), which is not a multiple of 3.

For x = 3, (x + 3) = 6, so 89 ≡ 2 (mod 4), which is not a multiple of 3.

For x = 4, (x + 3) = 7, so 89 ≡ 3 (mod 4), which is not a multiple of 3.

For x = 5, (x + 3) = 8, so 89 ≡ 0 (mod 4), which is a multiple of 3.

Therefore, the least positive value of x that satisfies the congruence is x = 5.

Alternatively, we can use algebraic manipulation to solve the congruence. We have:

89 ≡ (x + 3) (mod 4)

=> 89 ≡ x + 3 (mod 4)

=> 86 ≡ x (mod 4) (subtracting 3 from both sides)

Now we need to find the least positive value of x that satisfies this congruence and is a multiple of 3.

The solutions for this congruence are x = 2 (mod 4) and x = 6 (mod 4).

Plugging in x = 2, we get 89 ≡ 5 (mod 4), which is not a multiple of 3.

Plugging in x = 6, we get 89 ≡ 3 (mod 4), which is not a multiple of 3.

Therefore, the least positive value of x that satisfies the congruence and is a multiple of 3 is x = 10 (which is equivalent to x = 2 (mod 4) and x = 6 (mod 4)), but this is not the answer to the original question since x must be positive.

Plugging in x = 14, we get 89 ≡ 1 (mod 4), which is not a multiple of 3.

Plugging in x = 18, we get 89 ≡ 3 (mod 4), which is not a multiple of 3.

Finally, plugging in x = 22, we get 89 ≡ 1 (mod 4), which is not a multiple of 3.

Plugging in x = 26, we get 89 ≡ 3 (mod 4), which is not a multiple of 3.

Since we want x to be positive, we can stop here and conclude that the least positive value of x that satisfies the given congruence is x = 5.

Step-by-step explanation:

Answer:

[tex] \rm \: f(x) = \dfrac{5}{x} + \dfrac{7}{ {x}^{2} } [/tex]

Differentiating both sides with respect to x

[tex] \rm \dfrac{d}{dx} ( {f}( x) = \dfrac{d}{dx} \bigg( \dfrac{5}{x} + \dfrac{7}{ {x}^{2} } \bigg)[/tex]

Using u + v rule

[tex] \rm \: {f}^{ \prime} x = \dfrac{d}{dx} \bigg( \dfrac{5}{x} \bigg) + \dfrac{d}{dx} \bigg( \dfrac{7}{ {x}^{2} } \bigg)[/tex]

[tex] \rm \: {f}^{ \prime} x = 5. \dfrac{d}{dx} ( {x}^{ - 1} ) + 7. \dfrac{d}{dx} ( {x}^{ - 2} )[/tex]

[tex] \rm \: {f}^{ \prime} x = 5.( - 1. {x})^{ (- 1 - 1)} + 7.( - 2. {x})^{ - 2 - 1} [/tex]

[tex] \rm \: {f}^{ \prime} x = { - 5x}^{ - 2} { - 14x}^{ - 3} [/tex]

[tex] \rm \: {f}^{ \prime} x = - \dfrac{5}{ {x}^{2} } - \dfrac{14}{ {x}^{3} } [/tex]

[tex] \rm \: {f}^{ \prime} x = - \bigg(\dfrac{5}{ {x}^{2} } + \dfrac{14}{ {x}^{3} } \bigg)[/tex]

Hense The required Derivative is answered.

[tex]\boxed{\begin{array}{c|c} \rm \: \underline{function}& \rm \underline{Derivative} \\ \\ \rm \dfrac{d}{dx} ({x}^{n}) \: \: \: \: \: \: \: \: \: \ & \rm nx^{n-1} \\ \\ \rm \: \dfrac{d}{dx}(constant) &0 \\ \\ \rm \dfrac{d}{dx}( \sin x )\: \: \: \: \: \: & \rm \cos x \\ \\ \rm \dfrac{d}{dx}( \cos x ) \: \: \: & \rm - \sin x \\ \\ \rm \dfrac{d}{dx}( \tan x ) & \rm \: { \sec}^{2}x \\ \\ \rm \dfrac{d}{dx}( \cot x ) & \rm- { \csc }^{2}x \\ \\ \rm \dfrac{d}{dx}( \sec x ) & \rm \sec x. \tan x \\ \\\rm \dfrac{d}{dx}( \csc x ) & \rm \: - \csc x. \cot x\\ \\ \rm \dfrac{d}{dx}(x) \: \: \: \: \: \: \: & 1 \end{array}}[/tex]

The slope for the given point through which it passes through the graph is [tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex] = infinity .

What about slope?

The slope of a line is a measure of its steepness. It is defined as the ratio of the vertical change (rise) between two points on the line to the horizontal change (run) between the same two points. In other words, it is the rate at which the line rises or falls as we move along it in the horizontal direction.

Define graph:

In mathematics, a graph is a visual representation of a set of data or mathematical relationships between variables. Graphs can be used to display and analyze data in a variety of formats, including line graphs, bar graphs, scatter plots, and pie charts.

According to the given information:

As, we know that the slope =  [tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex]

⇒ In which the point given are (-1,8) and (-1,-4)

By putting the value of the given point we have that

⇒ [tex]\frac{-4-(8)}{-1-(-1)}[/tex] = infinity.

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Therefore, we can assume that the value of 10⁰ would be 1, based on the pattern of the other values in the table.

Celsius (symbol: °C) is a temperature scale used in the metric system. It is named after the Swedish astronomer Anders Celsius, who first proposed it in 1742. The Celsius scale is based on the properties of water, with 0°C defined as the freezing point of water, and 100°C defined as the boiling point of water at standard atmospheric pressure. Celsius is widely used in many countries around the world as a unit of temperature measurement, including in scientific and everyday contexts.

Given by the question.

In the table from part C, we see that as the power of 10 decreases by 1, the value of 10 raised to that power also decreases by a factor of 10. For example, we see that 10² = 100, 10¹ = 10, and 10⁰ = 1.

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The height of the pole to the nearest foot, given the angle of elevation and the surveyor distance, is C. 135 ft

To find the height of the pole, we can use the tangent function from trigonometry. The tangent of the angle of elevation (44°) is equal to the ratio of the height of the pole (h) minus the transit height (4 feet) to the distance between the transit and the pole (140 feet).

So, we have:

tan(44°) = (h - 4) / 140

h - 4 = 140 × tan(44°)

h = 140 × tan(44°) + 4

h = 140 × 0.965 + 4

h = 135.1

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If Point A is located at (4, 3). Point A is reflected.  the final location of A after both transformations is (-4, -7).

To reflect point A across the line y = -2, we need to find the point that is the same distance from the line but on the other side. The line y = -2 is a horizontal line that is 5 units above the point A (since the y-coordinate of A is 3). Therefore, the reflected point will be 5 units below the line, which gives us:

A' = (4, -7)

To rotate point A' 90 degrees clockwise about the origin, we can use the following rotation matrix:

| 0 1 |

| -1 0 |

Multiplying this matrix by the coordinates of A', we get:

| 0 1 | | 4 | | -7 |

| -1 0 | * | -7 | = | -4 |

So the final location of A after both transformations is (-4, -7).

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Answer:

4/9

Step-by-step explanation:

4 sandwiches in between 9 friends

4 sandwiches must be divided in between 9 people

- > [tex]\frac{4 sandwiches}{9 people}[/tex] = [tex]\frac{4}{9}[/tex]

Answer:

Step-by-step explanation:

2/3

Answer:

Step-by-step explanation:

subtract 20-3.50 you will get 16.50

Multiply 2.50x6 you will get 15

Subtract 16.50-15.00 you will get 1.50

The side lengths of such a triangle with a leg length of 5 and angles of 45, 45, and 90 are not listed in any of the options.

The right angle is established when 2 straight lines cross at a 90° angle or when they are perpendicular at the intersection. The symbol is used to indicate a right angle.

For the given question:

Triangle with degree angles - 45-45-90

It means two legs are of same length 5 units.

Then, hypotenuse H will be:

H² = 5² + 5²

H² = 25 + 25

H² = 50

H = 5√2

Thus, the correct third side of the triangle will be 5√2.

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Answer:

112

Step-by-step explanation:

According to the circle graph, the mystery books make up 20% of all books sold. So, we can calculate the number of mystery books sold as follows:

Number of mystery books = 20% of 560

= (20/100) x 560

= 112

Therefore, the number of mystery books sold at the book fair was 112.

The unit rate in minutes per meter is 10 minutes/meter.

What is unit conversion?

Unit conversion is the method of converting a quantity from one unit of measurement to another. In many cases, it is necessary to convert units of measurement to make them more meaningful, or to enable comparisons between different units.

We can start by finding the total time it would take to fill the aquarium to a depth of 1 meter.

If it takes 4 minutes to fill the aquarium to a depth of 2/5 meters, then we can find how long it would take to fill the aquarium to a depth of 1 meter by setting up the following proportion:

2/5 meters / 4 minutes = 1 meter / x minutes

To solve for x, we can cross-multiply and simplify:

(2/5) * x = 4 * 1

2x = 20

x = 10

Therefore, it would take 10 minutes to fill the aquarium to a depth of 1 meter.

The unit rate in minutes per meter is the time it takes to fill 1 meter of the aquarium, which we just calculated to be 10 minutes.

So, the unit rate in minutes per meter is 10 minutes/meter.

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Complete question :  it takes 4 minutes to fill an empty aquarium to a depth of 2/5 meters. what is the unit rate in minutes per meter?

Let's start by assigning variables to represent the unknowns in the problem.

Let's use "c" to represent the number of crayons Manuel had in the beginning and "m" to represent the number of markers he had in the beginning.

From the problem, we know that:

Manuel had 4 times as many crayons as markers in the beginning, so:

c = 4m

After he bought 250 crayons and 100 markers, he had 3 times as many crayons as markers, so:

c + 250 = 3(m + 100)

Now we can use algebra to solve for c:

c + 250 = 3m + 300 // distribute the 3

c = 3m + 300 - 250 // simplify by combining like terms

c = 3m + 50

Substitute c = 4m from the first equation into the second equation:

4m = 3m + 50 // subtract 3m from both sides

m = 50

So Manuel had 4 times as many crayons as markers in the beginning, which means he had:

c = 4m = 4(50) = 200 crayons in the beginning.

Using function concepts, we have that:1. Non-linear2.B)x y0 11 22 53 103. Linear4.: Linear: Linear: Non-Linear: Linear5. LinearIn a linear function, the rate of change is constant.A linear function is also of the first degree.Item 1:From -3 to -1, the rate of change is of From -1 to 1, the rate of change is of .Different rates of change, so non-linear.Item 2:At function b, from 0 to 1, the rate of change is of 1, from 1 to 2 of 3, different rates of change, so non-linear.Item 3:Highest degree of x is 1, so first degree, and thus linear.Item 4:The only non-linear is , which is of the second degree. is a constant function, with a rate of change of 0, so linear.The last function is written as:Highest degree of x is 1, so also linear.Item 5:In all cases, the rate of change is constant, so linear.

Answer:

10.01

Step-by-step explanation:

13 defective and 15 non defective calculators mean the total amount of calculators is 28

So there is a 13/28 chance that if we pull one calculator out, it will be defective

Since 3 calculators are pulled out, we cube 13/28

= (13/28)*(13/28)*(13/28)

= 2197/21952

=0.10008199708

as a percentage this is 10.008199708%

rounded -> 10.01

Answer:

The formula for compound interest is given by:

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

Where:

A = the amount of money accumulated after n years

P = the principal (initial investment)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the number of years

In this case, P = $2,500, r = 0.05 (since 5% = 0.05), n = 2 (since interest is compounded semiannually), and t = 15. Substituting these values into the formula, we get:

A = 2500(1 + 0.05/2)^(2*15)

A ≈ $5,551.33

Therefore, Lyla's investment will be worth approximately $5,551.33 after 15 years if interest is compounded semiannually.

Answer:

Null Hypothesis (H0): There is no significant difference in life span for people in the two towns.

Alternative Hypothesis (H1): There is a significant difference in life span for people in the two towns.

Answer:

The null and alternative hypotheses that should be used to test this claim are:

Null hypothesis: There is no significant difference in lifespan for people in the two towns. Symbolically, this can be represented as H0: μ1 = μ2, where μ1 and μ2 are the population mean lifespans of Town A and Town B, respectively.

Alternative hypothesis: There is a significant difference in lifespan for people in the two towns. Symbolically, this can be represented as Ha: μ1 ≠ μ2.

To test this claim, the researcher can conduct a two-sample t-test using the data collected from the two towns. The test statistic can be calculated as:

t = (x1 - x2) / (s1^2/n1 + s2^2/n2)^0.5

where x1 and x2 are the sample mean lifespans of Town A and Town B, respectively, s1 and s2 are the sample standard deviations of Town A and Town B, respectively, and n1 and n2 are the sample sizes of Town A and Town B, respectively.

Using the given data, the test statistic can be calculated as:

t = (78.5 - 74.4) / (11.2^2/20 + 12.3^2/20)^0.5 = 1.02

At a significance level of 0.05 with 38 degrees of freedom (df = n1 + n2 - 2), the critical value for a two-tailed test is ±2.024. Since the calculated t-value (1.02) falls within the acceptance region (-2.024 < t < 2.024), the null hypothesis cannot be rejected. Therefore, we do not have enough evidence to conclude that there is a significant difference in lifespan for people in the two towns.

Step-by-step explanation:

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