Find the missing side lengths, put answer as radical in simplest form
Look at picture for reference

Answer:

S₂₇ = 1188

Step-by-step explanation:

using the given formula for [tex]S_{n}[/tex] , that is

[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] (a₁ + [tex]a_{n}[/tex] )

with a₁ = 5 and [tex]a_{n}[/tex] = a₂₇ = 83 , then

S₂₇ = [tex]\frac{27}{2}[/tex] (5 + 83) = 13.5 × 88 = 1188

Answer:

Step-by-step explanation:

The compound statement "If there are penguins and the aquarium is full of fish, then this is an octopus" can be written in symbolic form as: r ∧ q → o

Let's assign variables to represent the statements:

q: The aquarium is full of fish.

r: There are penguins.

The compound statement "If there are penguins and the aquarium is full of fish, then this is an octopus" can be written in symbolic form as: r ∧ q → o

∧ represents the logical operator "and" which connects the statements r and q.

→ represents the logical operator "implies" or "if...then".

o represents the statement "this is an octopus".

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Answer:

AC = 3 + 2√15

Step-by-step explanation:

Call the center of the circle O.

(OB)² = 2² + 3²

(OB)² = 13

OD = OB

(OD)² + (DC)² = (CO)²

13 + 51 = (CO)²

(CO)² = 64

(CX)² + 2² = (CO)²

(CX)² = 64 - 4

(CX)² = 60

CX = √60 = 2√15

AX = BX = 3

AC = AX + CX

AC = 3 + 2√15

The total amount in the fund after 129 months will be approximately $59,392.90.

To calculate the total amount in the fund after 129 months, we need to use the formula for simple interest:

A = P(1 + rt)

Where:

A = Total amount (including interest)

P = Principal amount (initial investment)

r = Interest rate per period

t = Number of periods

Given:

P = $27,000

r = 11.15% per year (or 0.1115 as a decimal)

t = 129 months

First, we need to convert the interest rate from a yearly rate to a monthly rate since the time period is given in months. We divide the yearly rate by 12 to get the monthly rate:

monthly interest rate = 0.1115 / 12 = 0.0093 (rounded to four decimal places)

Now, we can plug in the values into the formula and calculate the total amount (A):

A = $27,000(1 + 0.0093 * 129)

A = $27,000(1 + 1.1997)

A = $27,000 * 2.1997

A ≈ $59,392.90

Therefore, the total amount in the fund after 129 months will be approximately $59,392.90.

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If the height of the cone were increased to 16 inches and the diameter stayed the same, the volume of the cone would increase by approximately 12π or around 37.7 cubic inches.

The formula for the volume V of a cone is given by:

V = 1/3 * π * r² * h

The height of an object is denoted as "h" and its base radius as "r", while the constant π is approximately equal to 3. 1416

Initially, it is crucial to determine the initial capacity of the triangular object.

The radius r is half the diameter, so r = 6/2 = 3 inches. Substituting the given height h = 12 inches and r = 3 inches into the formula, we get:

V1 = 1/3 * π * (3²) * 12 = 36π cubic inches

Then, calculate the new volume of the cone, using the same radius but the new height of 16 inches:

V2 = 1/3 * π * (3²) * 16 = 48π cubic inches

The increase in volume is the difference between the new volume and the original volume:

ΔV = V2 - V1 = 48π - 36π = 12π cubic inches

Therefore, if the height of the cone were increased to 16 inches and the diameter stayed the same, the volume of the cone would increase by approximately 12π or around 37.7 cubic inches.

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The Complete Question

Suppose you have a cone with a height of 12 inches and a base diameter of 6 inches. If the height of the cone were increased to 16 inches and the diameter stayed the same, by approximately how much would its volume increase?

The ratio of the side length of the side opposite any 30° angle to the length of the hypotenuse is 1/2.

A right triangle has one angle measuring 90°. This angle is also known as the right angle. The side opposite to the right angle is known as the hypotenuse. The other two sides of a right triangle are known as the adjacent side and the opposite side.

The adjacent side is the side adjacent to the angle that is not the right angle, while the opposite side is the side opposite the angle that is not the right angle. A ratio is a comparison of two quantities. In a right triangle, some ratios are of great importance. These ratios are known as trigonometric ratios. The three important trigonometric ratios are sine, cosine, and tangent.

The sine of an angle is the ratio of the opposite side to the hypotenuse. The cosine of an angle is the ratio of the adjacent side to the hypotenuse. The tangent of an angle is the ratio of the opposite side to the adjacent side.  Changing the scale factor of a right triangle results in the creation of similar right triangles. Two triangles are similar if they have the same shape but different sizes.  

When the scale factor is 1, the ratio of the side length of the side opposite angle A to the length of the hypotenuse is 1/2. Change the scale factor to 3. The ratio of the side length of the side opposite angle A to the length of the hypotenuse is still 1/2.What is the ratio of the side length of the side opposite any 30° angle and the length of the hypotenuse?

Therefore, the ratio of the side length of the side opposite any 30° angle to the length of the hypotenuse is 1/2.

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De acuerdo con la información, podemos inferir que un ejemplo de experimento aleatorio con orden y reemplazo: Lanzar un dado y registrar el resultado en cada lanzamiento.

Los experimentos aleatorios tienen diferentes formas de aplicación según su tipo, a continuación explicamos algunos:

De acuerdo con lo anterior, UN ejemplo de experimento aleatorio con orden y reemplazo: Lanzar un dado y registrar el resultado en cada lanzamiento.

Question in English

Give examples of randomized experiments with order, replacement, and no repetition.

Answer in English

Based on the information, we can infer that an example of a random experiment with order and replacement: Throw a die and record the result on each throw.

Random experiments have different forms of application depending on their type, some of which are explained below:

Order involve recording the results in sequence, while experiments. Orderless do not take into account the order in which the results occur. With replacement, items can be repeated in each selection, while in experiments. Without replacement, the selected items they are no longer available for subsequent selections.

Based on the above, ONE example of a random experiment with order and replacement: Roll a die and record the result on each roll.

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Answer:

[tex]f=\frac{N^2-H}{6}[/tex]

Step-by-step explanation:

[tex]N^2=6f+H\\\mathrm{or,\ }6f=N^2-H\\\mathrm{or,\ }f=\frac{N^2-H}{6}[/tex]

The equation of the midline of the graph is y = -2

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

The graph

Where we have

Max = 1

Min = -5

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

Midline, y = (Max + MIn)/2

substitute the known values in the above equation, so, we have the following representation

y = (1 - 5)/2

Evaluate

y = -2

Hence, the equation of the midline of the graph is y = -2

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Rounding to the nearest liter, the volume of the liquid in the pipe is approximately 0.01 liters. Therefore, none of the options A, B, C, or D provided is the correct answer.

To calculate the volume of the liquid in the cylindrical steel pipe, we need to find the difference in volume between the solid cylinder (hollow part) and the hollow cylinder.

Given:

Length of the cylindrical steel pipe (hollow part) = 21 cm

Radius of the solid cylinder = 0.4 cm

Radius of the hollow cylinder = 0.1 cm

First, let's calculate the volume of the solid cylinder (hollow part):

V1 = π × [tex]r1^2[/tex] × h

V1 = π × [tex](0.4 cm)^2[/tex] × 21 cm

Next, let's calculate the volume of the hollow cylinder:

V2 = π × [tex]r2^2[/tex] × h

V2 = π × [tex](0.1 cm)^2[/tex] × 21 cm

Now, we can find the volume of the liquid in the pipe by subtracting V2 from V1:

Volume of liquid = V1 - V2

Let's calculate these values:

V1 = π ×[tex](0.4 cm)^2[/tex] × 21 cm ≈ 10.572 cm³

V2 = π × [tex](0.1 cm)^2[/tex] × 21 cm ≈ 0.693 cm³

Volume of liquid = V1 - V2 ≈ 10.572 cm³ - 0.693 cm³ ≈ 9.879 cm³

To convert the volume from cubic centimeters (cm³) to liters (L), we divide by 1000:

Volume of liquid in liters ≈ 9.879 cm³ / 1000 ≈ 0.009879 L

Rounding to the nearest liter, the volume of the liquid in the pipe is approximately 0.01 liters.

Therefore, none of the options A, B, C, or D provided is the correct answer.

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In the class, Lucy's score is 6.

In statistics, the median is the middle value in a sorted, ascending or descending list of numbers. Thus, it represents the midpoint of the data.

The median is often compared with other descriptive statistics such as the mean (average), mode, and standard deviation.

Since Lucy is one of the 29 students in the class and her score was the same as the median score for her class.

Thus, the median score will be 15th score (i.e. the midpoint of 29). Using the frequency in of each score from the figure, let's keep adding the frequency until we reach 15 and we will then check the corresponding score at the 15th position:

2 + 2 + 4 + 7 = 15

Thus, the the corresponding score at the 15th position is 6.

Therefore, the median score is 6 and consequently Lucy's score is 6.

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Step-by-step explanation:

I'm pretty sure you just use the law of cosines in this case.

c= sqrt(a^2+b^2﹣2ab (cos γ))
where a is a Side, b is a Side, γ is Gamma or the angle. Sqrt means square root

In your case, a is 20, b is 13, and  γ is 93. Just plug it into the formula now :). I got 24.41751 ish.

627 x 26 equals 16,302 when solved using the standard algorithm.

The steps below can be used to solve the multiplication issue 627 x 26 using the conventional algorithm:

627 and 26 should be written one above the other, with the larger number appearing above and the smaller number beneath.

Start by adding each digit of the top number to the ones digit of the bottom number (six). Write the answers below the line after multiplying 6 by 7 and then by 2.

1254 -- a 6 x 7 partial product

1254 -- a half 6 x 2 product

Repeat the process by moving the bottom number one position to the left after that. Multiply 2 by 7 and then by 2, and then write the results one space to the left of the line below the line.

1254 x 627

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The larger number is 35.Let's assume the two numbers are represented by the variables "x" and "y," where "x" is the smaller number.

Equation 1: x + y = 59 (The sum of the two numbers is 59)

Equation 2: x - y = 11 (The difference between the two numbers is 11, with the smaller number being y)

According to the given information, the sum of the two numbers is 59, which can be expressed as:

x + y = 59

The difference between the two numbers is 11, with "x" being the smaller number. This can be expressed as:

y - x = 11

We can now solve these two equations simultaneously to find the values of "x" and "y."

Rearranging the first equation to solve for "y" gives us:

y = 59 - x

Substituting this value of "y" into the second equation:

(59 - x) - x = 11

Simplifying the equation:

59 - 2x = 11

Subtracting 59 from both sides:

-2x = 11 - 59

-2x = -48

Dividing both sides by -2:

x = -48 / -2

x = 24

Therefore, the smaller number is 24. To find the larger number, we substitute the value of "x" into the first equation:

24 + y = 59

y = 59 - 24

y = 35.

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The equation in standard form for the circle passing through (5/2, -6) and centered at the origin is [tex]4x^2 + 4y^2 = 169.[/tex]

To write the equation in standard form for the circle passing through (5/2, -6) and centered at the origin, we'll use the general equation of a circle:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

where (h, k) represents the center of the circle and r represents the radius.

Given that the circle is centered at the origin (0, 0), we can substitute these values into the equation:

[tex](x - 0)^2 + (y - 0)^2 = r^2\\x^2 + y^2 = r^2[/tex]

Now, we need to determine the radius of the circle. Since the circle passes through the point (5/2, -6), we can find the distance between the origin (0, 0) and this point, which represents the radius.

Using the distance formula:

d = √([tex](x2 - x1)^2 + (y2 - y1)^2)[/tex]

Substituting the values of the point (5/2, -6) and the origin (0, 0):

r = √([tex](5/2 - 0)^2 + (-6 - 0)^2)[/tex]

r = √(([tex]5/2)^2 + (-6)^2)[/tex]

r = √(25/4 + 36)

r = √(25/4 + 144/4)

r = √(169/4)

r = 13/2

Now, we can substitute the value of the radius (r = 13/2) into the equation:

[tex]x^2 + y^2 = (13/2)^2\\x^2 + y^2 = 169/4[/tex]

Multiplying both sides of the equation by 4 to eliminate the fraction:

[tex]4(x^2 + y^2) = 4(169/4)\\4x^2 + 4y^2 = 169[/tex]

Therefore, the equation in standard form for the circle passing through (5/2, -6) and centered at the origin is:

[tex]4x^2 + 4y^2 = 169.[/tex]

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Answer:Yes

Step-by-step explanation:

By the communitive property, they are equal to each other.

The mean, mean deviation mode, median standard deviation, and coefficients of variation, CV, are as follows;

1. The mean deviation about the mean, median mode are;

2.653, 3.709, and 3.709, respectively

The coefficients of variation are;

0.364, 0.337, 0.337, respectively

The variance is about 9.462

The standard deviation is about 3.076

2. Section A has a higher relative dispersion than section B

3. City 2

4. a. Mean ≈ 43.85

b. Mode 40 - 54 inches

c. Median ≈ 45.16

The variance ≈ 125.8056

Standard deviation ≈ 11.216

Coefficient of variation ≈ 0.268

The coefficient of variation of a data quantifies or is a measure of the variability of the values in the set of data.

The mean for the data can be calculated as follows;

Mean  = (3×10 + 6×15 + 11×25 + 2×6 + 4×4 + 10×3 + 5×2 + 7×8 + 8×9 + 9×4)/(10+15+25+6+4+3+2+8+9+4) ≈ 7.29

The median is the 86/2 value = 43rd value

10 + 15 + 25 = 50

Therefore, the median is in the class with a frequency of 25, which is 11

The mode of the data, which is the value with the highest frequency in the data set is 11

The mean deviation from the mean is therefore;

(|3 - 7.29|×10 + |6 - 7.29|×15 + |11 - 7.29| ×25 + |2 - 7.29| ×6 + |4 - 7.29| ×4 + |10 - 7.29| ×3 + |5 - 7.29| ×2 + |7 - 7.29|×8 + |8 - 7.29| ×9 + |9 - 7.29| ×4)/(10+15+25+6+4+3+2+8+9+4) ≈ 2.653

The coefficient for the mean deviation about the mean is therefore; 2.653/7.29 ≈ 0.364

The mean deviation about the median is therefore;

(|3 - 11|×10 + |6 - 11|×15 + |11 - 11| ×25 + |2 - 11| ×6 + |4 - 11| ×4 + |10 - 11| ×3 + |5 - 11| ×2 + |7 - 11|×8 + |8 - 11| ×9 + |9 - 11| ×4)/(10+15+25+6+4+3+2+8+9+4)

The mean deviation about the median = 319/86 ≈ 3.709

The coefficient ≈ 3.709/11 ≈ 0.337

The median = The mean deviation about the mode ≈ 3.709

The coefficient is about 0.337

The variance = (10×(3 - 7.29)² + 15×(6 - 7.29)² + 25×(11 - 7.29)² + 6×(2 - 7.29)² + 4×(4 - 7.29)² + 3×(10 - 7.29)² + 2×(5 - 7.29)² + 8×(7 - 7.29)² + 9×(8 - 7.29)² + 4×(9 - 7.29)²)/(10+15+25+6+4+3+2+8+9+4) ≈ 9.462

The standard deviation, s ≈ √(9.462) ≈ 3.076

2. The coefficient of variation, CV, can be used to compare the variability of the datasets comprising of different scales as follows;

CV = Standard deviation/( The mean of the data)

Therefore; CV for section A = 23/79 ≈ 0.2911

CV for section B = 11/64 ≈ 0.172

The higher CV value for section A indicates that the scores of section A have a higher relative dispersion than the scores for the section B

3. The consistency values can be obtained by finding the standard deviation for the values in the data for each city, using a web based tool as follows;

City 1; Mean = 23

Variance = 50/4

Standard deviation, City 1 = (√(50))/2 = 5·√2/2

City 2; Mean = 21.8

Sample variance = 2.2

Standard deviation, City 2 = √(2.2) ≈ 1.483

City 3; Mean = 29.2

Sample variance = 18.7

Standard deviation, City 3 = √(18.7) ≈ 4.324

The standard deviation of city 2, which is the smallest compared tom the other cities, indicates that the City 2, has the most consistent temperature

4. The mean of the data obtained from the data using the midpoint of the data, is; Mean = ∑fx/∑f

Therefore; The mean obtained using an online tool is; 43.85

The mode is the data that occurs most frequently, which is the data with the value; 40 - 54 inches

The median is the value at the middle of the data set, which is the value with a frequency of 53, and in the interval 40 - 54 inches

The median is; 39.5 + (54.5 - 39.5) × (50 - 30)/53 ≈ 45.16

The standard deviation found for the grouped data, using an online tool is as follows;

Variance, σ² ≈125.8056

The standard deviation, σ ≈ √(125.8056) ≈ 11.216

The coefficient of variation is; 11.216/41.83 ≈ 0.268

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Answer:

y - 3 = 1(x - 6)  or y - 4 = 1(x - 7)

Step-by-step explanation:

to find the slope: m = (3-4) / (6-7) = -1 / -1 = 1

next, substitute the values into the equation: y2 - y1 = m(x2 - x1)

you can use any x or y value from the given.

y - 3 = 1(x - 6)

or

y - 4 = 1(x - 7)

when the scale factor is 1, we have:

(side length of side opposite ∠A) / (length of hypotenuse) = sin A

When the scale factor is 1, the ratio of the side length of the side opposite ∠A to the length of the hypotenuse remains the same as in the original triangle. In a right triangle, the side opposite ∠A is referred to as the "opposite" side, and the hypotenuse is the longest side.

The ratio of the side length of the side opposite ∠A to the length of the hypotenuse is commonly known as the sine of angle A (sin A). So, when the scale factor is 1, the ratio of the side length of the side opposite ∠A to the length of the hypotenuse is equal to sin A.

In mathematical terms, when the scale factor is 1, we have:

(side length of side opposite ∠A) / (length of hypotenuse) = sin A

It's important to note that this ratio holds true for any right triangle, regardless of its size or dimensions, as long as the angle A remains the same.

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The yearly installment that Thabo has to pay to the bank is c. R202.

We shall use the formula for calculating the amortized loan payment to estimate the yearly installment that Thabo has to pay to the bank:

Installment = (Loan amount (L) * Interest rate (I)) / (1 - (1 + Interest rate)^(-Number of years (N))):

In short: Installment = (L * I) / (1 - (1 + I)⁽⁻ⁿ⁾)

Given:

L = R700

I = 6% = 0.06

N = 4

Plugging the values into the formula, we have the yearly installment:

Installment = (700 * 0.06) / (1 - (1 + 0.06)⁽⁻⁴⁾)

= 42/(1 - (1 + 0.06)⁽⁻⁴⁾ )

= 42/ (1 - (1.06)⁽⁻⁴⁾ )

= 42/(1 - 0.792)

= 42 / 0.2079 ≈ 202.01

Installment ≈ 202.01

Therefore, the yearly installment that Thabo has to pay to the bank is  R202.

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Answer:

3

Step-by-step explanation:

tan(A) = y-coordinate / x-coordinate

tan(A) = -3 / -1

tan(A) = 3/1

tan(A) = 3

Therefore, the value of tan(A) is 3.

Por lo tanto, los primeros 10 términos de la sucesión son:

1/2, 5/6, 7/6, 3/2, 11/6, 13/6, 17/6, 19/6, 23/6, 25/6.

La sucesión que se describe tiene un primer término de 1/2 y aumenta constantemente en 1/3 en cada término subsiguiente. Podemos encontrar los primeros 10 términos de la sucesión calculando cada término de manera sucesiva.

El primer término es 1/2.

Para encontrar el segundo término, sumamos 1/3 al primer término:

1/2 + 1/3 = 3/6 + 2/6 = 5/6

El segundo término es 5/6.

Para encontrar el tercer término, sumamos 1/3 al segundo término:

5/6 + 1/3 = 10/12 + 4/12 = 14/12 = 7/6

El tercer término es 7/6.

Podemos continuar este proceso para encontrar los siguientes términos:

4to término: 7/6 + 1/3 = 14/12 + 4/12 = 18/12 = 3/2

5to término: 3/2 + 1/3 = 9/6 + 2/6 = 11/6

6to término: 11/6 + 1/3 = 22/12 + 4/12 = 26/12 = 13/6

7mo término: 13/6 + 1/3 = 26/12 + 8/12 = 34/12 = 17/6

8vo término: 17/6 + 1/3 = 34/12 + 4/12 = 38/12 = 19/6

9no término: 19/6 + 1/3 = 38/12 + 8/12 = 46/12 = 23/6

10mo término: 23/6 + 1/3 = 46/12 + 4/12 = 50/12 = 25/6

Por lo tanto, los primeros 10 términos de la sucesión son:

1/2, 5/6, 7/6, 3/2, 11/6, 13/6, 17/6, 19/6, 23/6, 25/6.

Si tienes más preguntas, ¡no dudes en hacerlas!

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Given statement solution is :-  Box-and-whisker plot, the box represents the interquartile range (IQR), which is the range between Q1 and Q3 (13 to 18). The line inside the box represents the median (15). The whiskers (lines extending from the box) represent the minimum and maximum values (12 and 20).

To construct a box-and-whisker plot using the given data, you first need to find the minimum, maximum, median, and quartiles. Here are the steps to create the box-and-whisker plot for the given data set:

Sort the data in ascending order:

12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 17, 18, 20

Find the median (middle value) of the data set. Since the data set has an odd number of values, the median will be the middle value:

Median = 15

Find the lower quartile (Q1), which is the median of the lower half of the data set.

Counting from the start, the lower half of the data set is:

12, 12, 13, 14, 14

Q1 = 13

Find the upper quartile (Q3), which is the median of the upper half of the data set.

Counting from the end, the upper half of the data set is:

17, 18, 20

Q3 = 18

Find the minimum and maximum values in the data set:

Minimum = 12

Maximum = 20

Now that we have the necessary values, we can construct the box-and-whisker plot:

lua

Copy code

       |                 |

 12  |      -------------|    

 13  |                 |

 14  |              ----  

 15  |           -------

 16  |      -------------|    

 17  |                 |

 18  |              ----  

 20  |                 |

       |_________________|

In this box-and-whisker plot, the box represents the interquartile range (IQR), which is the range between Q1 and Q3 (13 to 18). The line inside the box represents the median (15). The whiskers (lines extending from the box) represent the minimum and maximum values (12 and 20).

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The length of x is 3.42cm

To determine the value, we have to know the different trigonometric identities.

These identities are listed thus;

From the information shown in the diagram, we have that;

Opposite side = x

The hypotenuse side = 10

The angle = 20 degrees

Now, using the sine identity, we get;

sin 20 = x/10

cross multiply the values, we get;

x = 10(0.3420

Multiply the values, we have;

x = 3.42cm

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Answer:

[tex]6,10[/tex]

Step-by-step explanation:

Method 1:

[tex]\mathrm{Let\ two\ numbers\ be\ x\ and\ y\ such\ that:}\\\mathrm{x:y=3:5\ \ \ \ and\ \ \ \ x+y=16}\\\mathrm{or,\ \frac{x}{y}=\frac{3}{5}}\\\\\mathrm{or,\ 5x=3y..........(1)}\\\mathrm{Also\ we\ have}\\\mathrm{x+y=16}\\\mathrm{or,\ 5(x+y)=5(16)}\\\mathrm{or,\ 5x+5y=80}\\\mathrm{or,\ 3y+5y=80\ [From\ equation\ 1]}\\\mathrm{or,\ 8y=80}\\\mathrm{or,\ y=10}\\\mathrm{From\ equation\ 1,}\\\mathrm{5x=3y}\\\mathrm{or,\ 5x=3(10)=30}\\\mathrm{\therefore x=6}[/tex]

[tex]\mathrm{So,\ the\ two\ numbers\ are\ 6\ and\ 10.}[/tex]

Alternative method 1:

[tex]\mathrm{Let\ the\ two\ numbers\ be\ x\ and\ 16-x.}\\\mathrm{Then,\ we\ have}\\\mathrm{x:(16-x)=3:5}\\\\\mathrm{or,\ \frac{x}{16-x}=\frac{3}{5}}\\\\\mathrm{or,\ 5x=3(16-x)=48-3x}\\\mathrm{or,\ 5x+3x=48}\\\mathrm{or,\ 8x=48}\\\mathrm{\therefore x=6}\\\mathrm{So,\ the\ other\ number=16-x=16-6=10}[/tex]

[tex]\mathrm{So,\ the\ two\ numbers\ are\ 6\ and\ 10.}[/tex]

Alternative method 2:

[tex]\mathrm{Let\ the\ two\ numbers\ be\ 3x\ and\ 5x.}\\\mathrm{Then,}\\\mathrm{3x+5x=16}\\\mathrm{or,\ 8x=16}\\\mathrm{or,\ x=2}\\\mathrm{So,\ first\ number=3x=3(2)=6}\\\mathrm{Second\ number=5x=5(2)=10}[/tex]

[tex]\mathrm{So,\ the\ two\ numbers\ are\ 6\ and\ 10.}[/tex]

The horizontal distance needed for the ramp is 518.4 cm to ensure a slope of 5/32 and reach the entrance of the school, which is 81 cm above the ground.

To find the horizontal distance needed for the ramp, we can use the formula:

Horizontal distance = Vertical distance / Slope

Given that the slope must be 5/32 and the entrance to the school is 81 cm above the ground, we can substitute these values into the formula:

Horizontal distance = 81 cm / (5/32)

To divide by a fraction, we can multiply by its reciprocal:

Horizontal distance = 81 cm * (32/5)

Simplifying:

Horizontal distance = (81 cm * 32) / 5

Horizontal distance = 2592 cm / 5

Horizontal distance = 518.4 cm

Therefore, the horizontal distance needed for the ramp is 518.4 cm to ensure a slope of 5/32 and reach the entrance of the school, which is 81 cm above the ground.

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