Find the distance from P to l (Lesson 3-6)

Line l contains points (0,3) and (-4,-9) . Point P has coordinates (-6,-5) .

The statement that Study 2, with a sample size of 100 bottles, is more likely to provide a more precise estimate of the mean caffeine content of the energy drink compared to Study 1 is true.

Based on the given information, the statement that is true is that Study 2, which uses a larger sample size of 100 bottles, is more likely to provide a more precise estimate of the mean caffeine content of the energy drink compared to Study 1, which uses a smaller sample size of 25 bottles.

When estimating a population parameter, such as the mean, using a sample, a larger sample size generally leads to a more accurate and precise estimate.

In this case, Study 2 has a larger sample size, which means it provides more information about the variability of the caffeine content in the energy drink.

With a larger sample, the estimate of the mean caffeine content is likely to have a smaller margin of error and be more representative of the true population mean.

On the other hand, Study 1 with a smaller sample size is more susceptible to sampling variability and may have a larger margin of error. This means that the estimate obtained from Study 1 may have more uncertainty and be less reliable compared to Study 2.

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Answer:

0.285 * 10^2

0.285 *100

28.5

Answer:

28.5

Step-by-step explanation:

In a certain market, research suggests that reducing the price of widgets from $8 to $3 increases sales from 2000 to 10,000 units.

The law of demand states that as the price of a product number decreases, the quantity demanded tends to increase.

In this case, the In a certain market, research suggests that reducing the price of widgets from $8 to $3 increases sales from 2000 to 10,000 units.

Initially, at a price of $8 per widget, the market demand for widgets is limited to 2000 units.

However, when the price is reduced to $3, the quantity demanded surges to 10,000 units. This inverse relationship between price and quantity demanded is due to consumer behavior.

Lowering the price makes widgets more affordable and attractive to a larger number of buyers, leading to an increase in demand.

The research findings highlight the market's responsiveness to price changes and illustrate the importance of pricing strategies in influencing consumer demand.

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The intersection point of lines q and r is (-2, -4).

To determine the solutions to the given system of linear equations using only the equations of the lines, you can set up and solve pairs of equations.

For example, let's determine the intersection point of lines q and r:

1. Set the y-values of the two equations equal to each other: 3x + 2 = 0.5x - 3.

2. Simplify the equation by combining like terms: 2.5x = -5.

3. Divide both sides of the equation by 2.5 to solve for x: x = -2.

4. Substitute the value of x back into either of the original equations to solve for y. Using line q: y = 3(-2) + 2 = -4.

Therefore, the intersection point of lines q and r is (-2, -4).

Similarly, you can determine the intersection points of other pairs of lines by setting their equations equal to each other and solving for x and y.

By finding the intersection points of each pair of lines, you can determine the solutions to the system of linear equations. The solutions will be the ordered pairs that are solutions to all the equations simultaneously.

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Answer:

<1 = 140

<2 = 40

<3 = 65

<4 = 75

<5 = 115

Step-by-step explanation:

<1 = 60 + 80 = 140

<2 = 40  180 -40

<3 = 65 (180- 75 - 40)

<4 = 75  (180-104)

<5 = 115 (40 + 75)

Answer:

16x

Step-by-step explanation:

the product in mathematics means multiplication of quantities

here 16 and x are being multiplied together then their product is

16 × x = 16x

the expression (sinθ - cosθ)(sinθ - cosθ)^-1 / sinθcosθ is equivalent to 1 / sinθcosθ. We can start by expanding the numerator and simplifying the expression to simplify

Expanding the numerator:

(sinθ - cosθ)(sinθ - cosθ)^-1 = (sinθ - cosθ) / (sinθ - cosθ)

Simplifying the expression:

(sinθ - cosθ) / (sinθ - cosθ) = 1

Now, we can substitute this simplified expression into the original expression:

1 / sinθcosθ

Therefore, the simplified expression is 1 / sinθcosθ.

To simplify the given expression, we first expand the numerator, which is (sinθ - cosθ)(sinθ - cosθ)^-1. The denominator is sinθcosθ.

Expanding the numerator, we apply the concept of multiplying binomials. The expression (sinθ - cosθ)(sinθ - cosθ) is equivalent to (sinθ - cosθ) squared. Expanding this expression yields sin^2θ - 2sinθcosθ + cos^2θ.

Next, we simplify the expression by noticing that (sinθ - cosθ) / (sinθ - cosθ) is equal to 1. Dividing any number by itself gives a result of 1.

Therefore, the numerator simplifies to 1, and the denominator remains as sinθcosθ.

Combining the simplified numerator and the original denominator, we obtain 1 / sinθcosθ as the final simplified expression. This means that

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Enantiomer A represents 82.5% of the mixture, while enantiomer B represents 17.5%.

Enantiomeric excess (ee) is a measure of the difference in concentration between two enantiomers in a mixture. It is expressed as a percentage. In this case, the given 65% ee represents that one enantiomer is present in excess, while the other is present in a lower amount.
To calculate the percentage of each enantiomer, we need to consider the relationship between the enantiomeric excess and the individual enantiomer concentrations. Let's denote %A as the percentage of the excess enantiomer and %B as the percentage of the other enantiomer.
Given that %A + %B = 100% (since they represent the total composition of the mixture), and %A - %B = 65% ee, we can solve these equations simultaneously.
Adding the two equations together, we get:
2%A = 100% + 65% ee
Dividing both sides by 2, we find:
%A = (100% + 65% ee) / 2
Substituting the given 65% ee value into the equation, we can calculate the percentage of the excess enantiomer:
%A = (100% + 65%) / 2 = 82.5%
Since %A represents the excess enantiomer, %B can be obtained by subtracting %A from 100%:
%B = 100% - %A = 100% - 82.5% = 17.5%
Therefore, based on the given 65% ee, the percentage of the excess enantiomer (%A) is 82.5%, while the percentage of the other enantiomer (%B) is 17.5%.

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It is given that, quadrilateral WXYZ is inscribed in a circle with m∠W = 50° and m∠X = 108°. So, the value of m∠Z is 72°.

The opposite angles in an inscribed quadrilateral are supplementary, which means their measures sum up to 180°. So, by deducting the measurements of angles W and X from 180 degrees, we may find the measure of angle Z.

Given,

m∠W = 50° and m∠X = 108°

m∠Z = 180° - (m∠W + m∠X)

m∠Z = 180° - (50° + 108°)

m∠Z = 180° - 158°

m∠Z = 22°

It is crucial to keep in mind that angle Z cannot be 22 degrees in the context of an inscribed quadrilateral because it must be an exterior angle of the triangle produced by the other three angles. Any polygon's outside angle measurements added together will always equal 360 degrees, so to determine the accurate measurement, we subtract the estimated ∠Z from 360 degrees.

m∠Z = 360° - 22°

m∠Z = 338°

This value, however, is outside the acceptable range for ∠Z's measurement in an inscribed quadrilateral. Therefore, the complementary angle to the specified ∠X, which is 180° - 108° = 72°, must be used as the accurate measurement for ∠Z.

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The complete question-

Quadrilateral WXYZ is inscribed in a circle with m∠W = 50° and m∠X = 108°. What is m∠Z?

The biggest decrease in waiting time between two consecutive eruptions is 18 minutes.

Here, we have,

To assign the biggest decrease to the biggest decrease in waiting time between two consecutive eruptions, we need to compare the decreases in waiting time for each pair of consecutive eruptions and identify the pair with the largest decrease.

Let's consider a sequence of eruption waiting times as an example:

Eruption 1: 60 minutes

Eruption 2: 70 minutes

Eruption 3: 74 minutes

Eruption 4: 62 minutes

Eruption 5: 80 minutes

To find the biggest decrease in waiting time, we compare the differences between consecutive eruptions:

Decrease between Eruption 1 and Eruption 2: 70 - 60 = 10 minutes

Decrease between Eruption 2 and Eruption 3: 74 - 70 = 4 minutes

Decrease between Eruption 3 and Eruption 4: 62 - 74 = -12 minutes

Decrease between Eruption 4 and Eruption 5: 80 - 62 = 18 minutes

From the calculations, we can see that the biggest decrease in waiting time occurs between Eruption 4 and Eruption 5, with a decrease of 18 minutes.

Therefore, the biggest decrease in waiting time between two consecutive eruptions is 18 minutes.

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In order to assign the biggest decrease to the biggest decrease in waiting time between consecutive eruptions, calculate the difference in waiting times for each pair of eruptions, and the biggest decrease will be the largest difference you find.

To assign the biggest decrease to the biggest decrease in waiting time between two consecutive eruptions, you need to first identify the waiting times between each pair of consecutive eruptions. Then, calculate the difference in waiting times between each pair. For example, if the third eruption occurred after 74 minutes and the fourth after 62 minutes, the decrease in waiting time was 74 - 62 = 12 minutes. You repeat this process for all pairs of consecutive eruptions, and the biggest decrease in waiting time will be the largest difference you calculate.

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To determine if a high school's claim of having unusually high math SAT scores is justified, we can compare the sample mean with the population mean using a z-score.

The average math SAT score is given as 514 with a standard deviation of 113. A random sample of 60 students from the high school yielded a sample mean of 531. By calculating the z-score and comparing it to the range of usual events, we can assess the validity of the high school's claim. To determine if the high school's claim is justified, we calculate the z-score using the formula: z = (x - μ) / (σ / sqrt(n))

Where:
x is the sample mean (531),
μ is the population mean (514),
σ is the population standard deviation (113),
and n is the sample size (60).

Substituting the values into the formula: z = (531 - 514) / (113 / sqrt(60))
Calculating the z-score gives us a value. By comparing the z-score to the range of usual events, we can determine if the high school's claim is justified. The range of usual events is typically within ±2 standard deviations from the mean. If the z-score falls within this range, it suggests that the sample mean is not significantly different from the population mean, and the claim of unusually high scores may not be justified.

Please note that the provided explanation assumes a normal distribution and the use of a one-sample z-test.

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The solution to the system of equations is x = 1 and y = 0.

To solve the system of equations using elimination, we can start by multiplying the first equation by 2 to make the coefficients of x in both equations equal.

1) Multiply the first equation by 2:

  4x + 6y = 8

2) Now, we can subtract the second equation from the modified first equation to eliminate the variable x:

  (4x + 6y) - (4x + 6y) = 8 - 9

  0 = -1

The result of the subtraction is 0 = -1, which is not a true statement.

This implies that the system of equations is inconsistent and does not have a solution. In other words, the lines represented by the equations do not intersect and are parallel.

Therefore, the given system of equations has no solution.

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To find the number of sides in a regular polygon when the measure of an interior angle is given, we can use the formula:

Number of sides = 360 degrees / Measure of each interior angle

Let's say the measure of an interior angle is given as x degrees. Then, the number of sides in the polygon can be calculated as:

Number of sides = 360 degrees / x degrees

By dividing 360 degrees by the measure of each interior angle, we can determine the number of sides in the regular polygon.

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The parabola with vertex (h, k) and focus (h, k+c) has the equation (x-h)² = 4c(y-k).This equation represents a parabola with a horizontal axis of symmetry and its vertex at (h, k), focusing at (h, k+c).

To prove that the equation of the parabola with vertex (h, k) and focus (h, k+c) is given by (x-h)² = 4c(y-k), we can start with the definition of a parabola.A parabola is defined as the set of all points that are equidistant from the focus and the directrix. Let's denote a general point on the parabola as P(x, y).

1. Distance from P to the focus (h, k+c):

  The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by:

  √((x₂ - x₁)² + (y₂ - y₁)²)

  Applying the distance formula, the distance from P(x, y) to the focus (h, k+c) is:

  √((x - h)² + (y - (k + c))²)

2. Distance from P to the directrix:

  The directrix of a parabola with vertex (h, k) is a horizontal line located at y = k - c.

  The distance from P(x, y) to the directrix y = k - c is given by:

  |y - (k - c)|

According to the definition of a parabola, these two distances are equal:

√((x - h)² + (y - (k + c))²) = |y - (k - c)|

To simplify the equation, we'll square both sides:

((x - h)² + (y - (k + c))²) = (y - (k - c))²

Expand the squared terms:

(x - h)² + (y - (k + c))² = y² - 2y(k - c) + (k - c)²

Rearrange the terms to isolate the squared term:

(x - h)² = y² - 2y(k - c) + (k - c)² - (y - (k + c))²

(x - h)² = y² - 2y(k - c) + (k - c)² - (y² - 2y(k + c) + (k + c)²)

Simplify further:

(x - h)² = y² - 2y(k - c) + (k - c)² - y² + 2y(k + c) - (k + c)²

(x - h)² = - 2y(k - c) + (k - c)² + 2y(k + c) - (k + c)²

(x - h)² = - 2y(k - c) + 2y(k + c) + (k - c)² - (k + c)²

(x - h)² = - 2y(k - c + k + c) + (k - c)² - (k + c)²

(x - h)² = - 2y(2k) + (k - c)² - (k + c)²

(x - h)² = - 4yk + (k - c)² - (k + c)²

(x - h)² = - 4yk + k² - 2kc + c² - (k² + 2kc + c²)

(x - h)² = - 4yk + k² - 2kc + c² - k² - 2kc - c²

(x - h)² = - 4yk - 4kc

Finally

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The value of the given trigonometric expression is 0.

Given is a trigonometric expression cos 50° cos 40° - sin 50° sin 40°, we need to solve it,

To find the value of the trigonometric expression cos 50° cos 40° - sin 50° sin 40°, we can use the trigonometric identity for the cosine of the difference of two angles:

cos(A + B) = cos A cos B - sin A sin B

Here, A = 50° and B = 40°,

So, we get,

= cos 50° cos 40° - sin 50° sin 40°

= cos(50° + 40°)

= Cos 90°

Now, we know that, Cos 90° = 0.

Hence the value of the given trigonometric expression is 0.

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Victoria charged her friend $112.50 for the pair of jeans she sold .To find out how much money Victoria charged her friend, we need to calculate 25% of the total amount she spent on the four jeans.

Victoria bought four jeans for a total of $450. To find 25% of $450, we multiply $450 by 25/100, which is the decimal representation of 25%.

25% of $450 = (25/100) * $450

                      = $0.25 * $450

                      = $112.50

Therefore, Victoria received $112.50 as a refund after selling one pair of jeans. This means she essentially got 25% of her money back.

Since Victoria sold one pair of jeans to her friend, she would charge her friend the original cost of that pair, which is equal to the price of the jeans she bought minus the refund she received.

Original cost of one pair of jeans = Cost of four jeans - Refund received

Original cost of one pair of jeans = $450 - $112.50 = $337.50

Therefore, Victoria charged her friend $337.50 for the pair of jeans she sold.

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The standard-form equation of the hyperbola with vertices (0, ±4) and foci (0, ±5) is:

x²/16 - y²/9 = 1.

The standard-form equation of a hyperbola with vertices (0, ±4) and foci (0, ±5) can be determined using the following formula:

For a hyperbola centered at the origin (h, k), the standard-form equation is given by:

(x - h)²/a² - (y - k)²/b² = 1, where a represents the distance from the center to the vertices and b represents the distance from the center to the foci.

In this case, since the center of the hyperbola is at (0, 0), the equation becomes:

x²/a² - y²/b² = 1.

To find the values of a and b, we can use the given information about the vertices and foci. Since the distance from the center to the vertices is 4, we have a = 4. Similarly, the distance from the center to the foci is 5, so we have c = 5.

We can use the relationship between a, b, and c for a hyperbola:

c² = a² + b²,

(5)² = (4)² + b²,

25 = 16 + b²,

b² = 25 - 16,

b² = 9,

b = 3.

Now we can substitute the values of a and b into the equation to get the standard-form equation of the hyperbola:

x²/4² - y²/3² = 1.

Therefore, the standard-form equation of the hyperbola with vertices (0, ±4) and foci (0, ±5) is:

x²/16 - y²/9 = 1.

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To determine the volume of alcohol in liters that weighs 30 kg, we can use the formula V = W / D, where V is the volume, W is the weight, and D is the density. 30 kg of alcohol will have a volume of 37.5 liters.

Given that the density of alcohol is 0.8 g/cm³ and 1 kilogram is equal to 1000 grams, we can convert the weight from kilograms to grams and then calculate the volume in cubic centimeters (cm³). Finally, we convert the volume from cm³ to liters.
First, we convert the weight from kilograms to grams by multiplying it by 1000:
Weight = 30 kg × 1000 g/kg = 30,000 g
Next, we can calculate the volume using the formula V = W / D:
Volume = 30,000 g / 0.8 g/cm³ = 37,500 cm³
Since 1 liter is equal to 1000 cm³, we can convert the volume from cm³ to liters by dividing by 1000:
Volume in liters = 37,500 cm³ / 1000 = 37.5 L
Therefore, 30 kg of alcohol will have a volume of 37.5 liters.

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Ella spent 34 minutes making the birthday card.

To calculate the time duration Ella spent making the birthday card, we need to subtract the starting time from the finishing time. Let's perform the calculation:

Finishing Time: 7:53 p.m.

Starting Time: 7:19 p.m.

To calculate the minutes, we can convert both times to minutes past midnight (assuming it is a 24-hour clock) and then find the difference.

Starting Time in Minutes: 7 * 60 + 19 = 439 minutes

Finishing Time in Minutes: 7 * 60 + 53 = 473 minutes

Now, we can find the duration by subtracting the starting time from the finishing time:

Duration = Finishing Time - Starting Time = 473 minutes - 439 minutes = 34 minutes

Therefore, Ella spent 34 minutes making the birthday card.

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The equation a = √(c² - b²) is a valid expression based on the Pythagorean Theorem.

Proof:

Given: In right triangle ABC, according to the Pythagorean Theorem, a² + b² = c².

To prove: a = √(c² - b²).

Proof Steps:

1. Start with the given equation from the Pythagorean Theorem: a² + b² = c².

2. Subtract b² from both sides of the equation to isolate a²: a² = c² - b².

3. Take the square root of both sides of the equation: √(a²) = √(c² - b²).

4. Apply the Square Root Property of Equality, which states that if a² = b², then a = ±√b². This allows us to simplify the equation further: a = ±√(c² - b²).

Hence, we have successfully verified that a = √(c² - b²) based on the Pythagorean Theorem.

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The solutions to the equation 2x² - 5 = -3x using the quadratic formula are x = 1 and x = -5/2.

To solve the equation 2x² - 5 = -3x using the quadratic formula, we need to first rewrite the equation in the standard form, which is ax² + bx + c = 0.
Given equation: 2x² - 5 = -3x
Let's bring all the terms to one side to obtain the standard form:
2x² + 3x - 5 = 0
Now, we can use the quadratic formula, which states that for an equation of the form ax² + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b² - 4ac)) / (2a)
Applying this formula to our equation, we have:
a = 2, b = 3, c = -5
x = (-3 ± √(3² - 4(2)(-5))) / (2(2))
Simplifying further:
x = (-3 ± √(9 + 40)) / 4
x = (-3 ± √49) / 4
Taking the square root of 49 gives us two possibilities:
x = (-3 + 7) / 4 or x = (-3 - 7) / 4
Simplifying these:
x = 4 / 4 or x = -10 / 4
Finally, simplifying further:
x = 1 or x = -5/2

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The mean, variance, and standard deviation of the data set 12, 3, 2, 4, 5, 7 are 5, 2.25, and 1.5, respectively.

The mean is the average of the data set. To find the mean, we add up all the numbers in the data set and then divide by the number of numbers in the data set. In this case, the mean is (12 + 3 + 2 + 4 + 5 + 7) / 6 = 5.

The variance is a measure of how spread out the data is. To find the variance, we first find the squared deviations from the mean for each number in the data set. In this case, the squared deviations from the mean are (7 - 5)² = 4, (2 - 5)² = 9, (3 - 5)² = 4, (4 - 5)² = 1, and (5 - 5)² = 0. We then add up all the squared deviations from the mean and divide by the number of numbers in the data set. In this case, the variance is (4 + 9 + 4 + 1 + 0) / 6 = 2.25.

The standard deviation is the square root of the variance. In this case, the standard deviation is √2.25 = 1.5.

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The mean number of bedrooms per house is 2.83

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

The table of values

The mean number in the housing estate is calculated as

Mean = Sum/Count

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

Mean = (1.9 + 3.1 + 3.5)/3

Evaluate

Mean = 2.83

Hence, the mean is 2.83

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The variable, w in the expression given represents the number of bottles.

Given that :

The whole expression 20w represents the amount of water in a given number of bottles .

Since the size of each bottle is 20 ounces, then the value of 'w' would represent the number of bottles.

Therefore, the constant value , 20 = size of water bottle while the variable, w represents the number of bottles.

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Amy will pay a total of approximately $30,007.20 in interest over the life of the loan.

Amy bought a car for $29,000, making a 10% down payment and financing the remaining balance for 60 months with an APR of 5.5%. The question asks for the total interest Amy will pay over the life of the loan.

To calculate the total interest paid, we need to determine the monthly payment amount and then multiply it by the number of months. The monthly payment can be calculated using the formula for a fixed-rate loan: P = (r × PV) / (1 - (1 + r)⁽⁻ⁿ⁾)

where P is the monthly payment, r is the monthly interest rate, PV is the present value or loan amount, and n is the number of months.

First, we calculate the loan amount after the down payment: $29,000 - ($29,000 × 10%) = $26,100.

Next, we calculate the monthly interest rate: 5.5% / 12 = 0.00458.

Using the formula, we can find the monthly payment amount: P = (0.00458 × $26,100) / (1 - (1 + 0.00458)⁽⁻⁶⁰⁾) ≈ $500.12.

Finally, we multiply the monthly payment by the number of months: $500.12 × 60 = $30,007.20.

Therefore, Amy will pay a total of approximately $30,007.20 in interest over the life of the loan.

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The amount of the payment is $2,705.88.

The payment amount is rounded to the nearest penny, and it is specified to use dollar signs and commas where needed. The total payment is $2,705.88, which indicates the monetary value of the transaction. The precision of the payment amount is provided, ensuring that it is accurate up to the nearest penny. The formatting guidelines for using dollar signs and commas are followed, adding clarity to the monetary value presented.

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The bag contains 40 kg of peanuts and 60 kg of almonds.

Let's assume the bag contains x kg of peanuts and y kg of almonds. According to the given information, the price of peanuts is $2 per kg, and the price of almonds is $2.04 per kg. The average price of the mixture is $2.08 per kg.

To find the solution, we need to set up an equation based on the prices and quantities. The equation can be written as:

(2x + 2.04y) / (x + y) = 2.08

Simplifying the equation, we get:

2x + 2.04y = 2.08(x + y)

2x + 2.04y = 2.08x + 2.08y

0.04y = 0.08x

y = 2x

Substituting this value of y in terms of x back into the equation, we have:

2x + 2.04(2x) = 2.08x + 2.08(2x)

2x + 4.08x = 2.08x + 4.16x

6.08x = 6.24x

0.16x = 0

x = 0

This means that x, the weight of peanuts, is equal to zero. However, since the total weight of the bag is 100 kg, there must be some peanuts in the bag. Therefore, there must be an error in the given information or calculation.

If we assume that the total weight of peanuts and almonds is 100 kg, we can solve for the quantities. Let's assign x as the weight of peanuts and y as the weight of almonds.

x + y = 100  (Total weight of the bag)

2x + 2.04y = 2.08 * 100  (Price equation)

Simplifying the equations, we have:

x + y = 100

2x + 2.04y = 208

Multiplying the first equation by 2, we get:

2x + 2y = 200

Subtracting this equation from the second equation, we have:

2x + 2.04y - (2x + 2y) = 208 - 200

0.04y = 8

y = 8 / 0.04

y = 200

Substituting the value of y into the first equation, we can solve for x:

x + 200 = 100

x = 100 - 200

x = -100

Since negative weight is not possible, we can conclude that there is an inconsistency or error in the given information or calculation.

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The equation of the ellipse in standard form is [tex]\frac{x^2}{279841} + \frac{y^2}{203401} = 1[/tex] and the length of the ellipse corresponds to twice the value of the semi-major axis and width of the ellipse corresponds to twice the value of the semi-minor axis.

To calculate the  equation of the ellipse in standard form, we need to determine the values of major and minor axis. The major axis corresponds to the longer dimension of the ellipse which according to the question is 1058 ft. The minor axis corresponds to the shorter dimension of the ellipse which according to the question is 902 ft.

The equation of the standard form of the ellipse is given as :

[tex]\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1[/tex]

where, (h, k) is the center of the ellipse, a is the semi major axis while b is semi minor axis. The origin is at the center of President's Park South, so the center of the ellipse is (0, 0).

So, the equation becomes:

[tex]\frac{(x - 0)^2}{(1058/2)^2} + \frac{(y - 0)^2}{(902/2)^2} = 1[/tex]

[tex]\frac{x^2}{529^2} + \frac{y^2}{451^2} = 1[/tex]

[tex]\frac{x^2}{279841} + \frac{y^2}{203401} = 1[/tex]

The length and width of the ellipse determine the values of the semi-major axis (a) and the semi-minor axis (b) in the equation. So, the length of the ellipse corresponds to twice the value of the semi-major axis and width of the ellipse corresponds to twice the value of the semi-minor axis.

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Statement II and III are not true, while statement I is true. Therefore, the correct answer is c. II and III only.

I. 68% of the values are between 37.6 and 50.4: This statement is true. In a normal distribution, approximately 68% of the values fall within one standard deviation of the mean. In this case, the range of 37.6 to 50.4 falls within one standard deviation of the mean (44 ± 3.2).

II. 13.5% of the values are less than 40.8: This statement is not true. In a normal distribution, approximately 50% of the values fall below the mean. Since the mean is 44, it is not possible for only 13.5% of the values to be less than 40.8.

III. 5% of the values are lower than 37.6 or higher than 50.4: This statement is not true. In a normal distribution, approximately 2.5% of the values fall below the mean minus one standard deviation, and approximately 2.5% of the values fall above the mean plus one standard deviation. Since the range of 37.6 to 50.4 falls within one standard deviation of the mean, it is not possible for 5% of the values to be outside this range.

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To prove that  ΔFED ≅ ΔBDC, we will make use of properties of parallel lines. As, we know that ΔACE is equilateral triangle, so all the three sides are equal. This implies that AC = CE. Now, we know that FB || EC, so  by the alternate interior angles theorem we can say that ∠FBD = ∠CEB.

In the question, it has been given that FDB || BC, so ∠FDB = ∠BCD. Similarly, BD || EF so by alternate interior angles theorem we can say that ∠BDC = ∠FED. We know that D is the midpoint of EF, so DE = DF. So, now ∠FED ≅ ∠BDC by alternate interior angles, DE ≅ BD as D is midpoint of EF and BD || EF, and DF ≅ DC because DE = DF and ΔACE is equilateral triangle.

Thus, we can say that ΔFED ≅ ΔBDC by Side Angle Side congruence.

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