Ms. Follis and Mr. Jackamonis start in the middle of the football field in the exact same spot. Ms. Follis walks 5 feet directly north. Mr. Jackamonis walks directly east. Ms. Follis and Mr. Jackamonis are now exactly 13 feet apart. How far did Mr. Jackamonis walk?

The volume of the solid obtained by rotating the region bounded by the curves y = x^2, x = 0, and y = 1 about the y-axis is π/2 cubic units.

The solid obtained by rotating the region bounded by the curves y = x^2, x = 0, and y = 1 about the y-axis can be found using the method of disks or washers.

To use the method of disks, we can imagine dividing the region into thin vertical strips, each with width dx. The volume of each disk is then πy^2dx, where y is the distance from the strip to the axis of rotation (which is the y-axis in this case). We can find y in terms of x by solving y = x^2 for x, which gives x = sqrt(y).

Thus, the volume of each disk is π(sqrt(y))^2dx = [tex]\pi ydx[/tex]. Integrating from y = 0 to y = 1, we get the total volume of the solid as π∫(0 to 1) [tex]ydx[/tex].

Using the power rule of integration, this simplifies to π[x^2/2] from 0 to 1, which equals π/2.

To use the method of washers, we can imagine dividing the region into thin horizontal strips, each with height [tex]dy[/tex]. The volume of each washer is then π(R^2 - r^2)[tex]dy[/tex], where R is the outer radius (which is 1 in this case) and r is the inner radius. The inner radius is simply the distance from the strip to the y-axis, which is x.

Using the equation y = x^2, we can solve for x in terms of y to get x = sqrt(y).

Thus, the inner radius is sqrt(y) and the volume of each washer is π(1^2 - (sqrt(y))^2)[tex]dy[/tex] = π(1 - y)[tex]dy[/tex]. Integrating from y = 0 to y = 1, we get the total volume of the solid as π∫(0 to 1) (1 - y)[tex]dy[/tex].

Using the power rule of integration, this simplifies to π[y - y^2/2] from 0 to 1, which equals π/2.

Therefore, the volume of the solid obtained by rotating the region bounded by the curves y = x^2, x = 0, and y = 1 about the y-axis is π/2 cubic units, regardless of which method we use.

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The union symbol (∪) means that we are combining all three intervals into a single set.

The set of points from -6 to 0 but excluding -5 and 0 can be expressed as a union of intervals as follows:

(-6, -5) U (-5, 0) U (0, 6)

This means that the set includes all real numbers from -6 to 6, except for -5 and 0, which are excluded. The parentheses indicate that the endpoints (-6, -5, 0, and 6) are not included in the set. The union symbol (∪) means that we are combining all three intervals into a single set.

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Answer: Yes, dilation centered at the origin with scale factor of 3.

Step-by-step explanation:

Two figures are similar if they have the same shape, but possibly different sizes. In order to check whether two triangles are similar, we need to check whether their corresponding angles are congruent and whether their corresponding side lengths are proportional.

We can see that triangle ACAT and triangle ABUG have corresponding angles that are congruent, and their corresponding side lengths are proportional with a ratio of 3 (i.e., AC = 3AB, AT = 3AU, and CT = 3UG). Therefore, triangle ACAT is similar to triangle ABUG.

To map triangle ACAT onto triangle ABUG, we need a transformation that scales each point by a factor of 3 about the origin (since the dilation is centered at the origin). Therefore, the correct answer is option (C).

Answer: yes

Step-by-step explanation THE Answer the length of the median

is 11.4 units.

Step 1. Given Information.

Given triangle CAT has vertices

,

and

. M is the midpoint of

.

The length of the median

is to be determined.

Step 2. Explanation.

The midpoint of two points

is given by

.

Plugging the values in the equation to find the point M:

The distance between two points

is given by

.

Plugging the given values in the equation to find the distance between C and M:

Step 3. Conclusion.

Hence, the length of the median

is 11.4 units.

Answer:

x = 2

Step-by-step explanation:

y = 3x - 7

y = 7 - x

следовательно,

3x - 7 = 7 - x

=> 2x = 14

=> x = 14/7

=> x = 2

The kayakers are traveling faster downstream (with the current) than upstream (against the current).

1) Variables:

- Speed of the kayaker (unknown, let's call it x)

- Speed of the current = 3 mph (given)

- Distance kayaked one way = 1 mile (given)

- Total distance covered (round trip) = 2 miles (given)

- Total time of the trip = 3 hours 20 minutes = 3.33 hours (converted to hours for convenience)

Table:

Photo attached.

2) The equation to model the problem is:

distance = rate × time

Using this equation for each kayaking portion, we get:

1 = (x - 3) t

1 = (x + 3) t

We also know that the total time of the trip is 3.33 hours:

t + t = 3.33

2t = 3.33

t = 1.665

3) Now we can solve for x by substituting t = 1.665 in either of the above equations:

1 = (x - 3) (1.665)

x - 3 = 0.599

x = 3.599

Thus, the kayakers are paddling at a speed of 3.599 miles per hour.

4) The kayakers are paddling at a speed of 3.599 miles per hour. This solution is obtained by calculating the average speed of the kayakers over the entire trip, taking into account the opposing speed of the river current.

The kayakers are traveling faster downstream (with the current) than upstream (against the current).

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Miss Edwards paid about $17.88 for the gasoline.

Price is the amount of money or other valuable consideration that a buyer is willing to pay to acquire a good or service from a seller. In a market economy, prices are determined by the forces of supply and demand.

To estimate how much Miss Edwards paid for the gasoline, we can round the price per gallon to the nearest cent and multiply it by the number of gallons purchased.

The price per gallon is $1.49 9/10, which is closest to $1.50 when rounded to the nearest cent.

So, we can estimate the cost of 1 gallon of gasoline as $1.50.

Therefore, the cost of 11.92 gallons of gasoline would be approximately:

11.92 x $1.50 = $17.88

So, we can estimate that Miss Edwards paid about $17.88 for the gasoline.

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Answer: B

Step-by-step explanation: 21+12= 33.

45 - 33 = 12.

red came 21 times, which is too much. blue came 12 times, which is exactly the same as green.

hope this helps, please make me the brainliest answer

Answer:

D

Step-by-step explanation:

The expression 1/1×2 + 1/2×3 + ... + 1/n(n+1) can be written as Σ(k=1 to n) 1/k(k+1). After examining the values of this expression for small values of n, we can conjecture that the formula for this expression is 1 - 1/(n+1).

To prove this formula, we can use mathematical induction.

We need to prove that 1/1×2 + 1/2×3 + ... + 1/n(n+1) = 1 - 1/(n+1) for all positive integers n.

First, we can show that the formula is true for n = 1:

1/1×2 = 1 - 1/2

Next, we assume that the formula is true for some positive integer k, and we want to prove that it is also true for k+1.

Assuming the formula is true for k, we have:

1/1×2 + 1/2×3 + ... + 1/k(k+1) = 1 - 1/(k+1)

Adding (k+1)/(k+1)(k+2) to both sides, we get:

1/1×2 + 1/2×3 + ... + 1/k(k+1) + (k+1)/(k+1)(k+2) = 1 - 1/(k+1) + (k+1)/(k+1)(k+2)

Simplifying the right side, we get:

1/1×2 + 1/2×3 + ... + 1/k(k+1) + (k+1)/(k+1)(k+2) = 1 - 1/(k+2)

Therefore, the formula is true for k+1 as well.

By mathematical induction, the formula is true for all positive integers n.

Thus, we have proved that 1/1×2 + 1/2×3 + ... + 1/n(n+1) = 1 - 1/(n+1) for all positive integers n.

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The domain of g(x,y) = 1/xy is all real numbers except for x=0 and y=0.

This is because division by zero is undefined and would result in an error or undefined value. In other words, x and y cannot be zero in order for the function to have a valid output.

To understand this further, we can think about the function graphically. The function g(x,y) represents a three-dimensional surface where the height of the surface at any point (x,y) is given by 1/xy. However, we can see that the function is undefined at the points where x=0 or y=0.

At these points, the surface would have a "hole" or a "break" since the height of the surface is undefined. Therefore, the domain of the function is all real numbers except for x=0 and y=0.

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To evaluate this iterated integral, we first need to integrate with respect to z from 0 to sin(y). This will give us an expression involving sin(y). Next, we integrate this expression with respect to x from 0 to y/7. Finally, we integrate the resulting expression with respect to y from 0 to π/2.

The integration steps involve some trigonometric substitutions and u-substitutions, which make the process quite lengthy. However, by carefully following the steps and simplifying the expressions, we can arrive at the final answer. In summary, the iterated integral evaluates a complicated expression involving sine and cosine functions, which can be obtained through a long explanation involving multiple integration steps. Overall, the process involves calculating the integral with respect to z, then x, and finally y, and simplifying the resulting expressions at each step.

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To evaluate this iterated integral, we first need to integrate with respect to z from 0 to sin(y). This will give us an expression involving sin(y). Next, we integrate this expression with respect to x from 0 to y/7. Finally, we integrate the resulting expression with respect to y from 0 to π/2.

The integration steps involve some trigonometric substitutions and u-substitutions, which make the process quite lengthy. However, by carefully following the steps and simplifying the expressions, we can arrive at the final answer. In summary, the iterated integral evaluates a complicated expression involving sine and cosine functions, which can be obtained through a long explanation involving multiple integration steps. Overall, the process involves calculating the integral with respect to z, then x, and finally y, and simplifying the resulting expressions at each step.

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Answer:

8x+48>280

Step-by-step explanation:

first do your mom then do your dad

To determine which hybrid plant to introduce, we need to calculate the expected profits for each option. The profits for each option depend on the demand for the plant and the cost of producing it. Let's assume that the cost of producing each hybrid plant is the same and equal to $5,000.

a) Expected profits for Hybrid 1:

If demand is low: profit = (10,000 - 5,000) = $5,000

If demand is medium: profit = (10,000 - 5,000) = $5,000

If demand is high: profit = (30,000 - 5,000) = $25,000

Expected profit for Hybrid 1 = (4/10)*5,000 + (3/10)*5,000 + (3/10)*25,000 = $13,000

Expected profits for Hybrid 2:

If demand is low: profit = (15,000 - 5,000) = $10,000

If demand is medium: profit = (10,000 - 5,000) = $5,000

If demand is high: profit = (35,000 - 5,000) = $30,000

Expected profit for Hybrid 2 = (4/10)*10,000 + (3/10)*5,000 + (3/10)*30,000 = $16,500

Therefore, if Venus wants to maximize expected profits, they should introduce Hybrid 2.

b) To determine the most that Venus would pay for a highly reliable demand forecast, we need to calculate the expected value of perfect information (EVPI). The EVPI is the difference between the expected profits with perfect information and the expected profits under uncertainty.

With perfect information, Venus would know exactly which hybrid plant to introduce based on the demand. The expected profits with perfect information would be:

If demand is low: profit = (15,000 - 5,000) = $10,000

If demand is medium: profit = (10,000 - 5,000) = $5,000

If demand is high: profit = (35,000 - 5,000) = $30,000

Expected profit with perfect information = (4/10)*10,000 + (3/10)*5,000 + (3/10)*30,000 = $16,500

The expected profits under uncertainty for Hybrid 2 (the preferred option) are $16,500. Therefore, the EVPI is:

EVPI = Expected profit with perfect information - Expected profit under uncertainty = $16,500 - $16,500 = $0

This means that Venus should not be willing to pay anything for a highly reliable demand forecast since it would not increase their expected profits.

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The probability of drawing two jacks in a row from a deck of 52 cards without replacement is equal to option(B) 0.005.

Total number of cards in a deck = 52

Number of Jack in deck of cards = 4

Probability of getting a jack on the first draw is

= 4/52

Now, there are only 3 jacks remaining in a deck of 51 cards.

This implies,

Probability of drawing another jack on the second draw given that the first card was a jack

= 3/51

Probability of drawing two jacks in a row,

Multiply the probability of drawing

= (a jack on first draw by another jack on second draw given first card was a jack)

⇒ P(two jacks)

= (4/52) × (3/51)

= 1/13 × 1/17

= 1/221

= 0.00452489

= 0.005 (rounded to three decimal places).

Therefore, the probability of drawing two jacks in a row is equal to option(B) 0.005.

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The probability of drawing two jacks in a row from a deck of 52 cards without replacement is equal to option(B) 0.005.

Total number of cards in a deck = 52

Number of Jack in deck of cards = 4

Probability of getting a jack on the first draw is

= 4/52

Now, there are only 3 jacks remaining in a deck of 51 cards.

This implies,

Probability of drawing another jack on the second draw given that the first card was a jack

= 3/51

Probability of drawing two jacks in a row,

Multiply the probability of drawing

= (a jack on first draw by another jack on second draw given first card was a jack)

⇒ P(two jacks)

= (4/52) × (3/51)

= 1/13 × 1/17

= 1/221

= 0.00452489

= 0.005 (rounded to three decimal places).

Therefore, the probability of drawing two jacks in a row is equal to option(B) 0.005.

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Answer: 4 km

Step-by-step explanation:

You use the Pythagorean theorem: [tex]a^{2} +b^{2} =c^{2}[/tex]

So 2^2 + 3^2 = c^2

C=[tex]\sqrt{13}[/tex] or 3.60555

Then you round to the nearest km to get 4

Answer:  Let's use "x" to represent the unknown quantity in grams:

We have:

3/10 x = 25% of 120g

Converting 25% to a fraction:

3/10 x = 25/100 * 120g

Simplifying 25/100:

3/10 x = 0.25 * 120g

Multiplying 0.25 by 120g:

3/10 x = 30g

To solve for "x", we can multiply both sides by the reciprocal of 3/10:

x = (10/3) * 30g

Simplifying:

x = 100g

Therefore, 3/10 of 100g is equal to 25% of 120g.

Step-by-step explanation:

Answer:

3/10 of 100g = 25% of 120g

Step-by-step explanation:

Let us assume that,

→ Missing quantity = x

Now we have to,

→ Find the required value of x.

Forming the equation,

→ 3/10 of x = 25% of 120

Then the value of x will be,

→ 3/10 of x = 25% of 120

→ (3/10) × x = (25/100) × 120

→ 3x/10 = 25 × 1.2

→ 3x/10 = 30

→ 3x = 30 × 10

→ 3x = 300

→ x = 300/3

→ [ x = 100 ]

Hence, the value of x is 100.

The answer of the given question based on the differential equation,  the answer is (C) 20,885.

A differential equation is equation that involves derivatives or differentials of unknown function. These equations describe how aquantity changes over time, space or some other variable. They are used to model a wide variety of phenomena in physics, engineering, biology, economics, and many other fields.

We are given the differential equation:

dy/dt = 0.4y

Utilising the separation of variables, we can resolve this.

dy/y = 0.4 dt

Integrating both sides:

ln|y| = 0.4t + C

where C is the constant of integration. To find C, we use the initial condition that the colony initially had 2000 bacteria, so when t = 0, y = 2000.

ln|2000| = 0 + C

C = ln|2000|

So the equation becomes:

ln|y| = 0.4t + ln|2000|

Simplifying, we get:

|y| = [tex]e^{(0.4t+ln|2000|)}[/tex] = [tex]2000e^{(0.4t)}[/tex]

Since the population of bacteria cannot be negative, we can drop the absolute value signs.

So, the solution to the differential equation is:

[tex]y = 2000e^{(0.4t)}[/tex]

To find the approximate number of bacteria at time t=7, we substitute t=7 into the equation:

[tex]y = 2000e^{(0.4(7)) }[/tex] = 20,885

Consequently, 20,885 bacteria are present at time t=7, which is a rough estimate.

So, the answer is (C) 20,885.

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Part A :f has relative maxima at x = 1 and x = 3, and a relative minimum at x = 8, on the interval [0, 9].

Part B: the graph of f is concave up on the intervals (−∞, 1) and (8, ∞) and is concave down on the interval (1, 8).

Part C: x = 1 and x = 8 these are the points of inflection of f.

To find relative extrema, we want to find the basic places of the capability and afterward decide if they are greatest or least by really looking at the indication of the subsequent subordinate.

The point is a relative minimum when the second derivative is positive, and it is a relative maximum when the second derivative is negative. To determine the nature of the critical point, we must employ alternative strategies if the second derivative is zero.

Part A:

The values of x at which f'(x) is zero or undefined are the critical points of f. From the table, we see that f'(x) = 0 at x = 1 and x = 8, and f'(x) is unclear at x = 3.

At each critical point, we must now examine the sign of the second derivative to determine whether it is a maximum or a minimum.

Negative f''(1) = -2 is the case when x = 1. As a result, the relative maximum is the critical point at x = 1.

Because f''(3) is not defined, the second derivative test cannot be used to determine the nature of the critical point at x = 3. However, as the table demonstrates, f shifts from increasing to decreasing at x = 3, indicating that x = 3 is the relative maximum.

We have a positive value of f'(8) = 2 when x = 8. As a result, the relative minimum at x = 8 is the critical point.

Therefore, f has relative maxima at x = 1 and x = 3, and a relative minimum at x = 8, on the interval [0, 9].

Part B:

To determine where the graph of f is concave up and where it is concave down, we need to find the intervals where f''(x) > 0 and where f''(x) < 0, respectively.

From the table, we see that f''(x) > 0 for x < 1 and for x > 8, and f''(x) < 0 for 1 < x < 8. Therefore, the graph of f is concave up on the intervals (−∞, 1) and (8, ∞) and is concave down on the interval (1, 8).

Part C:

We must determine the values of x at the points on the graph where the concavity changes in order to locate any points of inflection of f. From part B, we see that the concavity changes at x = 1 and x = 8. Accordingly, these are the marks of expression of f.

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The probability of getting exactly 3 tails when tossing a coin 4 times is 25%.

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

{HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}.

There are 16 equally likely outcomes in the sample space.

Out of these, there are 4 outcomes that have exactly 3 tails: TTTH, TTHT, THTT, and HTTT.

Therefore, P(3 tails) = 4/16 = 1/4 = 0.25.

So, the probability of getting exactly 3 tails when tossing a coin 4 times is 0.25 or 25%.

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The partial fraction decomposition of the integrand for the integral of x² - 2x - 1 / (x - 3)²(x² + 9) dx is of the form: A / (x - 3) + B / (x - 3)² + (Cx + D) / (x² + 9), where A, B, C, and D are constants.

First, factorize the denominator: (x - 3)²(x² + 9). This gives us three distinct linear factors: (x - 3), (x - 3), and (x² + 9).

Write the partial fraction decomposition using the distinct linear factors as denominators. In this case, we have two (x - 3) terms and one (x² + 9) term:

x² - 2x - 1 / (x - 3)²(x² + 9) = A / (x - 3) + B / (x - 3)² + (Cx + D) / (x² + 9).

Multiply both sides of the equation by the common denominator (x - 3)²(x² + 9) to eliminate the denominators.

Simplify the numerators and equate the corresponding coefficients on both sides of the equation.

Solve the resulting system of equations to find the values of A, B, C, and D.

Substitute the values of A, B, C, and D back into the original partial fraction decomposition.

Therefore, the partial fraction decomposition of the integrand for the given integral is: A / (x - 3) + B / (x - 3)² + (Cx + D) / (x² + 9), where A, B, C, and D are constants

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The quantity of ounces for each bag would be as follows:

walnut= 8.82

almonds = 4.29 and

cashew = 11.89 ounces.

The quantity of walnut used = 21 ounces

The quantity of almonds used = 10.2 ounces

The quantity of cashew used = 28.3 ounces

Total = 59.5ounces.

The total ounces makes up = 25 bags

The ounce of each bag;

walnut = 21/59.5×25/1

= 8.82

almonds= 10.2/59.5 × 25/1

= 4.29

cashew = 28.3/59.5 × 25/1

= 11.89

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The median class size, would be 40–50.

First, find the total frequency to be :

= 4 + 12 + 24 + 36 + 20+ 16 + 8 + 5

= 125

The median would be located at :

= (125 + 1 ) / 2

= 63 rd position

                  Cumulative frequency

10–20                4

20–30              16

30–40               40

This means that the median class would be 40–50 as this interval has a cumulative frequency of 76 which means the 63 rd number is there.

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The rest of the question is :
Find Median:- Class interval : 10–20 20–30 30–40 40–50 50–60 60–70 70–80 Frequency: 4 12 24 36 20 16 8 5​

To find the restricted range for the inverse function of sine, we represent y as follows: (y = sin⁻¹x) is [-π/2, π/2].

To find the restricted range for the inverse function of cosine, we would have;  (y = cos⁻¹x) is [0, π]

The restricted range refers to the ability to specify a function criterion such that the population data would meet that criterion. For the above problem, we are to get the inverse functions of sine and cosine by restricting the range.

This can be obtained as follows:

[-π/2, π/2] for the function sine and

[0, π] for the function cosine.

Thus, the restricted ranges are  [-π/2, π/2] and  [0, π] respectively.

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By rigid transformations, the images of the parent function are, respectively:

Case 1: f(x) = - |x| + 3

Case 2: f(x) = |x - 2| - 2

Case 3: f(x) = |x + 15|

Case 4: f(x) = - |x + 3| + 3

In this problem we must derive the absolute value functions by means of rigid transformations used on parent function f(x) = |x|. We proceed to apply any of the following rigid transformations:

Horizontal translation

f(x) → f(x - k), where k > 0 for a right translation.

Vertical translation

f(x) → f(x) + k, where k > 0 for a upward translation.

Reflection around x-axis

f(x) → - f(x)

Case 1 (Reflection - Vertical translation)

f(x) = - |x| + 3

Case 2 (Horizontal translation - Vertical translation)

f(x) = |x - 2| - 2

Case 3 (Horizontal translation)

f(x) = |x + 15|

Case 4 (Reflection - Horizontal translation - Vertical translation)

f(x) = - |x + 3| + 3

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The limit of the sequence as n approaches infinity is 4.

Therefore, lim n→[infinity] an = 4.

To determine whether the sequence converges or diverges, we need to find the limit of the sequence as n approaches infinity.

We can simplify the expression for the nth term of the sequence as follows:

an = 4 + 2n^2 / (2n + 3n^2)

= 4 + n / (1.5 + n)

As n approaches infinity, the denominator of the fraction approaches infinity much faster than the numerator. Therefore, the fraction approaches zero and the value of the nth term approaches 4.

In other words, as n becomes very large, the terms of the sequence become arbitrarily close to 4. Therefore, the sequence converges to 4.

Hence, the limit of the sequence as n approaches infinity is 4.

Therefore, lim n→[infinity] an = 4.

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The length of the side of the triangle BC is 0. 97 cm

It is important to note that their are different trigonometric identities. These identities are;

From the information given, we have that;

cos θ = adjacent/hypotenuse

sin θ = opposite/hypotenuse

tan θ = opposite/adjacent

Then, we have;

sin 17 = BC/√11

cross multiply the values, we have;

BC = sin 17 (√11)

BC = 0. 97 cm

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The Taylor series for f(x) = e^(7x) centered at c = -1 is e^(-7) - 7e^(-7)(x+1) + 49e^(-7)(x+1)^2/2! - 343e^(-7)(x+1)^3/3! + 2401e^(-7)(x+1)^4/4! - ...

The Taylor series for a function f(x) centered at c is given by

f(x) = f(c) + f'(c)(x-c)/1! + f''(c)(x-c)^2/2! + f'''(c)(x-c)^3/3! + ...

To find the Taylor series for f(x) = e^(7x) centered at c = -1, we will need to calculate the derivatives of f(x) at c = -1.

f(x) = e^(7x)

f'(x) = 7e^(7x)

f''(x) = 49e^(7x)

f'''(x) = 343e^(7x)

f''''(x) = 2401e^(7x)

...

Evaluating these derivatives at c = -1, we get

f(-1) = e^(-7)

f'(-1) = -7e^(-7)

f''(-1) = 49e^(-7)

f'''(-1) = -343e^(-7)

f''''(-1) = 2401e^(-7)

...

Substituting these values into the Taylor series formula, we get:

e^(7x) = e^(-7) - 7e^(-7)(x+1) + 49e^(-7)(x+1)^2/2! - 343e^(-7)(x+1)^3/3! + 2401e^(-7)(x+1)^4/4! - ...

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The given question is incomplete, the complete question is:

Find the Taylor series centered at c= -1, f(x) = e^(7x)

The joint PDF which is uniform over the triangle with vertices  is:

f(x,y) = 2, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1-x.

Since the joint PDF is uniform over the triangle with vertices at (0,0), (0,1), and (1,0), the joint PDF can be expressed as:

f(x,y) = c, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1-x,

where c is a constant that ensures that the total probability over the entire region is equal to 1. To find c, we can integrate the joint PDF over the entire region:

[tex]\int \int f(x,y)dydx = \int ^1_0 \int _0^{(1-x)} c dydx[/tex]

[tex]= \int ^1_0 cx - cx^2/2 dx[/tex]

= c/2

Since the total probability over the entire region must be equal to 1, we have:

∫∫f(x,y)dydx = 1

Thus, we have:

c/2 = 1

c = 2

Therefore, the joint PDF is:

f(x,y) = 2, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1-x.

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Answer:

  H  (8, -3)

Step-by-step explanation:

You want a possible vertex of rectangle KLMN if the midpoint of the diagonals is (2, 1) and one of the vertices is (-4, 5).

The diagonals of a rectangle are congruent and bisect each other, so the given point of intersection is the midpoint of the diagonals. If one of the diagonal end points is K = (-4, 5), then the other end of that diagonal is ...

  X = (K+M)/2

  M = 2X -K = 2(2, 1) -(-4, 5) = (2·2+4, 2·1-5)

  M = (8, -3)

Another vertex on the rectangle is (8, -3).

__

Additional comment

Segment KM will be the diameter of the circumcircle of the rectangle. Other possible vertices will lie on that circle. As it happens, none of the offered choices is the same distance from X as point K is.

The attached figure shows the given diagonal and one of an infinite number of possible rectangles KLMN.

The choice of naming the given vertex K is arbitrary.

The theorems that are necessary when proving that the opposite angles of a parallelogram are congruent are;

B. Angle Addition Postulate.

D. Corresponding parts of congruent triangles are congruent.

The pairs of alternate internal angles are congruent when two parallel lines are intersected by a transversal, such as a line that crosses through both lines.

When two sides of a parallelogram are parallel, the angles created by the transversal that cuts through them are alternate internal angles.

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Missing parts;

Which of the following are necessary when proving that the opposite angles of a parallelogram are congruent?

Check all that apply.

A. Corresponding parts of similar triangles are similar.

B. Angle Addition Postulate.

C. Segment Addition Postulate.

D. Corresponding parts of congruent triangles are congruent.

We need to check if a = 1 satisfies the original condition that AB = CB. If a = 1, then CB = √[(1 - 3[tex])^2[/tex] + (3 - 1[tex])^2[/tex]] = √[4 + 4] = 2√(2), which is equal to AB. Therefore, the solution is a = 1 or 5.

First, we need to find the length of AB and CB using the distance formula:

AB = √(1 - (-1)[tex])^2[/tex] + (3 - 1[tex])^2[/tex]] = √[4 + 4] = 2√(2)

CB = √(a - 3[tex])^2[/tex] + (3 - 1[tex])^2[/tex]]

Since AB = CB, we can set the two expressions equal to each other:

2√(2) = √[(a - 3[tex])^2[/tex] + 4]

Squaring both sides, we get:

8 = (a - 3[tex])^2[/tex] + 4

Subtracting 4 from both sides, we get:

4 = (a - 3[tex])^2[/tex]

Taking the square root of both sides, we get:

2 = a - 3 or -2 = a - 3

Solving for a, we get:

a = 5 or a = 1

However, we need to check if a = 1 satisfies the original condition that AB = CB. If a = 1, then CB = √[(1 - 3[tex])^2[/tex] + (3 - 1[tex])^2[/tex]] = √[4 + 4] = 2√(2), which is equal to AB. Therefore, the solution is a = 1 or 5.

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