Real-life Problems Question 6

The first car consumed 15 gallons of gas

The second car consumed 30 gallons of gas during the week

Let's denote the number of gallons consumed by the first car as x, and the number of gallons consumed by the second car as y.

We know that the total gas consumption is 45 gallons, so:

x + y = 45

We also know that the total distance traveled is 1725 miles, and that the fuel efficiency (miles per gallon) of the first car is 35, and of the second car is 40. Using this information, we can write another equation:

35x + 40y = 1725

We now have two equations with two unknowns, which we can solve simultaneously to find x and y.

Multiplying the first equation by 35, we get:

35x + 35y = 1575

Subtracting this equation from the second equation, we get:

5y = 150

y = 30

Substituting y = 30 into the first equation, we get:

x + 30 = 45

x = 15

Therefore, the first car consumed 15 gallons of gas, and the second car consumed 30 gallons of gas during the week.

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a. Apples:  Lily needs 10 cups of apples for her fruit bowl.

b. Watermelons: Lily needs 13-3/4 cups of watermelons for her fruit bowl.

c. Grapes: The amount of grapes that Lily has is not given

What is arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed constant number, called the common difference, to the preceding term.

To make her fruit bowl, Lily needs 2½ times the amount of each fruit she currently has.

a. Apples:

2½ × 4 cups = 10 cups

Therefore, Lily needs 10 cups of apples for her fruit bowl.

b. Watermelons:

2½ × 5-½ cups = (2 × 5 + 1/2) × 5/2 cups = 13-3/4 cups

Therefore, Lily needs 13-3/4 cups of watermelons for her fruit bowl.

c. Grapes:

The amount of grapes that Lily has is not given, so we cannot calculate how much she needs for her fruit bowl.

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The bar model that represents the number of cousins that Sawyer has, given the cousins Hazel has is the fourth bar model.

The letter " c " is meant to represent the number of cousins that Sawyer has, in relation to those that Hazel has. This means that bar models A and C are out because they represent the cousins as B.

The question shows that Hazel has twice the number of cousins that Sawyer has. This means that if Sawyer's cousins are represented as " c " then Hazel would be 2 c. This is shown in the last bar model so this is the correct bar model.

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First part is:

Hazel has 8 cousins. Hazel has 2 times as many cousins as Sawyer.

How many cousins does Sawyer have?

Answer: 52-9 = 43 units

Step-by-step explanation:

Answer:

2.362

Step-by-step explanation:

Use calculator: 2.361727836

Round: 2.362

Answer:

13.75 cm³

Step-by-step explanation:

In order to find the area of a rectangular prism you only need to multiply the width height and length together:

2.75 * 1.25 = 3.4375

3.4375 * 4 = 13.75

Hope this helps!

Answer:

a) x > -7

Step-by-step explanation:

an open circle (blank) means the greater than or less than symbols (< or >) and a close circle (filled in) means ≤ or ≥. the lining to your right looks more bold than the lining to your left so it's an; open circle, on -7, with the lining bold to your left.

in conclusion, x (the open circle) > -7

Answer:

Using order of operations:

[tex] \frac{12a}{2} - 3a \times 5[/tex]

[tex]6a - 3a(5)[/tex]

[tex]6a - 15a = - 9a[/tex]

[tex] - 9 \times 4 = - 36[/tex]

The expression 0.20×0.30 could be modeled by the given decimal grid option (A) is correct.

What is a mathematical expression?

A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself. This mathematical operation may be addition, subtraction, multiplication, or division. An expression's basic components are as follows: Expression: (Math Operator, Number/Variable, Math Operator).

We have a total of 10 columns and 10 rows.

Total boxes = 10×10 ⇒ 100

Number of shaded boxes = 3×10 ⇒ 30

Each box contains X's out of 100 that is:

              = 30/100

                 = 0.30

We can represent a number of columns in decimal such that:

            = 20/100  = 0.20

The expression becomes 0.20×0.30.

Thus, the expression 0.20×0.30 could be modeled by the given decimal grid option (A) is correct.

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he two numbers that multiply to -35 and add to get -18 are -5 and 3.

To see why, you can use the factoring method. First, find two numbers that multiply to give you -35. The factors of -35 are -1, 1, -5, and 5. So, the two numbers that multiply to -35 are either -5 and 7 or 5 and -7.

Next, find which pair of numbers adds up to -18. It's clear that 5 and -7 add up to -2, so they don't work. However, if we choose -5 and 3, we get:

-5 + 3 = -2

So, -5 and 3 are the two numbers that multiply to -35 and add to -18.

The elements of the sets  A ∩ (B ∪ C) is  {b, c, i}.

A universal set (U) is a set which has elements of all the related sets, without any repetition of elements.

Therefore,

U = {a, b, ….. i, j}

A = {b, c, i, j}

B = {a, c, g, i}

C = {a, b, d, i}

Therefore, let's find the elements A ∩ (B ∪ C).

Hence,

(B ∪ C) = {a, b, c, d, i, g}

A =  {b, c, i, j}

Finally,

A ∩ (B ∪ C). = {b, c, i}

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The subtraction expression for each length is given as follows:

Suppose that we have two points of the coordinate plane, and the ordered pairs have coordinates [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex].

The shortest distance between them is given by the equation presented as follows, derived from the Pythagorean Theorem:

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Hence the distance OB is given as follows:

OB = sqrt((3 - (-3))² + (4 - 4)²)

OB = 3 - (-3).

The distance AB is given as follows:

AB = sqrt((-2 - (-2))² + (4 - (-2))²)

AB = 4 - (-2)

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The annual depreciation expense for this car is $1,833.33.

Simply deduct the vehicle's current fair market value from the purchase price, less any applicable sales tax or fees, to determine how much value has been lost.

We must determine the total depreciation throughout the vehicle's useful life and divide it by the number of years to determine the annual depreciation expense:

Total depreciation = purchase price - resale value

= $28,500 - $17,500

= $11,000

Annual depreciation expense = total depreciation / useful life = $11,000 / 6 = $1,833.33 (rounded to the nearest hundredth)

Therefore, the annual depreciation expense for this car is $1,833.33.

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Using the Intersecting secants theorem, the length of JK in the given diagram is 17

From the question, we are to determine the length of JK in the given diagram.

The diagram shows a circle with intersecting secants.

From the Intersecting secants theorem which states that "if two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment".

Then,

We can write that

JK × KL = NM × ML

From the given diagram,

KL = 14

NM = 14

ML = 17

Substituting the values

JK × 14 = 14 × 17

JK = (14 × 17) / 14

JK = 17

Hence, the length of JK is 17

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The length of the missing base is approximately 6.67 cm.

By applying the formula for the area of a trapezoid, we will be able to solve for the missing base.

Recall that the formula for the area of a trapezoid is:

A = [tex]\frac{1}{2} * h * (b_{1} + b_{2})[/tex]

where

A = area

h = height

b₁ and b₂ are the lengths of the parallel bases.

From the question, we are given:

A = 64 m²

h = 12 cm

b₁ = 4 cm.

Plug these values into the formula above to solve the missing base

64 = [tex]\frac{1}{2} * 12 * (4+ b_{2})[/tex]

Multiplying both sides by 2:

128 = 12(4 + b₂)

Simplifying:

128 = 48 + 12b₂

80 = 12b₂

b₂ = 80/12 = 20/3

b₂ = 6.67

Therefore, the length of the missing base is approximately 6.67 cm.

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A graph of the line with slope 3 passing through the point (5, 3) is shown in the image attached below.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

At data point (5, 3) and a slope of 3, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 3 = 3(x - 5)  

y = 3x - 15 + 3

y = 3x - 12

In conclusion, we would use an online graphing calculator to plot the equation y = 3x - 12 as shown in the graph attached below.

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Answer:

Option A

Concept Applied:

(g-f)(x) = g(x)-f(x)

Step-by-step explanation:

Substitute and solve

The length of the square is 1.75 inches, while the king's height is 2.25 inches.

Define length.

The length of an object or the extent of a one-dimensional object along its longest dimension is measured physically. It is a basic idea in physics and geometry and is frequently expressed in terms of meters (m), centimetres (cm), feet (ft), or inches (in).

According to the information provided, the square length to king's height ratio is 3.5 to 4.5 inches. Let's use this ratio to determine the square's length when the king is 2.25 inches tall.

We may make a proportion to get the square's length:

(square length) / (king's height) = 3.5/4.5

king's height = 2.25

Let's substitute the given values:

(square length) / 2.25 = 3.5/4.5

To solve for the length of the square, we can cross-multiply:

(square length) = (2.25) x (3.5/4.5)

(square length) = 1.75

Therefore, the length of the square is 1.75 inches, while the king's height is 2.25 inches.

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Answer: (0, 3)

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Answer:

The roots of the quadratic equation x² + 4x + 7 = 0 can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

In this case, a = 1, b = 4, and c = 7. Substituting these values into the formula, we get:

x = (-4 ± √(4² - 4(1)(7))) / 2(1)

x = (-4 ± √(16 - 28)) / 2

x = (-4 ± √(-12)) / 2

x = (-4 ± 2i√3) / 2

x = -2 ± i√3

Therefore, the roots of the equation x² + 4x + 7 = 0 are: A) 2 + i√3 or 2 - i√3.

So, the correct answer is A

This is not true for all functions. To prove this, an example is given below. Let f(x) = x and g(x) = x².

A function is a relation that associates each element of a set, called the domain, to exactly one element of another set, called the range. It is a type of correspondence between two sets, where each element of one set is associated with exactly one element of another set.

The first derivative of f(x) is f ' (x) = 1 and the first derivative of g(x) is g ' (x) = 2x.

Therefore, ( f ⋅ g )' = ( x ⋅ x2 )'

= 1 ⋅ 2x + x ⋅ 2

= 2x + 2 ≠ 1 ⋅ 2x + 2x

= 4x

= f ' ⋅ g '

This example proves that the statement (f⋅g)' = f'⋅g' is incorrect. It is important to note that this does not mean that the statement is not true for all functions. It simply means that for the particular example given, the statement is not correct.

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The answer of the given question based on the probability is , the probability that at least 6 adults watch the news is approximately 0.755.

Probability is  measure of likelihood or chance of  event occurring. It is a number between 0 and 1, where 0 indicates that an event is impossible and 1 indicates that an event is certain. Probability is used to analyze random experiments and events, and it is an essential concept in statistics, mathematics, and many other fields.

This is a binomial probability problem, where X follows a binomial distribution with n = 8 and p = 0.60. We want to find the probability that at least 6 adults watch the news, which can be written as:

P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + ...

To solve , we use  binomial probability formula:

P(X = k) =(n choose k) *p^k *(1 - p)^(n - k)

where "n choose k" is the binomial coefficient, which represents the number of ways to choose k items from a set of n items.

Using this formula, we can find the probabilities for each value of k:

P(X = 6) = (8 choose 6) * 0.60⁶ * 0.40² = 0.311

P(X = 7) = (8 choose 7) * 0.60⁷ * 0.40¹ = 0.276

P(X = 8) = (8 choose 8) * 0.60⁸ * 0.40⁰ = 0.168

Note that we can also use  binomial probability table or  calculator with  binomial probability function to find the probabilities.

Then, we can add these probabilities to get the probability that at least 6 adults watch the news:

P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + ... = 0.311 + 0.276 + 0.168 + ...

This is a infinite series, but we can stop at P(X = 8) since the probabilities become very small after that.

P(X ≥ 6) = 0.755

Therefore, the probability that at least 6 adults watch the news is approximately 0.755.

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Required Answers :

1. 2x + 4 = 10

[tex]\: :\implies [/tex] 2x = 10 - 4

[tex]\: :\implies [/tex] 2x = 6

[tex]\: :\implies [/tex] x = 6/2

[tex]\: :\implies [/tex] x = 3.

Hence, Option (A) x = 3 is the answer.

2. 25/5 × 7 = ?

[tex]\: :\implies [/tex] 5 × 7

[tex]\: :\implies [/tex] 175/5

[tex]\: :\implies [/tex] 35

Hence, The required answer is 35.

3. 100 - 50 + 10 - 5 = ?

[tex]\: :\implies [/tex] 50 + 5

[tex]\: :\implies [/tex] 55

Hence, Option (A) 55 is the answer.

4. 24 / 6 = ?

[tex]\: :\implies [/tex] 24 ÷ 6

[tex]\: :\implies [/tex] 4

Hence, Option (A) 4 is the answer.

5. A triangle has base 6 inches and height 8 inches. What is the area?

Area of triangle = ½ × Base × Height

[tex]\: :\implies [/tex] 1/2 × 6 × 8

[tex]\: :\implies [/tex] 1/2 × 48

[tex]\: :\implies [/tex] 48/2

[tex]\: :\implies [/tex] 24 sq. in

Hence, Option (A) 24 sq in is the answer.

6. What is the greatest common factor of 36 and 56?

36 = 1,2,3,4,6,9,12, 18 and 36.

56 = 1,2,4,7,8,14,28 and 56.

Hence, Option (A) 4 is the answer.

7. What is the square root of 121?

[tex]\: :\implies [/tex] 11 × 11

[tex]\: :\implies [/tex] 121

Hence, (A) 11 is the answer

8. If 2x + 4 = 14, what is x?

[tex]\: :\implies [/tex] 2x + 4 = 14

[tex]\: :\implies [/tex] 2x = 14 - 4

[tex]\: :\implies [/tex] 2x = 10

[tex]\: :\implies [/tex] x = 10/2

[tex]\: :\implies [/tex] x = 5

Hence, Option (A) 5 is the answer

9. (3x - 5) + (7x + 1) = ?

[tex]\: :\implies [/tex] 3x - 5 + 7x + 1

[tex]\: :\implies [/tex] 10x - 5 + 1

[tex]\: :\implies [/tex] 10x - 4

Hence, Option (A) 10x - 4 is the answer.

10. What fraction is 5/12?

[tex]\: :\implies [/tex] 1/2 (approximately)

Hence, Option (A) 1/2 is the answer.

120 × 5 ÷ 20

= 120 × 5/20

= 6 × 15

= 90 Answer.

The number of dahlias should the florist expect to sell on a day with 120 customers is 90 bouquets.

The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value. Unitary method is a technique by which we find the value of a single unit from the value of multiple devices and the value of more than one unit from the value of a single unit. It is a method that we use for most of the calculations in math.

We are given that;

Sold bouquets=15

Total customers=120

Now,

In this case, the number of successful outcomes is 15, and the number of total outcomes is 20.

So the experimental probability is:

2015​=43​

To find the expected number of bouquets of dahlias to sell on a day with 120 customers, you need to multiply the experimental probability by the number of customers.

In this case, you need to multiply 3/4 by 120:

3/4​×120=3×120​/4

=360/4​

=90

Therefore, by the unitary method the answer will be 90 bouquets

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The probability of the sample mean being less than 25.3 is given as follows:

0.9713 = 97.13%.

The sample mean would not be considered unusual, as it has a probability that is greater than 5%.

The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is obtained by the equation presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The parameters for this problem are given as follows:

[tex]\mu = 25, \sigma = 1.3, n = 68, s = \frac{1.3}{\sqrt{68}} = 0.1576[/tex]

The probability of a score less than 25.3 is the p-value of Z when X = 25.3, hence:

Z = (25.3 - 25)/0.1576

Z = 1.9

Z = 1.9 has a p-value of 0.9713.

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The surface area of the cylinder is 1851.97 square meters

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

Radius, r = 13.1 meters

Height, h = 9.4 meters

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

Surface area = 2πr(r + h)

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

Surface area = 2π * 13.1 * (13.1 + 9.4)

Evaluate

Surface area = 1851.97

Hence, the surface area is 1851.97 square meters

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(a) This study is observational because the consultant is not manipulating any variables or imposing treatments on the machines.

(b) The factors are:

Machine (experimental factor): There are six machines, labeled A through F.

Carton (observational factor): There are 20 cartons selected randomly from each machine.

Deviation from 32 ounces (response variable): This is the weight difference between the actual fill amount and the target fill amount of 32 ounces.

The factor levels are:

Machine: A, B, C, D, E, F

Carton: 1, 2, 3, ..., 20

The factor-level combinations are all possible pairs of machine and carton, such as A-1, A-2, ..., F-20.

(c) This is a nested design because the 20 cartons selected from each machine are not interchangeable between machines.

(d) Here's an example code snippet in R to import the data and display the structure of the dataframe:

data <- read.csv("filling_machine_data.csv")

data$Machine <- factor(data$Machine)

levels(data$Machine)

Assuming the data is stored in a CSV file called "filling_machine_data.csv", the code reads in the data, coerces the Machine variable to a factor, and displays the factor levels of Machine

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