5. Find tn for the geometric sequence where t3 = 24 and t9 = 1536. Why are there 2 answers? (4 marks)

Answer:

[tex]\mbox{\large $0.5 + 0.16 \bar {3}$ = \boxed{\dfrac{199}{300}}}[/tex]

Step-by-step explanation:

[tex]0.16\bar{3} = 0.16333333333\\[/tex]

the 3 repeats until infinity

Let's multiply this by 100 to move the non-repeating part to the left of the decimal point

0.1633333333...... x 100 = 16.333333333333333...

.333333 = 1/3

Therefore the recurring number multiplied by 100 is
[tex]16 + \dfrac{1}{3} = \dfrac{16 \times 3 + 1}{3}=\dfrac{49}{3}[/tex]

Since this fraction was arrived at by multiplying by 100, divide this fraction by 100 to get back the fraction in its original decimal value of 0.163333

[tex]\dfrac{49}{3} \div 100[/tex]

To divide by a number, multiply by the reciprocal of that number

Reciprocal of 100 is 1/100:

[tex]\dfrac{49}{3} \div 100 = \dfrac{49}{3} \times \dfrac{1}{100} = \dfrac{49}{300}[/tex]

(You can verify this by dividing 49 by 300 in a calculator and seeing that the result is 0.163333333333333333333 i.e. 0.16333 recurring

We want to add 0.5 to this.

0.5 = [tex]\dfrac{1}{2}[/tex]

so
0.5 + 0.16333333 = 0.6633333

[tex]= \dfrac{1}{2} + \dfrac{49}{300}[/tex]

We can write [tex]\dfrac{1}{2}[/tex] as [tex]\dfrac{150}{300}[/tex]  by multiplying numerator and denominator by 150

So we get

[tex]0.5 + 0.1633333 = \dfrac{150}{300} + \dfrac{49}{300} = \dfrac{150+49}{300} = \dfrac{199}{300}[/tex]

If you perform this division on a calculator you will get the answer as 0.663333 = [tex]0.66\bar{3}[/tex] which is what you get when you add 0.5 and [tex]0.16\bar{3}[/tex]

Given point D is the centroid of triangle ABC, BD = 4x + 2 and DF = 3x and the value of x is -1

Centroid is a term used in triangles and it is equals to the reciprocal of three-second of the distance from each midpoint of the given triangle to each vertices.

Given point D is the centroid of triangle ABC, BD = 4x + 2 and DF = 3x and Centroid is a term used in triangles and it is equals to the reciprocal of three-second of the distance from each midpoint of the given triangle to each vertices.

we know that,

BD = 4x+2

DF = 3x

So by the given formulae, we used to solve the x value as,

BD = (1/(3/2)) [DF]

BD = (2/3)[DF]

4x+2 = (2/3)(3x)

4x+2 = 2(x)

4x+2 = 2x

4x-2x = -2

2x = -2

x = -1

Therefore, the value of x is x = -1

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Answer:

-5x-96/12

Step-by-step explanation:

Firstly we want ot combine the multiplied terms into one fraction:

(1x/3-3)+(-3/4 x-5)

Then we multiply by 1:

(x/3-3)+(-3/4 x-5)

Find common denominator:

(x/3+3(-3)/3)+(-3/4 x-5)

Combine fractions with common denominator:

(x+3(-3)/3)+(-3/4 x-5)

Multiply the numbers:

(x-9/3)+(-3/4 x-5)

Combine multiplied terms into a single fraction:

x-9/3+(-3x/4 -5)

Find common denominator:

x-9/3+(-3x/4 + 4(-5)/4)

Combine fractions with common denominator:

x-9/3+(-3x + 4(-5)/4)

Multiply the numbers:

x-9/3 + (-3x-20/4)

Find common denominator:

4(x-9)/12 + 3(-3x-20)/20

Combine fractions with common denominator:

4(x-9)+3(-3x-20)/12

Distribute:

4x-36+3(-3x-20)/12

Distribute:

4x-36-9x-60/12

Subtract the numbers:

4x - 96 - 9x/12

Combine like terms:

-5x - 96/12

Answer: -5x-96/12

Answer:

Step-by-step explanation:

Greetings!!!

The given two numbers ratio.

[tex] \frac{5}{4} [/tex]

The smaller number is 32

[tex] \frac{5}{4} \times 32[/tex]

convert elements into fraction

[tex] \frac{5}{4} \times \frac{32}{1} [/tex]

Apply fraction rule

[tex] \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} [/tex]

Multiply the numbers

[tex] \frac{5 \times 32}{4 \times 1} [/tex]

Divide the numbers

[tex] \frac{32}{8} = 8 \\ 8 \times 5 = 40[/tex]

Hope it helps!!!

The values and graph of the equation 3x - 2y = 6 is attached below while the slope is 3/2

The equation of a straight line can be represented in various forms. One of the most commonly used forms is the point-slope form, which is represented as:

y - y1 = m (x - x1)

where (x1, y1) is a point on the line, and m is the slope of the line, which is the rate of change of y with respect to x.

Another common form of the equation of a line is the slope-intercept form, which is represented as:

y = mx + b

where m is the slope of the line, and b is the y-intercept, which is the point where the line crosses the y-axis.

In this problem, we need to write the table of values and the find the slope of the line.

3x - 2y = 6

rewriting this equation;

2y = 3x - 6

y = 3x/2 - 3

when y = 0

0 = 3x/2 - 3

3x/2 = 3

3x = 2 * 3

3x = 6

x = 2

when x = 0

y = 3(0)/2 - 3

y = 0 - 3

y = -3

The slope of the line can be calculated using the two points from above.

slope (m) = y₂ - y₁ / x₂ - x₁

m = -3 - 0 / 0 - 2

m = 3/2

The slope is 3/2

The graph of the equation is attached below

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The measures of the angles formed by the cords and secant in the circle are found using circle theorems and presented as follows;

A secant is a line that has two points of intersection on a circle.

The specified parameters are;

1. The measure of arc  [tex]m\widehat{AB}[/tex] = 80°, and the measure of [tex]m\widehat{CD}[/tex] = 15°

Angle at the center of the circle = 2 × Angle at the circumference

Angle at the center of the circle = The arc angle of the circle

Therefore;

m∠DAC = 15°/2 = 7.5°

m∠ACO = 80°/2 = 40°

The external angle of a triangle theorem, indicates that we get;

m∠1 = 40° + 7.5° = 47.5°

m∠1 = 47.5°

2. m∠BOD = 100°, [tex]m\widehat{AC}[/tex] = 90°

The intersecting chords theorem indicates that we get;

m∠BOD = (1/2)×([tex]m\widehat{AC}[/tex] + [tex]m\widehat{BD}[/tex])

Therefore; 100° = (1/2)×(90° + [tex]m\widehat{BD}[/tex])

[tex]m\widehat{BD}[/tex] = 2 × 100° - 90° = 110°

[tex]m\widehat{BD}[/tex] = 110°

3. [tex]m\widehat{AB}[/tex] = 90°, [tex]m\widehat{CD}[/tex] = 20°

Therefore; m∠ACO = 90°/2 = 45°

m∠DAC = 20°/2 = 10°

The external angle theorem indicates that we get;

m∠COD = 45° + 10° = 55°

m∠COD = 55°

4. m∠1 = 35°, [tex]m\widehat{AB}[/tex] = 50°, therefore;

m∠ACO = 50°/2 = 25°

m∠CAO = 35° - 25° = 10°

[tex]m\widehat{CD}[/tex] = 2 × m∠CAO

Therefore; [tex]m\widehat{CD}[/tex] = 2 × 10° = 20°

[tex]m\widehat{CD}[/tex] = 20°

5. m∠AOC = (1/2)×([tex]m\widehat{AC}[/tex] + [tex]m\widehat{BD}[/tex])

Therefore; m∠AOC = (1/2)×(90° + 110°) = 100°

m∠AOC = 100°

6. [tex]m\widehat{CD}[/tex] = 20° and [tex]m\widehat{AB}[/tex] = 70°

The external secant, tangent theorem, indicates that we get;

m∠APB = (1/2) × ([tex]m\widehat{AB}[/tex] - [tex]m\widehat{CD}[/tex])

m∠APB = (1/2) × (70° - 20°) = 25°

m∠APB = 25°

7. [tex]m\widehat{AB}[/tex] = 50°,

The angle at the center of a circle is twice the angle subtended at the circumference, therefore;

m∠ADB = 50°/2 = 25°

m∠ADB = 25°

8. m∠CAD = 45°

The angle formed by an arc at the center of a circle theorem, indicates that we get;

[tex]m\widehat{CD}[/tex] = 2 × m∠CAD

[tex]m\widehat{CD}[/tex] = 2 × 45° = 90°

[tex]m\widehat{CD}[/tex] = 90°

9. m∠ACB = 70°

m∠ACB is the angle subtended by the arc [tex]m\widehat{AB}[/tex] on the circumference

Therefore;

[tex]m\widehat{AB}[/tex] = 2 × m∠ACB

[tex]m\widehat{AB}[/tex] = 2 × 70° = 140°

[tex]m\widehat{AB}[/tex] = 70°

10. m∠AOB = 80°, [tex]m\widehat{AB}[/tex] = 60°

m∠(AOB) = (1/2) × ([tex]m\widehat{AB}[/tex] + [tex]m\widehat{CD}[/tex])

Therefore; m∠(AOB) = 80° =  (1/2) × (60 + [tex]m\widehat{CD}[/tex])

80° =  (1/2) × (60 + [tex]m\widehat{CD}[/tex])

[tex]m\widehat{CD}[/tex] = 2 × 80° - 60° = 100°

[tex]m\widehat{CD}[/tex] = 100°

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The complete question:

A baker uses the equation y = 12x to predict the revenue generated from selling a cakes. She uses equation y= 450 +4.5x to

model the cost to produce a cakes:

What does (60, 720), the solution of the system -450 +4.5x &

y=12x represent?

Solution:-

The solution of the system (-450 + 4.5x, y = 12x) at (60, 720) represents :

a. The point where the revenue generated from selling 60 cakes is 720 dollars

b. The cost to produce those 60 cakes is 720 dollars.

In geometry, A system of equations is a collection of two or more equations with the same set of unknown variables. In solving a system of equations, we try to calculate values for each of the unknowns that will satisfy each and every equation in the system.

The given system of equations is:

y = 12x (equation 1)

y = 450 + 4.5x (equation 2)

where x represents the number of cakes produced and sold, y represents the revenue generated from selling x cakes, and 450 + 4.5x represents the cost to produce x cakes.

To find the solution of the system, we can substitute equation 1 into equation 2, and solve for x as follows:

12x = 450 + 4.5x

7.5x = 450

x = 60

Substituting this value of x into equation 1 gives us:

y = 12(60)

y = 720

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After 38 hours, there will be 43.1 mg of the radioactive substance remaining.This means that the amount of a radioactive substance decreases exponentially over time

Radioactive decay follows an exponential decay curve. This means that the amount of a radioactive substance decreases exponentially over time. The equation for this is: [tex]A(t) = A0*e^(-λ*t\\[/tex])Where A(t) is the amount of substance at time t, A0 is the initial amount of substance, and λ is the decay constant. In this problem, A0 = 100 mg, t = 29 hours, and A(t) = 50 mg. We can use this to solve for λ:50 mg = 100 mg * [tex]e^(-λ*29)λ =[/tex] -0.027Now that we have the decay constant, we can calculate the amount of substance remaining after 38 hours:A(38) = 100 mg * [tex]e^(-0.027*38[/tex]).A(38) = 43.1 mg.Therefore, after 38 hours, there will be 43.1 mg of the radioactive substance remaining.

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Answer: The length is 9 units and the width is 1 unit

Step-by-step explanation:

We can use the formula for the area of a rectangle to solve this problem. The formula is A = l * w, where A is the area, l is the length, and w is the width.

We have A = 9, and we know that w = l - 8. Substituting this into the formula, we get:

9 = l * (l - 8)

We can rearrange this to solve for l:

9 = l^2 - 8l

Adding 8l to both sides of the equation, we get:

9 + 8l = l^2

Subtracting 9 from both sides, we get:

8l = l^2 - 9

We can factor this to get:

l(l - 9) = 0

Since the product of two numbers is zero only when one of them is zero, we can set each factor equal to zero and solve:

l = 0 or l - 9 = 0

Adding 9 to both sides of the second equation, we get:

l = 9

Therefore, the length of the rectangle is 9 units.

Answer:

Y < (1/3) x - 5

There are 56 servings in a 28 pound bag of dog food.

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

Given:

Each day Fei yen's dog eat dog food = 8 ounces

Fei yen bought a 28 pound bag of dog food.

And we know that 1 pound = 16 ounces

Then ,the food in ounces

= 28 x 16

= 448 ounces

Now, the number of 8 ounces servings are in a 28 pound bag of dog food

= 448 / 8

= 56

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Answer:

The correct answer is C. Statement: m∠3 + m∠5 = 180° and m∠4 + m∠6 = 180° Reason: definition of supplementary angles.

This statement uses the definition of supplementary angles, which states that two angles are supplementary if their measures add up to 180 degrees. Since ∠3 and ∠6 are corresponding angles, and ∠4 and ∠5 are also corresponding angles, their measures must add up to 180 degrees in order for the parallel lines to be cut by the transversal.

Answer:

Option, D. Statement: ∠7 ≅ ∠6 and ∠8 ≅ ∠5 Reason: Vertical Angles Theorem

Step-by-step explanation:

In this question that you have put:

7 ≅ ∠6 and ∠8 ≅ ∠5 Reason =

Vertical Angles Theorem ( Or in similar terms VAT )

Make sure that you remember the following:

vertical angles are the angles that are opposite each other when the two line " meet together "

Thus, your answer to your questions is.

D. Statement: ∠7 ≅ ∠6 and ∠8 ≅ ∠5 Reason: Vertical Angles Theorem

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By applying the slope of the function at that point, linear approximation provides an estimate of the value of a function close to a beginning point.

When estimating the value of a function at a nearby location, linear approximation uses the slope of the function at a particular position. In order to arrive at the starting position, it first calculates the difference between the two points, multiplies it by the slope, and adds it. This provides a rough estimate of the function's value at the new location. A technique for calculating a function's value close to a beginning point is called linear approximation. It is predicated on the notion that a linear function can be used to roughly approximate the function close to the point of interest. This indicates that the value of the function at surrounding places can be inferred from the slope of the function at the point in question.

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A statement which best describes the graph of y ≥ -x² + 8x - 2 include the following: B. The vertex is at (4, 14). The parabola is a solid line that opens down. Shading is above the parabola.

Generally speaking, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. For this quadratic function, we can logically deduce that the graph is a downward parabola because the coefficient of x² is negative and the value of "a" is less than zero (0).

By critically observing the graph of the given quadratic function, we can logically deduce that the solutions (roots) and vertex are as follows;

x = 0.258, y = 0.

x = 7.742, y = 0.

Vertex = (4, 14).

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first one is a

Step-by-step explanation:

The side of the triangle from shortest to longest can be arranged as,

A triangle is a polygon with three sides such that the sum of all the interior angles is always 180°.

In a triangle, the side opposite to the biggest angle is always the lengthiest, while the side opposite to the smallest angle is the shortest.
Given that the measure of ∠A is 100°, therefore, the length of the side BC will be the lengthiest.
Also, the measure of ∠B is 20°, therefore, the length of the side AC will be the shortest.

Hence, BC > AB > AC.

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Answer:

2.3333 inches

Step-by-step explanation:

To find the appropriate scale factor, we need to determine how many times smaller the piece of paper is compared to the dog. We can start by finding the ratio of the length of the paper to the length of the dog:

3 inches / 9 inches = 1/3

Similarly, the ratio of the width of the paper to the height of the dog is:

5 inches / 7 inches = 5/7

So, the smaller of these two ratios, 1/3, is the appropriate scale factor for Meredith to use in her drawing. This means that the length of the dog in the drawing will be 9 inches * 1/3 = 3 inches, and the height will be 7 inches * 1/3 = 2.3333 inches.

The expected number of aces in a poker hand given that the first card drawn is an ace is 0.235.

If the first card drawn is an ace, then there are only 3 aces left in the deck of 51 cards from which to draw the remaining 4 cards. Therefore, we can model the number of aces in the poker hand as a binomial random variable with parameters n = 4 (the number of cards after the first ace) and p = 3/51 (the probability of drawing an ace from the remaining deck).

The expected number of aces in a poker hand given that the first card drawn is an ace can be calculated as follows:

E(X) = np = 4 * 3/51 = 12/51 = 0.235

Therefore, the expected number of aces in a poker hand given that the first card drawn is an ace is approximately 0.235.

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____ The given question is incorrect, the correct question is given below:

A poker hand consists of five cards drawn from a standard 52-card deck. find the expected number of aces in a poker hand given that the first card drawn is an ace.

Answer:

nth term = n²-6

Step-by-step explanation:

we know our nth-term formula is going to be of the form an² + bn + c. We just have to find a, b, and c.

In this series,

-5, -2, 3, 10, 19....

a=1, b=0, c=-6

nth term = n²-6

if n=1 then,

nth term=1²-6

⇒nth term=-5

The expression that is equivalent to 5^6 is A.125 x 125

Index forms are defined as mathematical forms of writing numbers that are too large or small in more convenient forms.

Other terms for index forms are called standard forms or scientific notations.

They are also described as numbers or variables that are raised to an exponent.

From the information given, we have that;

5⁶

This is replicating 5 in 6 places, we have the representation as;

5 × 5 × 5 × 5 × 5 × 5

Hence, the expression is 125 x 125

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Answer: The two shaded parts of the wall have a combined area of 20m * 12m + 16m * 15m = 240 + 240 = 480 square meters.

Therefore, the volume of yellow paint required is 480 * 0.1 = 48 litres.

Step-by-step explanation:

The function f(x) = 6log₂x - 3 increasing at the greatest rate in the interval [1/8, 1/2].

The maxima and minima of a function are the extreme values within the range; in other words, they are the maximum and minimum values of a function, respectively, at a given position.

Let the function be f(x) = 6log₂x - 3

For the interval x ∈ [2, 6]

f(x) ∈ [-0.678, 3]

For the interval x ∈ [1/8, 1/2]

f(x) ∈ [ -9.96, -5.32]

For the interval x ∈ [1, 2]

f(x) ∈ [-3, -0.67]

For the interval x∈ [1/2, 1]

f(x) ∈ [-5.32, -3]

The function increased most rapidly in the interval [1/8, 1/2], as can be seen from the function value and its interval values.

Therefore, the range [1/8, 1/2] is where the function f(x) = 6log₂x - 3 increases at the fastest pace.

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Answer: 135

Step-by-step explanation:

Counting by 5 The last point at the end of the box and whisker plot is 135

Answer:

1)10abc^2+6a^2bc

2)6.6x^2yz-2.8xy^2z+0.5xyz^2

Step-by-step explanation:

Answer:

(24) + (20) + (8) = 52 ft^2

Step-by-step explanation:

This figure is a compound of 3 rectangles that can be split up like in my attached picture.

The top, horizontal rectangle is 8x3 = 24 sq ft.

The bottom, almost square rectangle is 4x5 = 20 sq ft.

The middle rectangle has a height of 4 feet and a width of (8-6) = 2 feet.

So, its area is 4x2= 8 sq ft.

The compound figure is the sum of all of these areas:

24 + 20 + 8 = 52

The area of the compound figure is 52 sq ft.

Area of 1 = 8 * 3 = 24

Area of 2 = 4 * 2 = 8

Area of 3 = 5 * 4 = 20

Total area = 24 + 8 + 20 = 52

Answer:

x + 3y = 9

Step-by-step explanation:

Answer: 37 points

Step-by-step explanation:

Answer:

Answer is Below ------v

1. clockwise

2. 3

1. (y, -x)

2. (3x,3y)

Ashton has 4.5 bags of pecans. since you can't have half a bag, he might need to round up to 5 bags

A fraction represents a part of a whole.

If Ashton used 1/4 of the pecans in a soup recipe, he has 3/4 of the pecans left.

To find out how many pounds of pecans he has left.

we can multiply the original amount of pecans by 3/4:

3/4 x 6 pounds = 4.5 pounds

So Ashton has 4.5 pounds of pecans left to split into bags.

If he splits the remaining pecans into 1-pound bags, then he has:

4.5 pounds ÷ 1 pound/bag = 4.5 bags

Therefore, Ashton has 4.5 bags of pecans. However, since you can't have half a bag, he might need to round up to 5 bags

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The complete question

Ashton picked 6 pounds of pecans. He used 1/3 of the pecans in a soup recipe. Ashton puts the pecans that are left in 1/4 pound bags. How many bags of pecans does he have