A body moves on a coordinate line such that it has a position s​=f(t)=t^2-6t+5 on the interval 0≤t≤8​, with s in meters and t in seconds.
a. Find the​ body's displacement and average velocity for the given time interval.
b. Find the​ body's speed and acceleration at the endpoints of the interval.
c.​ When, if​ ever, during the interval does the body change​ direction?

Answer:

Y(0,13)

X(13,0)

Answer:

Bottom graph

Step-by-step explanation:

[tex]|4x+5|-7 > 12\\|4x+5| > 19\\\\4x+5 > 19\\4x > 14\\x > \frac{7}{2}\\\\4x+5 < -19\\4x < -24\\x < -6[/tex]

Therefore, the last graph is the correct answer

The value of (fog)(4) for the composition given is 48

Obtain the composition of the functions f and g by evaluating f(g(x)).

Evaluate g(x);

Here, g(4) becomes :

Evaluate f(g(x)) by making g(x) = 6

f(x) = x² + 2x

Inputting the values into the function :

Therefore, (fog)(4) = 48.

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This introduces some bias as it excludes individuals who haven't attended, potentially influencing the overall popularity of the tricks.

We can evaluate the sampling method used for the magician's survey.

Voluntary: The survey was sent out to people who had previously attended the magician's programs. It suggests that participation in the survey is optional, meaning respondents can choose whether or not to participate. Therefore, the survey can be considered voluntary.

Ranking Preference: The survey asks participants to rank their three favorite tricks. This allows respondents to express their preferences in a specific order. It provides a clear structure for respondents to indicate their choices, making it easier for the magician to analyze the data and determine the most popular tricks.

Email Survey: The survey was conducted through email, which can be an effective method for reaching a targeted audience. However, it's important to note that the sample is limited to people who have attended the magician's programs in the past. This may introduce some bias as it excludes individuals who haven't attended, potentially influencing the overall popularity of the tricks.

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The true balance of the Ledger Control Account for January 2019 is $3,447,600 (debit).

To correct the Ledger Control Account for the month of January 2019, we need to analyze the given entries and make appropriate adjustments to arrive at the true balance.

Step 1: Analyzing the given entries

From the information provided, we can identify the following entries in the Ledger Control Account:

- Opening Balance (bf): $318,000 (debit)

- Purchases: $1,364,300 (debit)

- Cheques paid to creditors: $41,200 (credit)

- Purchases Returns: $31,000 (debit)

- Transfer from Sales Ledger: $31,000 (credit)

- Discount Received: $31,000 (credit)

- Closing Balance (c/d): $1,724,100 (debit)

Step 2: Adjusting the entries

To arrive at the true balance, we need to make adjustments for any errors or omissions in the given entries. Let's correct each entry:

- Opening Balance (bf) remains unchanged.

- Purchases: No adjustment needed.

- Cheques paid to creditors should be debited instead of credited. Adjust the entry to $41,200 (debit).

- Purchases Returns should be credited instead of debited. Adjust the entry to $31,000 (credit).

- Transfer from Sales Ledger should be debited instead of credited. Adjust the entry to $31,000 (debit).

- Discount Received should be debited instead of credited. Adjust the entry to $31,000 (debit).

- Closing Balance (c/d) remains unchanged.

Adjusted entries:

- Opening Balance (bf): $318,000 (debit)

- Purchases: $1,364,300 (debit)

- Cheques paid to creditors: $41,200 (debit)

- Purchases Returns: $31,000 (credit)

- Transfer from Sales Ledger: $31,000 (debit)

- Discount Received: $31,000 (debit)

- Closing Balance (c/d): $1,724,100 (debit)

Step 3: Calculating the true balance

To calculate the true balance, we need to sum up the debit and credit entries separately and find the difference:

Debit total: $318,000 + $1,364,300 + $41,200 + $31,000 + $31,000 + $1,724,100 = $3,509,600

Credit total: $31,000 + $31,000 = $62,000

True balance: Debit total - Credit total = $3,509,600 - $62,000 = $3,447,600 (debit)

Therefore, the true balance of the Ledger Control Account for January 2019 is $3,447,600 (debit).

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The equations are R + P + D = 800, 3.5R + 4.25P + 4D = 2910 and R = P + D + 440

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

Let R represent the number of Regular gallons, P represent that of Premium and D represent that of Diesel.

A total of 800 gallons was sold, hence:

R + P + D = 800    (1)

Also:

$2910 was sold combined, therefore:

3.5R + 4.25P + 4D = 2910    (2)

440 more gallons of regular was sold, hence:

R = P + D + 440   (3)

The equations are R + P + D = 800, 3.5R + 4.25P + 4D = 2910 and R = P + D + 440

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(a) Probability that 3 balls are all white:

Step 1:

Total balls = Number of red balls + Number of white balls + Number of green balls = 7 + 12 + 4 = 23.

Step 2:

Total ways to choose 3 balls = C(23, 3) = 23! / (3 * (23-3) ) = 23 / (3 * 20) = (23 * 22 * 21) / (3 * 2 * 1) = 1771.

Step 3: Calculate the number of ways to choose 3 white balls out of 12:

Number of ways to choose 3 white balls = C(12, 3) = 12 / (3 * (12-3) ) = 12 / (3 * 9) = (12 * 11 * 10) / (3 * 2 * 1) = 220.

Step 4: Calculate the probability of selecting 3 white balls:

Probability = Number of ways to choose 3 white balls / Total ways to choose 3 balls = 220 / 1771 ≈ 0.1241.

Therefore, the probability that 3 balls drawn are all white is approximately 0.1241.

(b) Probability that 3 balls are one of each color:

Step 1: Calculate the total number of balls in the bag (same as above): Total balls = 23.

Step 2: Calculate the total number of ways to choose 3 balls out of 23 (same as above): Total ways to choose 3 balls = 1771.

Step 3: Calculate the number of ways to choose 1 ball of each color:

Number of ways to choose 1 red, 1 white, and 1 green ball = Number of red balls * Number of white balls * Number of green balls = 7 * 12 * 4 = 336.

Step 4: Calculate the probability of selecting 3 balls of different colors:

Probability = Number of ways to choose 1 ball of each color / Total ways to choose 3 balls = 336 / 1771 ≈ 0.1899.

Therefore, the probability that 3 balls drawn are one of each color is approximately 0.1899.

(c) Probability that 3 balls are of the same color:

Step 1: Calculate the total number of balls in the bag (same as above): Total balls = 23.

Step 2: Calculate the total number of ways to choose 3 balls out of 23 (same as above): Total ways to choose 3 balls = 1771.

Step 3: Calculate the number of ways to choose 3 balls of the same color:

Number of ways to choose 3 red balls = C(7, 3) = 7 / (3 * (7-3) ) = 7 / (3 * 4 ) = (7 * 6 * 5) / (3 * 2 * 1) = 35.

Number of ways to choose 3 white balls = C(12, 3) = 220.

Number of ways to choose 3 green balls = C(4, 3) = 4.

Step 4: Calculate the probability of selecting 3 balls of the same color:

Probability = (Number of ways to choose 3 red balls + Number of ways to choose 3 white balls + Number of ways to choose 3 green balls) / Total ways to choose 3 balls = (35 + 220 + 4) / 1771 ≈ 0.1416.

Therefore, the probability that 3 balls drawn are of the same color is approximately 0.1416.

Answer:

Step-by-step explanation:

(a) Probability that 3 balls are all white:

Total number of balls = 7 red + 12 white + 4 green = 23

Number of favorable outcomes = 12 white balls

Probability = Number of favorable outcomes / Total number of possible outcomes

Probability = 12 / 23

(b) Probability that 3 balls are one of each color:

Total number of balls = 7 red + 12 white + 4 green = 23

Number of favorable outcomes = 7 red balls * 12 white balls * 4 green balls

Probability = Number of favorable outcomes / Total number of possible outcomes

Probability = (7 * 12 * 4) / (23 * 22 * 21)

(c) Probability that 3 balls are the same color:

Total number of balls = 7 red + 12 white + 4 green = 23

Number of favorable outcomes = (7 red balls * 6 red balls * 5 red balls) + (12 white balls * 11 white balls * 10 white balls) + (4 green balls * 3 green balls * 2 green balls)

Step-by-step explanation:

The only thing that changes is the 'y' coordinate ....it changes fro 0 to minus  9  ..... distance is then 9  units

Dan will pay approximately $1231.87 in cash for the fence.

To determine how much Dan will pay in cash for the fence, we need to calculate the total cost of the fence, apply the tax, and then determine the portion that will be paid with a credit card.

Dimensions of the land to be fenced: 450 ft by 280 ft

Length of a roll of fence: 60 ft

Cost of a roll of fence: $60.00

Tax rate: 9.5%

Portion to be paid with a credit card: 1/4 (or 25%)

First, let's calculate the total perimeter of the land to determine how many rolls of fence are needed:

Perimeter = 2 [tex]\times[/tex] (Length + Width)

Perimeter = 2 [tex]\times[/tex] (450 ft + 280 ft)

Perimeter = 2 [tex]\times[/tex] 730 ft

Perimeter = 1460 ft

Since each roll of fence is 60 ft long, the number of rolls needed is:

Number of rolls = Perimeter / Length of a roll

Number of rolls = 1460 ft / 60 ft

Number of rolls ≈ 24.33 rolls

Since we can't purchase a fraction of a roll, we need to round up to the nearest whole number of rolls.

Thus, we need to purchase 25 rolls.

The total cost of the fence without tax is:

Total cost = Number of rolls [tex]\times[/tex] Cost per roll

Total cost = 25 rolls [tex]\times[/tex] $60.00 per roll

Total cost = $1500.00

Now, let's calculate the tax amount:

Tax amount = Total cost [tex]\times[/tex] Tax rate

Tax amount = $1500.00 [tex]\times[/tex] 9.5% (or 0.095)

Tax amount = $142.50

The total cost including tax is:

Total cost with tax = Total cost + Tax amount

Total cost with tax = $1500.00 + $142.50

Total cost with tax = $1642.50

Finally, let's calculate the portion to be paid with a credit card and the portion to be paid in cash:

Portion paid with credit card = Portion to be paid with a credit card * Total cost with tax

Portion paid with credit card = 25% [tex]\times[/tex] $1642.50

Portion paid with credit card = $410.63

Portion paid in cash = Total cost with tax - Portion paid with credit card

Portion paid in cash = $1642.50 - $410.63

Portion paid in cash = $1231.87

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Answer:

(a)  start value = $45,000

(b)  4.6875 months exactly

======================================================

Explanation

Part (a)

I'll use x in place of t, and y in place of 'a'. The reason for this will be mentioned in part (b)

The equation [tex]a = 45+75t-4t^2[/tex] turns into [tex]y = 45 + 75x - 4x^2[/tex] which rearranges to [tex]y = -4x^2+75x+45[/tex]

Let's plug in x = 0 so we can find the account value (variable y) at the very start.

[tex]y = -4x^2+75x+45\\\\y = -4*0^2+75*0+45\\\\y = 45\\\\[/tex]

The nice thing is any term involving x will go to zero, which leaves behind 45 at the end.

The initial value is $45,000. Recall that y is the value in thousands of dollars.

Think of it like saying 45*1000 = 45,000.

-------------------

Part (b)

The template for a general quadratic is [tex]y = ax^2+bx+c[/tex] which involves 'a', so that's why I swapped to x,y to avoid confusion.

That template is used to help complete the square to get a quadratic into vertex form.

Compare [tex]y = ax^2+bx+c[/tex] with [tex]y = -4x^2+75x+45[/tex] to find these values

Plug the first two items into the formula below

h = -b/(2a)

h = -75/(2*(-8))

h = 4.6875

This is the x coordinate of the vertex (h,k). It's the number of months it takes to reach the peak value.

-------------------

Extra info: If you plug x = 4.6875 back into the function, then you'll get y = 308.671875 which represents an account value of $308,671.88 (after rounding to the nearest penny). This is the investment's maximum value.

Answer:

A) [tex]x=-1\pm\sqrt{\frac{13}{6}}[/tex]

Step-by-step explanation:

[tex]\displaystyle x=\frac{-12\pm\sqrt{12^2-4(6)(-7)}}{2(6)}\\\\x=\frac{-12\pm\sqrt{144+168}}{12}\\\\x=\frac{-12\pm\sqrt{312}}{12}\\\\x=\frac{-12\pm2\sqrt{78}}{12}\\\\x=-1\pm\frac{\sqrt{78}}{6}\\\\x=-1\pm\sqrt{\frac{78}{36}}\\\\x=-1\pm\sqrt{\frac{13}{6}}[/tex]

When the Black-Scholes-Merton model is used  

the value of d1 is -0.0985,

the value of d2 is -0.2785,

the price of a call option is 2.9312,

and the price of a put option is 7.4279.

a) Calculation of the value of d1 in the Black-Scholes-Merton model:The formula to calculate the value of d1 is given by the expression:

d1=(ln⁡(S0/K)+(r+0.5σ2)T)/(σ√T)

Where,S0 is the current stock price,K is the strike price,r is the continuously compounded risk-free rate of return,σ is the annual volatility of the stock price, andT is the time to expiration of the option.

Using the above values, the value of d1 can be computed as:

d1=(ln⁡(85/90)+(0.06+0.5×0.18^2)×1)/(0.18×√1)=-0.0985

b) Calculation of the value of d2 in the Black-Scholes-Merton model:

The formula to calculate the value of d2 is given by the expression:d2=d1−σ√T

Using the above values, the value of d2 can be computed as:

d2=-0.0985−0.18×√1=-0.2785

c) Calculation of the price of a call option in the Black-Scholes-Merton model:

The formula to calculate the price of a call option is given by the expression:

C=S0N(d1)−Ke^(−rT)N(d2)

Where,C is the price of the call option,N(d) is the cumulative probability distribution function of the standard normal distribution evaluated at the value d.Using the above values, the price of a call option can be computed as:

C=85N(-0.0985)−90e^(−0.06×1)N(-0.2785)=2.9312

d) Calculation of the price of a put option in the Black-Scholes-Merton model:The formula to calculate the price of a put option is given by the expression:

P=Ke^(−rT)N(-d2)−S0N(-d1)

Where,P is the price of the put option, andN(d) is the cumulative probability distribution function of the standard normal distribution evaluated at the value d.

Using the above values, the price of a put option can be computed as:

P=90e^(−0.06×1)N(-(-0.2785))−85N(-(-0.0985))=7.4279

Therefore, the value of d1 is -0.0985, the value of d2 is -0.2785, the price of a call option is 2.9312, and the price of a put option is 7.4279.

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Answer:

<MON= 59°

Step-by-step explanation:

Look at the diagram what is <LON? It is the angle of the whole line. Remember when any three letters come the middle letter is the angle like here if it is asking MON it is requesting the angle O between the lines M and N. The angle between the line L and M is given so to find <MON we can simply minus <LOM with the total angle (<LON)

<MON= <LON - <LOM

= 142-83

<MON= 59°

The angle of intersecting chords theorem indicates that the value of x, obtained from the measure of the arc [tex]\widehat{KU}[/tex] is; x = 7

The angle of intersecting chords theorem states that the angle formed at the intersection of two chords is half the sum of the angles of the arcs intercepted by the two chords.

The angle of intersecting chords theorem indicates that we get;

The measure of the arc KU = 6·x + 8

122 = (1/2) × (6·x + 8 + 194)

Therefore; 6·x + 8 + 194 = 2 × 122 = 244

6·x = 244 - (194 + 8) = 42

x = 42/6 = 7

Therefore; m[tex]\widehat{KU}[/tex] = 6 × 7 + 8 = 50°

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The 9th term of the given sequence is 78732.

The given sequence is 12, 36, 108... is a geometric sequence with a common ratio of 3.To find the 9th term of the given sequence, we will use the formula for the nth term of a geometric sequence, which is given by:

aₙ = a₁rⁿ⁻¹

Here, a₁ = 12 and r = 3.

Therefore, the formula for the nth term becomes:

aₙ = 12(3)ⁿ⁻¹

Now, we need to find the 9th term of the sequence. Hence, n = 9. Substituting the values of a₁ and r, and n in the formula, we get:

a₉ = 12(3)⁹⁻¹= 12(3)⁸= 12(6561)= 78732

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Evaluating  this function at  f(3) = 5.

To evaluate the function f(x) = (x + 2)/(4 - x) at x = 3, we substitute x = 3 into the function:

f(3) = (3 + 2)/(4 - 3)

    = 5/1

    = 5

Therefore, f(3) = 5.

Evaluating  this function at  f(3) = 5.

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Answer:

3.

Step-by-step explanation:

The answer is 3, it says "THE HEIGHT OF ORNAMENT B IS 3 TIMES LARGER THAN THE HEIGHT OF ORNAMENT A.!!!"

The contractor has a maximum of 29 hours (rounded down) to complete the job while keeping the total cost under $2000.

To find the number of hours the contractor has for the job while keeping the total cost under $2000, we can use the given cost model equation: C = 43H + 750.

Since the owner wants the total cost to be under $2000, we can set up the inequality:

43H + 750 < 2000

Now, let's solve this inequality for H, the number of hours:

43H < 2000 - 750

43H < 1250

Dividing both sides of the inequality by 43:

H < 1250/43

To determine the maximum number of hours the contractor has for the job, we need to round down the result to the nearest whole number since the contractor cannot work a fraction of an hour.

Using a calculator, we find that 1250 divided by 43 is approximately 29.07. Rounding down to the nearest whole number, we get:

H < 29

Using the cost model equation C = 43H + 750, where C represents the total cost and H represents the number of hours, we set up the inequality 43H + 750 < 2000 to satisfy the owner's requirement of a total cost under $2000.

By solving the inequality and rounding down to the nearest whole number, we find that the contractor has a maximum of 29 hours to complete the job within the specified cost limit.

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Answer:

ok, here is your answer

Step-by-step explanation:

Using the formula for compound interest:

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

where:

A = the total amount

P = the principal amount

r = the annual interest rate (as a decimal)

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

t = the number of years

Given:

P = €250

r = 9% = 0.09

n = 1 (compounded annually)

t = 3 years

Now, substituting the given values in the formula:

A = 250 (1 + 0.09/1)^(1*3)

A = 250 (1 + 0.09)^3

A = 250 (1.09)^3

A = 250 (1.29503)

A = €323.76 (rounded to two decimal places)

Therefore, the balance in the account at the end of 3 years would be €323.76.

Willis's mistake is assuming that a constant difference in the y-values implies a constant rate of change, which is not necessarily true for non-linear functions.

While it is true that a linear function will always have a constant rate of change, the converse is not true. A non-linear function can also have a constant difference in the y-values over a certain interval, but the rate of change is not constant. This is because the rate of change of a non-linear function varies at different points along the curve.

In this case, Willis did not consider the possibility of a non-linear function with a constant difference in the y-values. Therefore, his conclusion that the function is linear based on the constant differences in the y-values is not necessarily correct. To determine whether the function is linear or non-linear, Willis should have examined the differences in the x-values as well, or plotted the points on a graph to see if they lie on a straight line.

The Algebraic expression for the costs of 4 pens and 6 rulers is:

Total cost = Cost of 4 pens + Cost of 6 rulersTotal cost = 4m + (6m - 30)Total cost = 10m - 30Hence, the algebraic expression for the costs of 4 pens and 6 rulers is 10m - 30.

Let the cost of a ruler be represented by r, then we have:

m Shillings is the cost of a pen and r Shillings is the cost of a ruler from the given information.Using the information, the cost of the ruler will be the cost of a pen minus 5.

This can be represented algebraically as:r = m - 5

To find the cost of 4 pens and 6 rulers, we can substitute the value of r from the equation above into the expression for the cost of 4 pens and 6 rulers.Cost of 4 pens = 4mCost of 6 rulers = 6r

Substituting r from the equation above into the expression for the cost of 6 rulers, we have:Cost of 6 rulers = 6(m - 5)Simplifying the expression above, we get:Cost of 6 rulers = 6m - 30

Therefore, the algebraic expression for the costs of 4 pens and 6 rulers is:

Total cost = Cost of 4 pens + Cost of 6 rulersTotal cost = 4m + (6m - 30)Total cost = 10m - 30Hence, the algebraic expression for the costs of 4 pens and 6 rulers is 10m - 30.

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The correct option is that B. Selena is correct because she simplified correctly and then evaluated correctly after substituting the values for the variables.

Selena is correct. She simplified correctly and then evaluated correctly after substituting the values for the variables. Drake is incorrect because he should have simplified first, before substituting the values for the variables.

When Drake substituted the values for the variables first, he ended up with the expression (StartFraction negative 8 Over 4 Endfraction) Superscript negative 1. This is incorrect because he did not simplify the expression first. If he had simplified first, he would have gotten the expression (StartFraction negative 2 Over 1 Endfraction) Superscript negative 1, which is equal to one-half.

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The distance to cover on the next training day is 29/36 mile

Sarah trained longer by 1/4 miles

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

Total distance = 7 1/4 miles

Days = 9

So, we have

Distance = (7 1/4)/9

Evaluate

Distance = 29/36

Here, we have

The dot plot

The total distance here is

Total = 1/4 * 2 + 1/2 * 1 + 1 * 5

Total = 6

7 1/4 miles is greater than 6 miles, by 1/4 miles

Hence, Sarah trained longer

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Answer:

Step-by-step explanation:

Answer:

The graph representing the function f(x) = −|x − 2| − 1 is a downward-opening V-shaped graph that is shifted 2 units to the right and 1 unit downward from the standard absolute value function |x|.

Step-by-step explanation:

Here is a description of the graph and its key points:

The vertex of the graph is located at the point (2, -1), which represents the minimum point of the graph.

The graph approaches negative infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.

The graph is symmetric with respect to the vertical line x = 2.

Please note that I cannot display images directly, but you can search for "graph of f(x) = −|x − 2| − 1" on an online graphing tool or use a graphing calculator to visualize the specific graph.

When we solve for theta, the result would be B. 20°

To solve for theta, we would first represent the expression in the following way;

sinθ = cos(θ + 50)

sinθ =sin 90° - (50 + θ) cos (90 - θ)

sinθ = 90° - 50° - θ

Collect like terms

2θ =  90° - 50°

2θ = 40°

Divide both sides by 2

θ = 20°

Therefore the solution to theta is 20°

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Answer:

B)  20°

Step-by-step explanation:

Given equation:

[tex]\sin \theta = \cos (\theta + 50^{\circ})[/tex]

To solve the given equation for θ, we can use a co-function identity.

Co-function identities are a set of trigonometric identities that relate the values of complementary angles.

[tex]\boxed{\begin{minipage}{5cm}\underline{Co-function Identities}\\\\$\sin \theta = \cos (90^{\circ}-\theta)$\\\\$\cos\theta = \sin(90^{\circ}-\theta)$\\\\\sin \theta=\cos\theta$,\;if\;$A+B=90^{\circ}$\\\end{minipage}}[/tex]

Using the cofunction identity sin θ = cos (90° - θ), we can say that:

[tex]\cos (90^{\circ} -\theta)=\cos (\theta + 50^{\circ})[/tex]

Therefore:

[tex]\begin{aligned} \theta + 50^{\circ} &= 90^{\circ} -\theta\\\theta + 50^{\circ} +\theta &= 90^{\circ} -\theta+\theta \\2 \theta+ 50^{\circ} -50^{\circ}&= 90^{\circ}\\2 \theta+ 50^{\circ} &= 90^{\circ} -50^{\circ}\\ 2 \theta&=40^{\circ}\\\theta&=20^{\circ}\end{aligned}[/tex]

Therefore, the value of θ is 20°.

The probability of choosing a different type of animal as the second toy is $\frac{7}{18}.

The problem is to draw a tree diagram to determine the probability of choosing a different type of toy animal when two toys are randomly selected. Imogen has two toy elephants and seven toy bears in a box.

It should be noted that a tree diagram is a visual tool that can be used to show the possible outcomes of a particular event. Each branch represents a possible outcome and the probabilities associated with each branch are assigned in the diagram. The steps involved in solving the problem are:

Step 1: Construct the tree diagram.

Step 2: Calculate the probability of choosing a different type of animal as the second toy. Step 3: Simplify the fraction. Solution:

Step 1: Construct the tree diagram. The tree diagram for the given problem is shown below. [asy] size(200); defaultpen(linewidth(0.7));   draw((0,0)--(2,-2),MidArcArrow(size=10)); draw((0,0)--(2,2),MidArcArrow(size=10)); draw((2,2)--(4,2),MidArcArrow(size=10)); draw((2,-2)--(4,-2),MidArcArrow(size=10)); draw((2,2)--(4,0),MidArcArrow(size=10)); draw((2,-2)--(4,0),MidArcArrow(size=10)); draw((4,2)--(6,2),MidArcArrow(size=10)); draw((4,0)--(6,0),MidArcArrow(size=10)); draw((4,-2)--(6,-2),MidArcArrow(size=10));  label("Elephant",(-1,0)); label("Bear",(-1,2)); label("Bear",(3,2)); label("Bear",(3,0)); label("Bear",(3,-2)); label("Elephant",(3,0)); label("Bear",(5,2)); label("Elephant",(5,0)); label("Bear",(5,-2));  [/asy] Step 2: Calculate the probability of choosing a different type of animal as the second toy. The total number of outcomes is 9, as there are 2 elephants and 7 bears. There are four possible ways in which Imogen can pick two different types of animal:

Elephant followed by bear, Bear followed by elephant, Elephant followed by elephant, and Bear followed by bear. The probability of choosing a different type of animal as the second toy is the sum of the probabilities of the first two outcomes, which is: $P(\text{different animal}) = \frac{2}{9}\times\frac{7}{8}+\frac{7}{9}\times\frac{2}{8}$ $=\frac{14}{72}+\frac{14}{72}$ $=\frac{28}{72}$

Step 3: Simplify the fraction. The fraction can be simplified by dividing the numerator and denominator by the highest common factor. The highest common factor of 28 and 72 is 4. Hence, $\frac{28}{72} = \frac{7}{18}$

Therefore, the probability of choosing a different type of animal as the second toy is $\frac{7}{18}$.

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The probability that the second toy Imogen chooses will be a different type of animal to the first toy is 7/18. This is calculated by determining the probability of picking an elephant then a bear, and the probability of picking a bear then an elephant, and adding these two probabilities.

Firstly, let's define the event: Picking an elephant (E) and picking a bear (B). At the beginning, Imogen has 2 elephants and 7 bears in the box, making a total of 9 toys.

When Imogen picks the first toy, the probabilities are: P(E) = 2/9 and P(B) = 7/9. Then, Imogen picks the second toy, not replacing the first one.

So, we have two scenarios for the second pick: Given that the first pick was an elephant, the probabilities for the second pick are: P(E) = 1/8 (because one elephant left) and P(B) = 7/8 (there are still 7 bears). If the first pick was a bear, the probabilities for the second pick are: P(E) = 2/8 (still 2 elephants) and P(B) = 6/8 (one bear left).

Now, we're interested in the probability of picking two different types of animal toys. That will be the sum of the probabilities of picking an elephant then a bear, and the probability of picking a bear then an elephant. So it's (P(E) * P(B|E)) + (P(B) * P(E|B)) = (2/9 * 7/8) + (7/9 * 2/8) = 14/72 + 14/72 = 28/72 which reduces to 7/18.

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