Solve for x.

5x - 1 = 26

x = 27/5
x = 5
x = -5

Final answer:

During the power stage of a four-cycle gas engine, both the intake and exhaust valves are closed, allowing the air-fuel mixture to expand and exert force on the piston, converting fuel's chemical energy into mechanical work.

Explanation:

The question concerns the position of the intake and exhaust valves during the power stage of a four-cycle gas engine, specifically within the context of the Otto cycle. During the power stroke, the intake valves are closed to allow the air-fuel mixture, which was ignited at the beginning of this stage, to expand without escaping. Meanwhile, the exhaust valves also remain closed to ensure that the expanding gases exert maximum force on the piston, driving it down.

This process converts the chemical potential energy of the fuel into mechanical work. The exhaust valve opens only after the completion of the power stroke, during the exhaust stage, to expel the combustion products and prepare the engine for the next cycle.

Final answer:

By setting up a system of equations based on the given information and solving for b, we find that side a is 40 meters, b is 20 meters, and c is 38 meters.

Explanation:

To solve for the lengths of the sides of the triangle, we can set up a system of equations based on the information given:

Substituting the expressions for a and c into the perimeter equation gives:

With the value for b, we can now find a and c:

The lengths of the sides of the triangle are 20 meters (b), 40 meters (a), and 38 meters (c).

Final answer:

To solve for the lengths of the triangle's sides, we formulate equations based on the given relationships and perimeter. We find that side b is 20 meters, side a is twice b (40 meters), and side c is 2 meters shorter than a (38 meters).

Explanation:

The student's question involves finding the length of each side of a triangle given specific relationships between the sides and the overall perimeter. The problem can be formulated using algebra. Let's denote side b as the baseline. From the question, we know that side a is twice side b (a = 2b), and side c is 2 meters shorter than side a (c = a - 2 = 2b - 2). The perimeter (P) is given as 98 meters, so we have P = a + b + c = 98 meters.

Substituting for a and c in terms of b we have P = 2b + b + (2b - 2) = 98, simplifying to 5b - 2 = 98. Solving for b we add 2 to both sides, 5b = 100, and then divide by 5 to get b = 20 meters. Now we easily find that a = 2(20) = 40 meters, and c = 40 - 2 = 38 meters. Therefore, the lengths of the sides of the triangle are 40 meters for side a, 20 meters for side b, and 38 meters for side c.

Answer:

The bucket hold 3x the capacity of the jar

Step-by-step explanation:

First we have to write down the amount of both, the jar and the bucket, to make a small equation to see the problem mathematically.

Bucket = c liters of water

Jar = c - 8

so we can say that:

c = c - 8

Now that we have the equation, we can determine that the variable 'c' on the right is substracting to the number 8. So we can pass the 'c' to the left of the equation by adding it to the elements on the left, like this:

c + c = 8

Now, two 'c' are the same, we can add them:

2c = 8

This two on the equation of the left is multiplying the variable 'c', so we can move it to the right by dividing the number 8 with this number:

c = 8/2

If we solve this operation, we'll get:

c = 4

That means that the amount of liters of water a jar can hold is 4 liters

Jar = 4 liters

Now that we have this value, we can solve the problen. If the jar holds 8 liters of water les than a bucket, we can now represent it like this:

Bucket = Jar capacity + 8 liters

Wich can be represented like this as well:

Bucket = 4 + 8

And the result is:

Bucket = 12 liters

If we try to fit the jar's capacity on the bucket, we find that the bucket hold 3x the capacity of the jar

Let me know if you have any doubts :D

Answer:

y=[tex]\frac{x^{2} }{18}[/tex]

Answer: 825.

Step-by-step explanation:

Answer:  Affect extremely.

Step-by-step explanation:

When a data set contains extreme values also known as outliers , then mean gets affected extremely.

For example : We have a data set with values : 1 , 2, 1,2,1, 2, 12

Here , 12 is much larger than the rest of the data values.

Mean of data : [tex]\dfrac{\text{Sum of observations}}{\text{Number of observations}}[/tex]

[tex]=\dfrac{1 +2+ 1+2+1+2+12}{8}=\dfrac{21}{8}=2.625[/tex]

Mean excluding outlier :

[tex]=\dfrac{1 +2+1+2+1+2}{7}=\dfrac{9}{7}=1.28571428571\approx1.29[/tex]

We can see that with outlier the mean is way higher than the mean of the rest of the values.

Hence, When an observation that is much larger than the rest of the data is added to a data set, the value of the mean will get affected.

The problem primarily involves the calculation and understanding of the concepts of perimeter and area in Mathematics. Given the length is 3 ft longer than the width, and the perimeter of the rectangle is 26 feet, first, we find that the width is 5 feet and the length is 8 feet. The area is thus, 40 square feet.

To solve this problem, we need to understand the formula for the perimeter of a rectangle, which is 2l + 2w, where 'l' stands for length and 'w' for width. Also, the area of a rectangle is calculated by l*w. The problem states that the length is 3 ft longer than the width and that the perimeter is 26 feet. Firstly, we can express the length as 'w + 3' and substitute into the perimeter formula:

2(w + 3) + 2w = 26

After simplifying, we get the width equals to 5 feet. Substituting 5 for width, we get the length as 8 feet, because length is 3 feet longer than the width. Then, the area of the rectangle can be calculated by multiplying the width by the length:

Area = l * w = 8 ft * 5 ft = 40 square feet

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x = number of pounds

5x<30

x<30/5

 so she can buy X<6 pounds

Answer:

The possible numbers of pounds she will buy is less than 6.

Step-by-step explanation:

Given : Donna buys candy that costs $5 per pound. She will spend less than $30 on candy.

To find : What are the possible numbers of pounds she will buy?

Solution :

Let x be the number of pounds.

Donna buys candy that costs $5 per pound.

i.e. cost of x candy is $5x.

She will spend less than $30 on candy.

i.e. [tex]5x<30[/tex]

Solve the inequality,

[tex]x<\frac{30}{5}[/tex]

[tex]x<6[/tex]

The possible numbers of pounds she will buy is less than 6.

To solve this problem, we have to use the formula:

P = 1 - e^(-λx)

where λ = 0.01382, and x is the  distance

A. when x = 100

(P < or equal to 100) = 1 - e^(-0.01382 * 100)

(P < or equal to 100) = 0.75 

B. when x = 200
(P < or equal to 200) = 1 - e^(-0.01382 * 200)
(P < or equal to 200) = 0.94

C. when 100 < x < 200
P (100 < x < 200) = P ((x < 100) + P(x > 200)) 
P (100 < x < 200) = 0.75 + (1 – 0.94)

P (100 < x < 200) = 0.81 

Final answer:

The probabilities for a banner-tailed kangaroo rat to move at most 100 m, at most 200 m, and between 100 and 200 m are 0.7487, 0.9389, and 0.1903 respectively, when rounded to four decimal places.

Explanation:

The question deals with exponential distribution, which is a type of continuous probability distribution often used to model the time or space between events in a Poisson process. Given that the parameter λ (lambda) for the banner-tailed kangaroo rats is 0.01382, the mean (μ) can be calculated as the reciprocal of λ, so μ = 1/λ. In this case, μ = 1/0.01382. The cumulative distribution function (CDF) for an exponential distribution is  [tex]P(X \leq x) = 1 - e^{-\lambda x}[/tex]

To find the probability that the distance is at most 100 m, we use the CDF:

[tex]P(X \leq 100) = 1 - e^{-(0.01382)(100)} = 1 - e^{-1.382} \approx 1 - 0.25132 = 0.74868[/tex]

For the distance at most 200 m:

[tex]P(X \leq 200) = 1 - e^{-(0.01382)(200)} = 1 - e^{-2.764} \approx 0.93894[/tex]

To find the probability that the distance is between 100 m and 200 m, we subtract the probability of being at most 100 m from the probability of being at most 200 m:

P(100 < X ≤ 200) = P(X ≤ 200) - P(X ≤ 100) = 0.93894 - 0.74868 = 0.19026

So, the probabilities are approximately 0.7487 for at most 100 m, 0.9389 for at most 200 m, and 0.1903 for between 100 and 200 m, when rounded to four decimal places.

Raina received 19 phone calls on the third evening, 26 phone calls on the first evening, and 38 phone calls on the second evening.

Let's consider the number of phone calls Raina received each evening.

Let x represent the number of phone calls on the third evening.

The first evening, Raina received 7 more calls than the third evening, so the number of phone calls on the first evening is x + 7.

The second evening, Raina received 2 times as many calls as the third evening, so the number of phone calls on the second evening is 2x.

Given that Raina received a total of 83 phone calls over the three evenings, we can set up the equation:  

x + (x + 7) + 2x = 83

Combining like terms, we have:

4x + 7 = 83

Subtracting 7 from both sides, we get:

4x = 76

Dividing both sides by 4, we find that x = 19.

Therefore, Raina received 19 phone calls on the third evening, 26 phone calls on the first evening, and 38 phone calls on the second evening.

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Final answer:

Pascal's Triangle demonstrates a pattern where the second term in each row relates to the row number raised to a specific power, with the power increasing by one as you move down the rows.

Explanation:

Pascal's Triangle is a mathematical concept that shows a triangular array of binomial coefficients. Each number in the triangle is the sum of the two directly above it.

The relationship between the row number and the second term in each row is that the second term in each row corresponds to the row number raised to the power of 1. For example, in the 3rd row, the second term is the row number (3) raised to the power of 1, which is 3.

As you move down the rows of Pascal's Triangle, the power to which you raise the row number to get the second term of each row increases by one. This pattern continues throughout the triangle.

The percentage of the students writing papers will be 13%.

The analysis of mathematical representations is algebra, and the handling of those symbols is logic.

The quantity of anything is stated as though it were a fraction of a hundred. A quarter of 100 can be used to express the ratio.

During study hall, 58% of the students are studying for tests, 29% of the students are doing homework, and the remaining students are writing papers.

The percentage of the students writing papers will be given by the difference of the 100% and the sum of 58% and 29%. Then we have

⇒ 100% - 58% - 29%

⇒ 100% - 87%

⇒ 13%

The percentage of the students writing papers will be 13%.

More about the Algebra link is given below.

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Final answer:

The percentage of students writing papers during study hall is 13%.

Explanation:

To find out what percentage of students are writing papers during study hall, we need to find the remaining percentage after accounting for the students studying for tests and doing homework.

Since 58% of the students are studying for tests and 29% of the students are doing homework, the percentage of students writing papers can be found by subtracting the sum of these two percentages from 100%:

Percentage of students writing papers = 100% - (58% + 29%) = 13%

Therefore, 13% of the students are writing papers during study hall.

Answer:

The total cost of the meal, rounded to the nearest cent, is approximately $82.30.

Explanation:

First, let's calculate the discount amount:

Discount = 15% of $78.38

         = 0.15 * 78.38

         ≈ $11.76

Now, let's find the discounted meal cost:

Discounted meal cost = Original meal cost - Discount

                    = $78.38 - $11.76

                    ≈ $66.62

Next, let's calculate the tip amount based on the original meal cost:

Tip = 20% of $78.38

   = 0.20 * 78.38

   ≈ $15.68

Now, let's find the total cost of the meal, including the discounted amount and the tip:

Total cost = Discounted meal cost + Tip

          ≈ $66.62 + $15.68

          ≈ $82.30

So, the total cost of the meal, rounded to the nearest cent, is approximately $82.30.

Answer:

Option B is correct

Left 1 unit.

Explanation:

According to the graph theory of transformation:

y = f(x+k)=[tex]\left \{ {{k>0 shift graph of y= f(x) left k unit} \atop {k<0} shift graph of y= f(x) right |k| unit} \right.[/tex]

Given the parent function: [tex]f(x)=x^2[/tex]

and the function [tex]g(x)=x^2+2x+1[/tex]

we can write it as:

g(x)= [tex](x+1)^2[/tex]   [ ∴[tex](a+b)^2 = a^2+2ab+b^2[/tex] ]

Therefore, vertex of the graph of the function [tex]g(x)=(x+1)^2[/tex] is 1 units to the left of the vertex of the graph of the function [tex]f(x)=x^2[/tex] .

Answer:

Shift 1 unit left

B is correct

Step-by-step explanation:

Given: The vertex of f(x) shift to g(x)

[tex]f(x)=x^2[/tex]

Vertex of f(x): (0,0)

[tex]g(x)=x^2+2x+1[/tex]

Vertex form: [tex]y=a(x-h)^2+k[/tex]

[tex]g(x)=(x+1)^2[/tex]

Vertex of g(x): (-1,0)

[tex](0,0)\rightarrow (-1,0)[/tex]

Only x-coordinate change and y-coordinate remain same.

[tex]0\rightarrow -1[/tex]

Hence, The vertex of f(x) shift 1 unit left to get vertex of g(x)

Answer:  The filled truth table for p ∨ q is shown below.

Step-by-step explanation:  We are given to fill the truth table for the p ∨ q.

We know that

If any one of p or q is true, then  p ∨ q is also true. And it is false, only when both p and q are false.

The filled truth table is as follows :

p                  q                     p ∨ q

T                  T                        T

T                  F                        T

F                  T                        T

F                  F                        F

Thus, a0 = T, a1 = T, a2 = T  and  a3 = F.

The angle bisectors of the same side interior angles, formed by two parallel lines and a transversal, intersect at a 90-degree angle.

In geometry, when you have two parallel lines intersected by a transversal, the same side interior angles are those which are in the same position on the two lines. If we bisect these angles (i.e., divide them in two equal parts), the angle bisectors will intersect forming an angle that is 90 degrees.

To understand why, note that the same-side interior angles are supplementary (i.e., they sum up to 180 degrees) because they are on the same side of the transversal. So, if we bisect these angles, each pair of congruent angles will add up to 90 degrees. When these bisectors intersect, they form four right angles. Therefore, the angle at the intersection of the bisectors is 90 degrees.

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The rate of return built into the annuity is approximately 4.38%, calculated using the present value formula and solving for [tex]\( r \).[/tex]

To determine the rate of return built into the annuity, we set the present value of the annuity equal to the lump sum option.

The present value formula for an annuity is used, equating the lump sum amount of $13,800,000 to the present value of the annuity.

The formula is:

[tex]\[ 13,800,000 = 1,050,000 \times \frac{1 - (1 + r)^{-20}}{r} \][/tex]

Solving this equation for [tex]\( r \)[/tex], the annual interest rate (rate of return), requires numerical methods or a financial calculator.

Using trial and error or a financial calculator, we find that [tex]\( r \)[/tex] is approximately 4.38%.

This implies that the annuity offers a rate of return of 4.38% per year over the 20-year period, making it equivalent to receiving $13,800,000 today. Thus, the correct answer is option (b) 4.38%.

Penny can arrange the 25 dimes into three piles in 6 different ways.

Penny has 25 dimes and wants to arrange them into three piles, with an odd number of dimes in each pile. In order to do this, let's consider the possible combinations:

Possible combinations:
1. Pile A: 1 dime, Pile B: 1 dime, Pile C: 23 dimes
2. Pile A: 1 dime, Pile B: 3 dimes, Pile C: 21 dimes
3. Pile A: 1 dime, Pile B: 5 dimes, Pile C: 19 dimes
4. Pile A: 1 dime, Pile B: 7 dimes, Pile C: 17 dimes
5. Pile A: 1 dime, Pile B: 9 dimes, Pile C: 15 dimes
6. Pile A: 1 dime, Pile B: 11 dimes, Pile C: 13 dimes

Therefore, Penny can arrange the dimes into three piles in 6 different ways.

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