The temperature in Madadeni one morning was -5°C at 08:00 and increased by 2°C every hour until 12:00. What will the temperature be at 11:30​

The two sides with lengths of 7 and 5 will never meet, which shows that the sum of the lengths of the two sides of a triangle must be equal to the length of the third side.

In Euclidean geometry, the Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than or equal (≥) to the third side of the triangle.

Mathematically, the Triangle Inequality Theorem is represented by this mathematical expression:

b - c < n < b + c

Where:

n, b, and c represent the side lengths of this triangle.

b - c < n < b + c

7 - 5 < n < 7 + 5

2 < n < 12

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Answer:

[tex]\theta = 60^\circ \;and\; \theta = 300^\circ[/tex]

Step-by-step explanation:

Given

[tex]\sin \left(2\theta\right)=\sin \left(\theta\right)[/tex]

Use the identity: [tex]\sin(2\theta) = 2 \sin(\theta)\cos(\theta)[/tex]

=> [tex]2 \sin(\theta)\cos(\theta) = \sin(\theta)[/tex]

Divide both sides by [tex]\sin(\theta)[/tex]

=> [tex]2 \cos(\theta) = 1[/tex]

[tex]= > \cos(\theta) = \dfrac{1}{2}[/tex]

[tex]= > \,\theta=\cos^{-1}\left(\dfrac{1}{2}\right)[/tex]

[tex]\cos^{-1}\left(\dfrac{1}{2}\right) \;is\: \dfrac{\pi}{3}} \text{ in the first quadrant and $\dfrac{5\pi}{3}$ in the fourth quadrant}}[/tex]

In degrees this corresponds to
[tex]\dfrac{\pi}{3} = \dfrac{\pi}{3} \times \dfrac{180^\circ}{\pi} = 60^\circ\\\\and\\\\\dfrac{5\pi}{3} = \dfrac{5\pi}{3} \times \dfrac{180^\circ}{\pi} = 300^\circ\\[/tex]

Answer
[tex]\theta = 60^\circ \;and\; \theta = 300^\circ[/tex]

The probability that the sample proportion is between 0.26 and 0.44 is 0.9998, or 99.98%.

The probability that the sample proportion is between 0.26 and 0.44 can be found using the normal distribution formula.

First, we need to find the mean and standard deviation of the sample proportion. The mean of the sample proportion is equal to the population proportion, which is 0.37. The standard deviation of the sample proportion can be found using the formula:

σ = √(p(1-p)/n)

Where p is the population proportion, and n is the sample size. Plugging in the given values, we get:

σ = √(0.37(1-0.37)/245) = 0.0196

Next, we need to find the z-scores for the given sample proportions. The z-score can be found using the formula:

z = (x - μ)/σ

Where x is the sample proportion, μ is the mean of the sample proportion, and σ is the standard deviation of the sample proportion. Plugging in the values for the lower bound of the sample proportion (0.26), we get:

z = (0.26 - 0.37)/0.0196 = -5.61

Similarly, for the upper bound of the sample proportion (0.44), we get:

z = (0.44 - 0.37)/0.0196 = 3.57

Now, we can use the standard normal table to find the probabilities corresponding to these z-scores. The probability for z = -5.61 is 0, and the probability for z = 3.57 is 0.9998.

Finally, to find the probability that the sample proportion is between 0.26 and 0.44, we subtract the lower probability from the upper probability:

P(0.26 < p < 0.44) = 0.9998 - 0 = 0.9998

Therefore, the probability that the sample proportion is between 0.26 and 0.44 is 0.9998, or 99.98%.

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The final answer is 3

The calculation you are looking to solve is 2.2 + (3)/(5). To solve this, we first need to convert the improper number 2.2 to a fraction. We can do this by multiplying the whole number by the denominator of the fraction and then adding the numerator. In this case, we would multiply 2 by 5 and then add 2, giving us 12/5. Now, we can add this fraction to the other fraction:

12/5 + 3/5 = 15/5

Next, we can simplify the fraction by dividing both the numerator and denominator by the greatest common factor. In this case, the greatest common factor is 5, so we can divide both the numerator and denominator by 5:

15/5 = 3

Therefore, the final answer is 3.

In summary, 2.2 + (3)/(5) = 3.

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Step-by-step explanation:

Lets find the slope of the line first so we can write the equation.

Counting the slope, we can see the slope of the line is [tex]\frac{3}{1}[/tex] or 3, so we have to write the equation of the line in slope intercept form [tex]y=mx+b[/tex] where m is the slope and b is the y intercept.

We know that the y intercept is -4 by looking at the graph, so we simply plug in our slope and y intercept.

[tex]y=3x-4[/tex]

To tell if equations are parallel or perpendicular:

Parallel:  The slope is the same

Perpendicular: The slope is the opposite reciprocal

Lets look at the equations and see if there parallel:

1. [tex]y=-3x+10[/tex] is neither.

2. The equation of the line is in point slope form, however we are already given the slope in the equation. The slope is [tex]-\frac{1}{3}[/tex], which is the opposite reciprocal of 3, therefore it is perpendicular.

3. [tex]\frac{1}{3}[/tex] is not the opposite reciprocal of 3, it is neither.

4. The equation of the line is in standard form, which means we must solve for y to get it in slope intercept form

[tex]-3x+y=1[/tex]
Subtract -3x on both sides

[tex]y=1-(-3x)[/tex]

Simplify

[tex]y=3x+1[/tex]

The equation has the same slope, so it is parallel.

The the distance between the two stations is 300km, then the speed of first motorcyclist is 63 km/h and speed of second-motorcyclist is 70 km/h.

The distance between the "two-stations" is given to be 300 km;

Let the speed of first-motorcyclists be x km/h

and let the speed of second-motorcyclists be (x + 7) km/h,

So, the Distance covered by first motorcyclist after 2 hours is = 2x km

and distance covered by second motorcyclist after 2 hours is = 2(x+7) km

⇒ 2x + 14 km,

So, the distance not covered by them after 2 hours is = 300 - (2x+2x+14) km.

The distance between the motorcyclist after 2 hours is 34 km,

Which means ,

⇒ 300-(4x+14) = 34

⇒ 300 - 4x - 14 = 34

⇒ 4x = 300 - 48

⇒ x = 63,

So, Speed of first-motorcyclists is 63 km/h, and

Speed of second-motorcyclist is = (63 + 7) = 70 km/h.

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The given question is incomplete, the complete question is

The distance between two stations is 300 km. Two motorcyclists start simultaneously from these stations and move towards each other. The speed of one of them is 7 km/h more than that of the other. If the distance between them after 2 hours of their start is 34 km, find the speed of each motorcyclist.

No, this relation does not define y as a function of x.

A function is a relation in which each input (x-value) is paired with exactly one output (y-value). In this relation, the x-value of 2 is paired with two different y-values (3 and -3), which violates the definition of a function.

Therefore, this relation does not define y as a function of x.

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To find the number of coins the statement made by Elena is correct.

The exponent of a number indicates how many times a number has been multiplied by itself. For instance, 34 indicates that we have multiplied 3 four times. Its full form is 3 3 3 3. Exponent is another name for a number's power. It might be an integer, a fraction, a negative integer, or a decimal.

Elena is on point. The formula 228 = 28 x 2 = 56 doesn't make sense in this situation since it suggests that they only counted for 28 days and received 228 coins, when the problem states that they counted for 28 days and received 228 coins. The fact that they counted for 28 days and came up with a total of 28 times 28 coins, or 784, makes the equation 228 = 28 x 28 make sense. Elena is accurate as a result.

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The correct answer is C. (256m^(4)n^(8)p^(4))/(81).

To simplify ((4m^(2)n^(2)p^(2))/(3mp))^(4), we need to first apply the power of 4 to each term inside the parentheses. This gives us:

(4^(4)m^(8)n^(8)p^(8))/(3^(4)m^(4)p^(4))

Next, we can simplify the terms with the same base by subtracting the exponents. This gives us:

(256m^(4)n^(8)p^(4))/(81)

Therefore, the correct answer is C. (256m^(4)n^(8)p^(4))/(81).

It is important to note that we assumed that the denominator does not equal zero, as dividing by zero is undefined.

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(a) By help of induction, it is proved the sequence is monotonically increasing for all n ≥ 1.

(b) The sequence is bounded above by 3 for all n ≥ 1.

(c)  Applying the Monotonic Sequence Theorem, it is proved that the limit of the sequence exists.

(d) limn→[infinity] an is 2.

(e) The series X[infinity] n=1 an is convergent

(a) To show that the sequence is monotonically increasing, we can use induction. Let's first consider the base case, n = 1. We have a1 = √2 and a2 = √2 + a1 = √2 + √2 > a1, so the sequence is increasing for n = 1. Now, let's assume that the sequence is increasing for n = k, so ak+1 > ak. Then, for n = k+1, we have ak+2 = √2 + ak+1 > √2 + ak = ak+1, so the sequence is also increasing for n = k+1. Therefore, by induction, the sequence is monotonically increasing for all n ≥ 1.

(b) To show that the sequence is bounded above by 3, we can also use induction. Let's first consider the base case, n = 1. We have a1 = √2 < 3, so the sequence is bounded above by 3 for n = 1. Now, let's assume that the sequence is bounded above by 3 for n = k, so ak < 3. Then, for n = k+1, we have ak+1 = √2 + ak < √2 + 3 = 3.2 < 3, so the sequence is also bounded above by 3 for n = k+1. Therefore, by induction, the sequence is bounded above by 3 for all n ≥ 1.

(c) By the Monotonic Sequence Theorem, if a sequence is both monotonically increasing and bounded above, then the limit of the sequence exists. Since we have shown that the sequence is monotonically increasing in part (a) and bounded above by 3 in part (b), we can conclude that the limit of the sequence exists.

(d) To find the limit of the sequence, we can use the fact that an+1 = √2 + an for all n ≥ 1. Taking the limit of both sides as n approaches infinity, we get limn→∞ an+1 = limn→∞ √2 + an. Since the limit of the sequence exists, we can write this as L = √2 + L, where L is the limit of the sequence. Solving for L, we get L = 2, so the limit of the sequence is 2.

(e) To determine whether the series X∞ n=1 an is convergent, we can use the fact that the limit of the sequence is 2. Since the sequence converges to 2, the terms of the sequence are getting closer and closer to 2 as n approaches infinity. This means that the terms of the series are getting smaller and smaller, and the series is convergent.

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The vector x = [2, -3, 1] is a basis for Row(A)⊥. This means that any vector in Row(A)⊥ can be written as a multiple of x.

To find a basis for Row(A)⊥ using the dot product, we need to find a vector that is orthogonal to all the rows of A. This means that the dot product of the vector and each row of A should be equal to 0.

Let's say the vector we are looking for is x = [x1, x2, x3]. Then we need to solve the following system of equations:

x1 * 1 + x2 * 4 + x3 * 2 = 0
x1 * 2 + x2 * 2 + x3 * 1 = 0
x1 * 3 + x2 * 1 + x3 * 1 = 0

We can write this system of equations in matrix form as:

[1 4 2] [x1] = [0]
[2 2 1] [x2] = [0]
[3 1 1] [x3] = [0]

We can use Gaussian elimination to solve this system of equations. After performing the necessary row operations, we get:

[1 0 -2] [x1] = [0]
[0 1  3] [x2] = [0]
[0 0  0] [x3] = [0]

From the last equation, we can see that x3 can be any value. Let's choose x3 = 1. Then, from the second equation, we get x2 = -3, and from the first equation, we get x1 = 2.

So, the vector x = [2, -3, 1] is a basis for Row(A)⊥. This means that any vector in Row(A)⊥ can be written as a multiple of x.

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Answer:

48

Step-by-step explanation:

We know

1/6 times a number is the same as 8. Let's x be the unknown number we have the equation

1/6x = 8

8 divided by 1/6

8 ÷ 1/6 = 8 × 6 = 48

So, the answer is 48

50 pizzas can be sold in 10 hours, according to this equation p = 5h.

In mathematics, an equation is a statement that asserts the equality of two expressions. An equation typically consists of two expressions separated by an equal sign (=). The expression on the left side of the equal sign is usually called the "left-hand side" (LHS) of the equation, while the expression on the right side of the equal sign is called the "right-hand side" (RHS) of the equation.

Let's use the variable p to represent the number of pizzas sold, and the variable h to represent the number of hours it takes to sell those pizzas. We can use the formula for finding the rate of sales (also known as the unit rate) to write an equation that represents the situation:

rate = amount of sales ÷ time

In this case, we know that 35 pizzas are sold in 7 hours. So, the rate of sales is:

rate = 35 pizzas ÷ 7 hours

Simplifying this expression, we get:

rate = 5 pizzas per hour

Therefore, the equation that represents the situation is:

p = 5h

This equation tells us that the number of pizzas sold (p) is equal to 5 times the number of hours it takes to sell those pizzas (h). For example, if we want to know how many pizzas can be sold in 10 hours, we can plug in h = 10 and solve for p:

p = 5h

p = 5(10)

p = 50

Therefore, 50 pizzas can be sold in 10 hours, according to this equation p = 5h

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We can determine that TC owns one cow using exponential calculations.

Exponent-based equations are those in which the exponent, or a portion of the exponent, is a variable.

For illustration, [tex]3^{x}[/tex] = 81.

In the question given,

TC is drinking no. of cups = 17

Let no. of cows TC has = x

Now according to the question, we can for the equation as:

[tex](\sqrt[5]{x} )^{2}[/tex] = 17 -1

⇒ x = [tex]\sqrt[5]{4}[/tex]

⇒ x = 1.319

⇒ x ≈ 1

Hence, TC owns 1 cow.

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The value of x to the nearest degree is 30°.

What is Triangle ?

A triangle is one that has three sides, three angles, and whose total angles is always 180 degrees.

Three line segments are linked end to end to make a triangle, a two-dimensional geometric structure with three angles. These line segments are known as sides, and their intersections are known as vertices.

The measurements of a triangle's sides and angles are used to categorize it.

We can use trigonometric ratios to get the value of x in the right triangle given:

sin(x) = opposite/hypotenuse

[tex]sin(x) = 8/17x = sin^{-1(8/17)}x = 29.74^{o} $ (approx.) $[/tex]

Therefore, the value of x to the nearest degree is 30°.

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The exponential function that represents a vertical compression by a factor of 2 of [tex]f(x) = 2^x[/tex] is given as follows:

[tex]g(x) = 0.5(2)^x[/tex]

The standard definition of an exponential function is given as follows:

[tex]y = a(b)^x[/tex]

In which:

When a function is vertically compressed by a factor of a, we have that the output of the function is multiplied by 1/a = divided by a, hence, considering the factor of 2, the function g(x) is given as follows:

g(x) = 1/2 x f(x)

[tex]g(x) = 0.5(2)^x[/tex]

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Answer:

Step-by-step explanation:

Let's assume that the number of child tickets sold is x and the number of adult tickets sold is y.

From the problem statement, we know that:

The total number of tickets sold is 412, so x + y = 412.

The total amount of money collected from selling these tickets is 2,725.50, so 3x + 7.50y = 2,725.50.

We can use these two equations to solve for x and y. One way to do this is to use substitution:

Solve the first equation for x: x = 412 - y.

Substitute this expression for x into the second equation: 3(412 - y) + 7.50y = 2,725.50.

Simplify and solve for y: 1,236 - 3y + 7.50y = 2,725.50, so 4.50y = 1,489.50, and y = 330.

Use the first equation to find x: x = 412 - y, so x = 82.

Therefore, Kell High School sold 82 child tickets and 330 adult tickets.

Answer:

Step-by-step explanation:

A cereal box manufacturer changes the sizeof the box to increase the amount of cereal itcontains. The equations 12 + 7.6n and 6 + 8n,where n is the number of smaller boxes, areboth representative of the amount of cereal thatthe new larger box contains. How many smallerboxes equal the same amount of cereal in thelarger box?

The probability of fraudulent applications are:

The given parameters are

n = 25

p = 0.12

The individual probability can be calculated as

P(x) = C(n, x) * p^x * (1 - p)^(n - x)

So, we have

Probability the number of fraudulent applications in the sample is 0

P(0) = C(25, 0) * 0.12^0 * (1 - 0.12)^(25 - 0)

P(0) = 0.0410

Probability the number of fraudulent applications in the sample is 3

P(3) = C(25, 3) * 0.12^3 * (1 - 0.12)^(25 - 3)

P(3) = 0.2387

Probability the number of fraudulent applications in the sample is 3 or less

P(x ≤ 3) = P(0) + ... P(3)

Using the formula above, we have

P(x ≤ 3) = 0.4088

Probability the number of fraudulent applications in the sample is 5 or more

P(x ≥ 5) = P(5) + ... P(25)

Using the formula above, we have

P(x ≥ 5) = 0.1734

Probability the number of fraudulent applications in the sample is more than 3

P(x > 3) = 1 - P(x ≤ 3)

By substitution, we have

P(x > 3) = 1 - 0.4088

P(x > 3) = 0.5912

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The required sinusoidal function be,

⇒ y = 25 sin(2π/15 (x - 1 + 15/4 + 30nπ)) + 38

                 or

⇒ y = 25 sin(2π/15 (x - 1 - 15/4 - 30nπ)) + 38

Since we know that,

The general formula of a sinusoidal function is,

⇒ y = A sin(B(x - C)) + D,

where,

A is the amplitude

B is the frequency (and related to the period by T = 2π/B)

C is the phase shift (the horizontal displacement from the origin)

D is the vertical shift (the midline)

Using the given information,

Amplitude = 25, so A = 25.

Period = 15, so T = 15.

We know that,

T = 2π/B, so we can solve for B,

⇒ 15 = 2π/B

⇒  B = 2π/15

Midline is y = 38, so D = 38.

⇒ f(1) = 63,

so we can also use this to find the phase shift:

⇒ 63 = 25 sin(B(1-C)) + 38

⇒ 25 sin(B(1-C)) = 25

⇒ sin(B(1-C)) = 1

⇒ B(1-C) = π/2 + 2nπ or 3π/2 + 2nπ,

where n is an integer.

Substituting B and solving for C in each case, we get,

⇒ B(1-C) = π/2 + 2nπ 2π/15 (1 - C)

              = π/2 + 2nπ 1 - C

              = 15/4 + 30nπ C

              = 1 - 15/4 - 30nπ

⇒ B(1-C) = 3π/2 + 2nπ 2π/15 (1 - C)

              = 3π/2 + 2nπ 1 - C

              = 15/4 + 60nπ/2 C

              = 1 - 15/4 - 30nπ

So we have two possible functions are,

⇒ y = 25 sin(2π/15 (x - 1 + 15/4 + 30nπ)) + 38

                 or

⇒ y = 25 sin(2π/15 (x - 1 - 15/4 - 30nπ)) + 38

where n is any integer.

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The distance between the ship and the base of the is approximately 359.32 feet.

The distance between the ship and the base of the lighthouse can be found using the tangent of the angle of elevation.

The tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the lighthouse (160 feet) and the adjacent side is the distance between the ship and the base of the lighthouse (x).

So, we can set up the equation:

tan(24) = 160/x

To solve for x, we can cross multiply and then divide:

x * tan(24) = 160

x = 160/tan(24)

Using a calculator, we can find that tan(24) is approximately 0.4452.

So, x = 160/0.4452

x = 359.32 feet

Therefore, the distance between the ship and the base of the lighthouse is approximately 359.32 feet.

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The requried value of the expression [tex]\sum_{n=11}^{30}n-\sum_{n=1}^{10}n[/tex] is 355.

Simplification involves applying rules of arithmetic and algebra to remove unnecessary terms, factors, or operations from an expression.

Here,
To evaluate the expression:

[tex]\sum_{n=11}^{30}n-\sum_{n=1}^{10}n[/tex]

we can first simplify each summation separately and then subtract the second summation from the first.

[tex]\sum_{n=11}^{30}n[/tex]= 11 + 12 + 13 + ... + 29 + 30

We can use the formula for the sum of an arithmetic series to simplify this expression:

S = (n/2)(a + l)

In this case, a = 11, l = 30, and n = 20 (since we're summing 20 terms).

So, we have:

S = (20/2)(11 + 30)

S= 410

Similarly,

[tex]\sum_{n=1}^{10}n[/tex] = 1 + 2 + 3 + ... + 9 + 10

S = 55

Finally, we can subtract the second summation from the first:

= 410 - 55 = 355

Therefore, the value of the expression is 355.

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Therefore , the solution of the given problem of slope comes out to be slope-intercept equation y = 2x + 1.

The y-intersection axis's with the slope of the line marks the inflection point in arithmetic where the y-axis intersects a line or curve. Y = mx+c, where m stands for the slope and c for the y-intercept, is the equation for the long line. The y-intercept (b) and slope (m) of the line are emphasised in the equation intercept form. An solution with the intersecting form (y=mx+b) has m and b as the slope and y-intercept, respectively.

Here,

Y = mx + b, where m is the line's slope and b is the y-intercept, is the slope-intercept version of a linear equation. Given two points (x1, y1) and (x2, y2), we can use the following method to determine the slope of the line:

=> m = (y2 - y1) / (x2 - x1) (x2 - x1)

For instance, if the two locations (2, 5) and (4, 9) are provided, we can determine the slope as follows:

=> m = (9 - 5) / (4 - 2) = 2

The y-intercept can then be determined by using one of the locations and the slope. Let's use points 2 and 5:

=> y = mx + b

=> 5 = 2(2) + b

=> 5 = 4 + b

=> b = 1

As a result, the line going through the points (2, 5) and (4, 9) has the slope-intercept equation y = 2x + 1.

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Th expression for total price of the supplies is $ 12.91 .

Any mathematical statement with variables, numbers, and an arithmetic operation between them is called an expression or an algebraic expression. For instance, the expression 4m + 5 has the terms 4m and 5 as well as the variable m of the supplied expression, all of which are separated by the arithmetic sign +.

Anything that is variable, or without a fixed value, is a variable. Alphabetic characters like a, b, c, m, n, p, x, y, z, and so on are typically used to denote expression variables. By combining several variables and numbers, we can create a wide range of expressions.

Given : price of nails, n = $1.29/b

           price of washers, w = $0.79/b

           price of bolts, b = $2.39/b

He bought 2 pounds of nails, 4 pounds of washers, and 3 pounds of bolts.

So, The total supplies will be :

= $1.29/b × 2 + $0.79/b × 4 + $2.39/b × 3

= 2.58 + 3.16 + 7.17

= $ 12.91

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For n≥2, the corresponding recursive function is f (1) =3f(n)=f(n1) +5.

In mathematics, a function is a rule that pairs each element from the domain with exactly one from the range or codomain of two sets.

In a recursive function, the output value at a certain input value is defined as a function of the output value at the previous input value. In this instance, we may use the definition to derive the recursive function from the explicit function:

f (n) = 5n - 2 f (n) = 5n - 3 f (n) = 5(2) - 8 f (n) = 5(3) - 13

The right response is: A. When n=2, f(1) = 2f(n)= f(n1) + 5.

As a result, the recursive function can be written as: f (1) =3f(n)=f(n1) +5 for n2.

Thus, for n≥2, the analogous recursive function is f (1) =3f(n)=f(n1) +5.

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a)18

b)0.023

1) The null mean for the disc width is assumed to be 18.
2) According to your null distribution, the probability of obtaining a sample estimate of 18.76 cm or more extreme is 0.023, and the p-value is 0.023.

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The number of sodas sold is 83 and the number of hot dogs sold is 52.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

Let's call the number of sodas sold "x".

According to the problem, the number of hot dogs sold is 31 less than the number of sodas sold, so we can write the number of hot dogs sold as "x - 31".

We also know that the combined total of sodas and hot dogs sold is 135, so we can write an equation:

x + (x - 31) = 135

Simplifying this equation:

2x - 31 = 135

Adding 31 to both sides:

2x = 166

Dividing both sides by 2:

x = 83

So the number of sodas sold is 83.

To find the number of hot dogs sold, we can use the equation we came up with earlier:

x - 31 = 83 - 31 = 52

So the number of hot dogs sold is 52.

Therefore, the number of sodas sold is 83 and the number of hot dogs sold is 52.

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The coordinates after the translation are (2, 6).

Here we start with the point (4, 3) and we want to apply the translation defined by (x - 2, y + 3)

This would be a translation of 2 units to the left and 3 units up, using a "coordinate-axis" notation.

So we just need to subtract 2 from the x-value and add 3 to the y-value, we will get the new coordinates:

(4 - 2, 3 + 3) = (2, 6)

These are the coordinates of point B after the translation, the correct option is the second one.

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Answer:

OB' (2,

Step-by-step explanation:

take your first point (B) (4,3) and plug it into the x and y in (x-2, y+3) so you get (4-2, 3+3) which will give you (2,

Answer:

The distributive property is used when finding the product of two polynomials by distributing one polynomial to each term of the other polynomial. For example, if we wanted to multiply (3x - 4)(2x + 5), we would use the distributive property by first multiplying 3x(-4) and 2x(5) and then adding the results together.

Polynomials are closed under multiplication, meaning that when two polynomials are multiplied together, the result is always another polynomial. This is true because a polynomial is a combination of constants and variables raised to non-negative integer powers, and when two polynomials are multiplied, the result is a combination of constants and variables raised to non-negative integer powers, which is a polynomial.