The length of a rectangle is 3cm
greater than it's width. The area of the rectangle is 180cm ². Find the
length and the width.

The angle measuring 280° is coterminal with the angle measuring -800° and lies between 0° and 360°.

To find the coterminal angle between 0° and 360° for an angle measuring -800°, follow these steps:
1. Divide the given angle (-800°) by 360° to determine how many full rotations are made:

-800° ÷ 360° = -2.22.
2. Since we are looking for a positive coterminal angle, round the result down to the nearest whole number:

-2.22 rounds down to -3.

This tells us there are three full negative rotations.
3. Multiply the whole number (-3) by 360° to find the total angle of the rotations:

-3 × 360° = -1080°.
4. Add the total angle of the rotations to the given angle to find the coterminal angle between 0° and 360°: -800° + 1080° = 280°.

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Answer:

Step-by-step explanation:

x = [tex]+-\sqrt[]{\frac{1}{2} }[/tex]

we use the definition of supplementary angles again to conclude that ∠3 and ∠5 are supplementary.

To prove that ∠3 and ∠5 are supplementary, we can use the following steps:

Given: w ∥ x and y is a transversal.

∠1 ≅ ∠5 by the corresponding angles postulate.

By definition, ∠3 and ∠1 are a linear pair, so they are supplementary.

Therefore, m∠3 + m∠1 = 180 by the definition of supplementary angles.

Substituting m∠5 for m∠1, we get m∠3 + m∠5 = 180.

Therefore, by the definition of supplementary angles, ∠3 and ∠5 are supplementary.

In step 1, we use the corresponding angles postulate, which states that when two parallel lines are cut by a transversal, the pairs of corresponding angles are congruent. In step 2, we use the definition of a linear pair, which states that when two angles form a line, they are supplementary. In step 3, we use the definition of supplementary angles, which states that the sum of two supplementary angles is 180 degrees. In step 4, we substitute m∠5 for m∠1, since we know that ∠1 ≅ ∠5. Finally, in step 5, we use the definition of supplementary angles again to conclude that ∠3 and ∠5 are supplementary.

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Answer:

Step-by-step explanation:

its doc sus

If -4 is a zero with multiplicity 1, then (x + 4) is a factor of the polynomial. Similarly, if 1 is a zero with multiplicity 2, then (x - 1)^2 is a factor of the polynomial. Therefore, we can write the polynomial in factored form as:

f(x) = a(x + 4)(x - 1)^2

where "a" is a constant that we need to determine.

To find "a", we use the fact that f(0) = -12. Substituting x = 0 into the equation above, we get:

f(0) = a(0 + 4)(0 - 1)^2

-12 = -4a

Solving for "a", we get:

a = 3

Therefore, the polynomial is:

f(x) = 3(x + 4)(x - 1)^2

Note that this polynomial has a zero at x = -4 (with multiplicity 1), a zero at x = 1 (with multiplicity 2), and f(0) = -12.

Answer and Explanation:

The value of a brand new car,

P

=

$

28000

The value depreciates every year,

r

=

19

%

Time,

t

=

1

4

Number of compounding period (Compounded quarterly),

n

=

4

Write a function to represent the value of the car after

t

years:

A

=

P

(

1

−

r

n

)

n

×

t

∴

A

=

28000

(

1

−

19

%

4

)

4

t

Let us evaluate the value of the car after one quarter:

A

=

28000

(

1

−

19

%

4

)

4

t

=

28000

(

1

−

19

100

4

)

4

×

1

4

=

28000

(

1

−

0.19

4

)

4

×

1

4

=

28000

(

1

−

0.19

4

)

1

=

28000

(

1

−

0.19

4

)

=

28000

(

1

−

0.0475

)

=

28000

(

0.9525

)

∴

A

=

26670

Therefore, the value of the car after one quarter is

$

26670

.

To calculate the percentage rate of change per quarter:

Percentage rate

=

P

−

A

P

×

100

%

=

28000

−

26670

28000

×

100

%

=

1330

28000

×

100

%

=

70

⋅

19

70

⋅

400

×

100

%

=

19

400

×

100

%

=

0.0475

×

100

%

∴

Percentage rate of change per quarter

=

4.75

%

Hence, the percentage rate of change per quarter is

4.75

%

.

The area of the shape made of two rectangles is 85 square cm.

To find the area of the shape made of two rectangles with dimensions 9cm, 5cm, 8cm, and 5cm, follow these.

As per the given information,
Rectangle 1 has dimensions 9cm and 5cm.
Rectangle 2 has dimensions 8cm and 5cm.
For calculating the area of each rectangle using the formula:

Area = length × width
[tex]{Area\: of\: Rectangle} _1[/tex] = 9cm × 5cm = 45 square cm
[tex]{Area of Rectangle}_ 2[/tex] = 8cm × 5cm = 40 square cm
Add the areas of both rectangles to find the total area of the shape.
Total Area = [tex]{Area\: of\: Rectangle} _1[/tex] + [tex]{Area\: of\: Rectangle} _2[/tex]
Total Area = 45 square cm + 40 square cm

Total Area = 85 square cm.

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Question: The shape below is made of two rectangles joined together. 5 cm 9 cm 8 cm 5 cm Find the total area of the shape.

The solution is greater than [tex]-2[/tex]

A mathematical assertion known as an inequality contrasts two values or expressions, demonstrating their relationship in terms of greater than[tex]\geq[/tex], less than (), greater than or equal to [tex]\geq[/tex], or less than or equal to ().

When two quantities are related in a manner other than being equal, it is referred to as an inequality.

[tex]8 > -3a+2[/tex]

Subtracting [tex]2[/tex] from both sides we get:

[tex]6 > -3a[/tex]

Dividing both sides by [tex]-3[/tex] and reverse the inequality sign when dividing by negative number

[tex]a > -2[/tex]

Since a cannot equal [tex]-2[/tex], we can plot an open circle at this value to represent the solution on a number line. We can also depict an arrow pointing up the number line to represent that an is larger than [tex]-2[/tex]

The arrow denotes any value higher than -2, and the open circle at -2 denotes that -2 is not part of the solution.

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Answer:

a>-2

explained:

-3a+2<8

−3+2−2<8−2

−3a<6

−[tex]\frac{3a}{3}[/tex]<[tex]\frac{6}{-3}[/tex]

a>−2

The statement which is "The binomial distribution table, the number of trials must be greater than 30." is the not correct statement related to the binomial and Poisson distributions. Hence option a is the right choice.

a. "When using the binomial distribution table, the number of trials must be greater than 30"This is an incorrect statement.

There is no such rule stating that the number of trials must be greater than 30 for the binomial distribution.

b. "The Poisson distribution represents the random arrival of events per unit of time or space" This is the correct statement.

The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, given a constant average rate of occurrence.

c. "Only two outcomes are possible in a binomial situation" This is correct statement.

A binomial distribution represents the number of successes in a fixed number of trials, each with only two possible outcomes (success or failure).

d. To use the Poisson distribution tables, mu must be known or be able to be calculated: This is correct.

In order to use the Poisson distribution tables, the mean number of events (represented by the symbol µ) must be known or calculable.

So, option a is right choice.

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Question :-

Which of the following is not a correct statement about the binomial and Poisson distributions?

a. When using the binomial distribution table, the number of trials must be greater than 30.

b. The Poisson distribution represents the random arrival of events per unit of time or space.

c. Only two outcomes are possible in a binomial situation.

d. To use the Poisson distribution tables, Mu must be known or be able to be calculated

Nonprobability sampling carries more risk of selection bias than probability sampling because nonprobability sampling does not use randomization.What is nonprobability sampling?A nonprobability sampling is a method of selecting participants for a study that does not allow the researcher to use a random selection process.

Nonprobability sampling, also known as purposive sampling, is a technique that involves choosing participants based on subjective criteria, such as availability or willingness to participate. This method is frequently used in research that investigates hard-to-reach populations, such as homeless individuals or drug addicts.

A key feature of probability sampling is that it allows the researcher to obtain a representative sample of the population, minimizing the risk of selection bias. In contrast, nonprobability sampling does not provide the same level of assurance that the sample will be representative. Because participants are chosen based on subjective criteria rather than random selection, there is a greater risk of selection bias.

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The problem involves a line segment divided into two parts in a specific ratio. The ratio of the length of the lesser part to the length of the greater part is found to be (√2 - 1).

Let the length of the whole line segment be x, and let y be the length of the greater part. Then the length of the lesser part is (x - y).

According to the problem statement, the ratio of the lesser part to the greater part is the same as the ratio of the greater part to the whole. Mathematically, we can write this as:

(x - y)/y = y/x

Simplifying this equation, we get:

x^2 - y^2 = y^2

x^2 = 2y^2

Taking the square root of both sides, we get:

x = y√2

Therefore, the value of the ratio of the lesser part to the greater part is:

(x - y)/y = (√2 - 1)

So, the answer will be (√2 - 1).

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Answer:

Base length=22

width=30

height=15

volume of pyramid= Lxwxh/3

=22x 30x 15/3 =3300in^3.

The findings of the research project suggest that regular exercise can be an effective tool for managing stress in university students.

A sample research project is given below:

Study of Data: The impact of exercise on stress levels in university students

Prediction: We predict that there will be a significant decrease in stress levels following a four-week exercise program.

Data Collection: Participants were recruited through flyers and social media. They were asked to complete a pre-test stress survey before beginning the exercise program, and a post-test stress survey after completing the program. The exercise program consisted of a combination of cardiovascular and resistance training, meeting three times a week for four weeks.

Table of Figures:

Participant Pre-Test Stress Score Post-Test Stress Score

         1                          32                                 20

         2                         28                                  17

         3                         36                                  25

         ...                         ...                                   ...

         30                          41                                  22

Measures of Central Tendency:

Graph: Line graph showing the change in stress levels over time, with pre-test scores in blue and post-test scores in red.

Written Analysis: The line graph shows a consistent decrease in stress levels across all participants following the four-week exercise program. The majority of participants showed a decrease in stress levels of 10 or more points, indicating a significant reduction in stress. There is a clear pattern of improvement in stress levels over time, with the post-test scores consistently lower than the pre-test scores.

Findings: Our study found a significant decrease in stress levels following a four-week exercise program in university students. Our prediction was accurate, as we anticipated a decrease in stress levels.

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As a result, the slope-intercept form equation for the line through points (5, 21) and (11, 3) is:

y = -3x + 36

Therefore, the equation of the line through (4, 8) and (12, 20) in slope-intercept form is:

y = (3/2)x + 2

The slope of the line must first be determined using the following formula to determine the equation of a line through two points in slope-intercept form:

slope[tex](m) =\frac {(y2 - y1)}{(x2 - x1)}[/tex]

where the coordinates for two points are (x1, y1) and (x2, y2).

The line's slope-intercept form is thus applicable, and it is as follows:

y = mx + b

where b is the y-intercept and m is the slope that we just discovered.

We possess

slope [tex](m) = \frac{(3-21)}{ (11-5)}[/tex],

m = -3

So, using one of our points, we can determine b:

y-21=-3(x-21)

As a result, the slope-intercept form equation for the line through points (5, 21) and (11, 3) is:

y = -3x + 36

9) (4, 8) and (12, 20)

The slope is

[tex]m= \frac{20-8}{12-4}\\m=\frac{12}{8}=\frac{3}{2}[/tex]

equation is (y-8)=1.5(x-4)

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Using equations,

8. The original dimensions of the rectangle are:

Length = 6m and Width = 2m.

9. The meter contains the following:

Value of nickels = 0.63

Value of dime = 2.26

Value of quarters = 0.21

10. Volume of the sphere is V. So, the value of r will be:

r = ∛ (3v/4π)

11. Solving for the linear equation we can get the value of m = -1

A mathematical statement that has two expressions with equal values separated by the symbol "equal to" is called an equation.

In the question,

1. Let the length of the rectangle be = x.

Let the width of the rectangle be = w.

Given,

l = 3w

When the width is increased by 4 meters, (w + 4) meters is the width now.

Then, the rectangle becomes a square.

That means length and width are equal.

l = w+4

Using l from the 1st equation,

3w = w + 4

Subtracting w from both sides,

⇒ 3w - w = 4

⇒ 2w = 4

Diving both sides by 2,

⇒ w = 2 meters.

We know, l = 3w

l = 3 × 2

l = 6 meters.

2. Here,

Let nickels be n.

Let dimes be d.

Let quarters be q.

Given,

Triple the number of quarters as nickels, so:

3q = n

Solving this for q,

q = n/3

Dimes is one than double the times of nickels, so:

d = 2n + 1

Total value of coins = $3.10

So, n + d + q = 3.10

Substituting the value of q and d,

⇒ n + 2n + 1 + n/3 = 3.10

⇒ 3n + n/3 = 3.10 - 1

⇒ (9n+n)/3 = 2.10

⇒ 10n = 6.3

⇒ n =0.63.

We can now find,

d = 2n + 1

= 2 × 0.63 + 1

= 1.26 + 1

= 2.26

q = n/3

= 0.63/3

= 0.21

3. Given,

Volume of sphere,

V = 4/3 × π × r³

Solving for r,

Multiply 3/4 on both sides,

⇒ 3v/4 = π r³

Divide both sides by π,

⇒ 3v/4π = r³

Now, taking cube root on both sides we get,

r = ∛ (3v/4π)

4. Now given a linear equation,

[tex]y=-2x^{m+2}[/tex]

As the equation is a linear equation,

The value of (x, y) we can take as (0,0)

So, 1 = m +2

⇒ m = -1

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the correct answer is: Wide Awake, with a median value of 4.5 gallons. Thus, option A is correct.

Based on the data given, the correct measure of center to determine which shop typically sells the most amount of coffee per hour would be the median.

Calculating the medians for both coffee shops:

For Coffee Ground:

[tex]1.5, 12, 11, 9.5[/tex] (sorted data)

Median [tex]= (11 + 12)/2 = 11.5[/tex]

For Wide Awake:

[tex]2.5, 18, 3, 6[/tex] (sorted data)

Median [tex]= (3 + 6)/2 = 4.5[/tex]

Comparing the medians, we can see that Wide Awake has a median value of 4.5 gallons, which is higher than the median value of 11.5 gallons for Coffee Ground.

Therefore, the correct answer is: Wide Awake, with a median value of [tex]4.5[/tex] gallons.

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Answer: Coffee Ground typically sells more coffee per hour than Wide Awake.

Step-by-step explanation:

Given that:

Coffee ground :

Hour 1: 1.5

Hour 2: 20

Hour 3: 3.5

Hour 4: 12

Hour 5: 2

Hour 6: 5

Hour 7: 11

Hour 8: 7

Hour 9: 2.5
We add up all of these numbers and divide by the total number of hours to find the mean:

(1.5 + 20 + 3.5 + 12 + 2 + 5 + 11 + 7 + 2.5) / 9 = 7.222

therefore mean is 7.2

Median:

Arranging them in Ascending or descending order, we get:

1.5,2,2.5,3.5,5,7,11,12,20

As we know, The median of discrete data is :

((n+1)/2)th element if the number of elements is odd

Average of (n/2) th and ((n+1)/2)th element if the number of elements is even. the value

Here the number of elements is odd, so the median is:

((n+1)/2)th =(9+1)/2 = 5th element

i.e., the median is 5.

Wide Awake:

Given that:

Hour 1: 2.5

Hour 2: 10

Hour 3: 4

Hour 4: 18

Hour 5: 4

Hour 6: 3

Hour 7: 6.5

Hour 8: 5

Hour 9: 15

We add up all of these numbers and divide by the total number of hours to find the mean:

(2.5 + 10 + 4 + 18 + 4 + 3 + 6.5 + 5 + 15) / 9 = 7.056

therefore mean is 7.1

Median:

Arranging them in Ascending or descending order, we get:

3,6.5,5,4,4,2.5,10,18,15

Here the number of elements is odd, so the median is:

((n+1)/2)th =(9+1)/2 = 5th element

i.e., the median is 4.

As we can observe, the mean and median of coffee ground is high.

So we can conclude that the Coffee Ground shop has sold more coffee.

The probability is approximately 17.43%.

Use Bayes' theorem. Bayes' theorem states that is:

P(A|B) = (P(B|A) * P(A)) / P(B),

Using the law of total probability: P(B) = P(B|A) * P(A) + P(B|A') * P(A')
P(B) = (0.01 * 0.95) + (0.9 * 0.05) = 0.0095 + 0.045 = 0.0545
Apply Bayes' theorem to find P(A|B):
P(A|B) = (P(B|A) * P(A)) / P(B) = (0.01 * 0.95) / 0.0545 ≈ 0.1743
So, the probability that the suspect is innocent given that the serum indicates guilt is approximately 17.43%.

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The surface area of the given right triangular prism is 84 in².

How to find the surface area of a right triangular prism?

To calculate the surface area of a right triangular prism, we need to find the area of all the faces and add them up.

First, let's find the area of the triangular bases. The base of the triangle is 8 in, and the height is 3 in, so the area of each triangular base is [tex]\frac{1}{2} \times base \times height = \frac{1}{2} \times 8 \times 3 = 12 \: {in}^{2}[/tex]

There are two triangular bases, so the total area of both bases is (2 × 12) in²= 24 in²

Now, let's find the area of the rectangular faces. The length of the rectangle is 5 in, and the breadth is 4 in.

So, the area of each rectangular face is length × breadth = 5 in × 4 in = 20 in²

There are three rectangular faces, so the total area of all three faces is 3 × 20 in² = 60 in².

Finally, we add the areas of the bases and the rectangular faces to get the total surface area.

Total surface area = 24 in² + 60 in² = 84 in²

Therefore, the surface area of this right triangular prism is 84 square inches.

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The value of x is given as 9.8

We have to solve for X using the Pythagorean theorem

The Pythagorean theorem is a mathematical formula that relates to the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In other words:

c² = a² + b²

Where:

c is the length of the hypotenuse

a and b are the lengths of the other two sides

14² = x² + 10²

196 = x² + 100

196 - 100 = x²

x = √96

x = 9.8

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Answer:

B and C are the answers according to the expression

Two parents are present in 69% of all families with children ages six to eleven. Option d is correct.

Option d is the correct choice, which states that 69 percent of all families with children aged six to 11 have two parents. This information is supported by various studies and statistics, which have consistently shown that the majority of families in the United States consist of two-parent households. While single-parent families do exist, they are less common than families with two parents.

This information is important in understanding the dynamics and structure of families in the US, as well as in developing policies and programs that support and strengthen families.

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--The complete question is, According to your text, 69 percent of all families with children aged six to 11:

a) are harmonious and stable.

b) have one parent.

c) are chronically poor.

d) have two parents.
Link to the text is,  open.maricopa.edu/devpsych/chapter/chapter-9-early-adulthood/--

Answer:

To verify the given identity:

(1 - cos a) (1 + cot a)

= (1 - cos a) (1 + cos a / sin a) [since cot a = cos a / sin a]

= 1 - cos^2 a / sin a + cos a - cos^2 a / sin a

= 1 - (cos^2 a + cos^2 a) / sin a + cos a

= 1 - 2 cos^2 a / sin a + cos a

= 1 - 2 (1 - sin^2 a) / sin a + cos a [since cos^2 a = 1 - sin^2 a]

= 1 - 2 / sin a + 2 sin a / sin a + cos a

= 1 - 2 / sin a + 2 + cos a

= 1 + 2 (1 - sin a) / sin a

= 1 + 2 cos^2 a / sin a

= 1 + 2 cot^2 a

= (1 + cot^2 a) + 2 cot^2 a

= cosec^2 a + 2 cot^2 a

= 1 + cot^2 a [since cosec^2 a = 1 + cot^2 a]

Therefore, (1 - cos a) (1 + cot a) = 1 is true.

The probability of prime number in a die would be ½.

If a die rolls, the total outcome would be 6.
Number of possible outcomes = 3( as there are only 3 prime numbers in the die that are 2,3 and 5).
P(prime number) = 3/6
So the probability of the prime number in the die would be ½.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will learn the potential outcomes for a random experiment using this fundamental theorem of probability, which is also applied to the probability distribution. Knowing the total number of outcomes is necessary before we can calculate the likelihood that a specific event will occur.

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Answer:

Step-by-step explanation:

The p-value is  0.046, or 4.6% if a sample of shoppers showed stating that the supermarket brand was as good as the national brand.

To calculate the p-value, we use null and alternative hypotheses, as well as the test statistic and its distribution under the null hypothesis.

By the null hypothesis, we can say that supermarket brands are as good as national brands, and by the alternative hypothesis, we can say that supermarket brands are not as good as national brands.

This hypothesis can be tested by performing a two-tailed z-test for proportions.

In a sample of n shoppers, say that x shoppers say that supermarket brands are as good as domestic brands. The sample percentage can be calculated as follows:

p hat = x/n

in the null hypothesis, the  sample proportion is equal to the hypothesized proportion, that is 0.5

The test statistic for a two-sided z-test for proportions is given by:

z = (p-hat - p0) / sqrt(p0(1-p0)/n)

where p0 is the hypothesized proportion under the null hypothesis.

In this case, we have:

p-hat = 0.5

p0 = 0.5

n = sample size

The null hypothesis states that the supermarket brand is as good as the national brand, so we would expect the proportion of shoppers who state this to be 0.5.

If the test statistic z falls in the rejection region (i.e., if |z| > 1.96 for a significance level of 0.05),

we would reject the null hypothesis and conclude that there is evidence to suggest that the supermarket brand is not as good as the national brand.

If the test statistic falls in the non-rejection region (i.e., if |z| <= 1.96 for a significance level of 0.05),

we would fail to reject the null hypothesis and conclude that there is insufficient evidence to suggest that the supermarket brand is not as good as the national brand.

To calculate the p-value, we need to find the probability of getting a test statistic as extreme or more extreme than the one we observed, assuming the null hypothesis is true.

For a two-tailed test, the p-value is twice the observed area of ​​the tails above the absolute value of the test statistic.

Assuming the sample size is large enough, the distribution of the test statistic can be approximated as a standard normal distribution under the null hypothesis.

Suppose in a sample of 100 shoppers, 50 said that supermarket brands are as good as domestic brands. after that:

p hat = 0.5

p0 = 0.5

n=100

The test statistic is:

z = (p hat - p0) / sqrt(p0(1-p0)/n) = 0 / sqrt(0.5 * 0.5 / 100) = 0

The p-value is the probability that the test statistic is extreme or more extreme than 0. This is the probability of getting further away from the 50/100 ratio or 0.5 in either direction.

p-value = P(p hat <= 0.4 or p hat >= 0.6) = 2 * P(p hat <= 0.4) = 2 * P(Z <= ( 0.4 - 0.5) / sqrt (0.5 * 0.5 / 100) ) = 2 * P(Z <= -2) ≈ 0.046

where Z is a standard normal random variable.

Therefore, the p-value is approximately 0.046, or 4.6%.

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Answer:

Step-by-step explanation:

1.) 1/5x - 2 = -3

Adding 2 to both sides, we get:

1/5x = -1

Multiplying both sides by 5, we get:

x = -5

Therefore, the solution is x = -5.

2.) 15 = 10 - x/4

Subtracting 10 from both sides, we get:

5 = -x/4

Multiplying both sides by -4, we get:

-20 = x

Therefore, the solution is x = -20.

3.) x/3 - 12 = -2

Adding 12 to both sides, we get:

x/3 = 10

Multiplying both sides by 3, we get:

x = 30

Therefore, the solution is x = 30.

4.) x/-2 + (-6) = 6

Adding 6 to both sides, we get:

x/-2 = 12

Multiplying both sides by -2, we get:

x = -24

Therefore, the solution is x = -24.

The radius r that produces the minimum surface area is 1.06 cm and the height h is 1.41 cm.

Let r be the radius of the cylinder and h be the height. You can find the equation for the volume of a solid by adding the volumes of the two hemispheres and the cylinder.

Volume = [tex](2/3)πr^3 + πr^2h[/tex] = 10 cubic centimeters

Now let's find the values ​​of r and h that minimize the surface area of ​​the solid. This area consists of three parts:

Curved surfaces of two hemispheres and sides of a cylinder. This can be expressed as:

surface area = [tex]2πr^2 + 2πrh[/tex]

To find the values ​​of r and h that minimize this expression, we can use the Lagrangian multiplier method. Given that the volume of the solid is 10 cubic centimeters, we want to minimize the surface area. So it looks like this:

[tex]f(r, h) = 2πr^2 + 2πrh + λ[(2/3)πr^3 + πr^2h - 10][/tex]

Taking the partial derivatives for r, h, and λ and setting them to zero gives

[tex]4πr + 2πh = 2πr^2λ + (4/3)πr^3λ[/tex]

[tex]2πr = πr^2λ + πrhλ[/tex]

Solving these equations simultaneously gives:

h = 4r/3

λ = 2/(3r)

Substituting these values ​​into the volume equation gives:

[tex]r^2h = 5/3[/tex]

Substituting h = 4r/3 from above, we get:

[tex]r^3 = 15/8pi[/tex]

Taking the cube root of both sides gives:

r=1.06cm

Substituting this value for r into the formula for h yields:

h=1.41cm

Therefore, the radius r that produces the minimum surface area is about 1.06 cm and the height h is about 1.41 cm.

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Given the similar triangles, Note that x = 51.2

Since both triangles are proportional,

64/60 = x/48

To solve for x in the equation:

64/60 = x/48

We can cross-multiply to get rid of the fractions:

64 * 48 = 60 * x

3072 = 60x

Finally, we isolate x by dividing both sides by 60:

x = 3072/60 = 51.2

Therefore, the solution is:

x = 51.2

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[tex]27000(0.77^t)[/tex]and the percentage rate of change per month is 3.13%.

What is the percentage rate?

The term annual percentage rate of charge refers to the interest rate for an entire year rather than just a monthly fee or rate as applied on a loan, home loan, credit card, etc. It can also be referred to as a nominal APR or an effective APR. It is an annual rate of a finance charge.

Here, we have

The value of a brand-new car is $27,000 and the value depreciates 23% every year.

we have to write a function to represent the value of the car after t years.

The coefficient of the function is 0.77

To find the rate of change per month, we need to find the rate at which the value of the car is decreasing each month.

we can use the rule of 72

72/r = t where r is the rate of change per month

r = 72/t

[tex]r = 72/23 = 3.13[/tex] (approx)

So the percentage rate of change per month is 3.13%

Hence, The function to represent the value of the car after t years is V(t) = [tex]27000(0.77^t)[/tex] and the percentage rate of change per month is 3.13%.

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