Using the Pythagorean theorem, the smallest diameter of the pipe that will fit the fiber optic line is approximately 4.61 cm (Option C).
To determine the smallest diameter of the pipe that will fit the fiber optic line in rectangular casing BCDE, we need to find the diagonal AC of the rectangle. Since the rectangle is inscribed within circle A, the diameter of the circle will be equal to the diagonal of the rectangle.
Using the Pythagorean theorem, we can find the length of AC:
AC^2 = DE^2 + BE^2
AC^2 = (3 cm)^2 + (3.5 cm)^2
AC^2 = 9 + 12.25
AC^2 = 21.25
AC = √21.25 ≈ 4.61 cm
Therefore, the smallest diameter of the pipe that will fit the fiber optic line is approximately 4.61 cm (Option C).
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To the nearest hundredth, what is the relative frequency of boys who want to go to the water park
To find the relative frequency of boys who want to go to the water park, we need to divide the number of boys who want to go to the water park by the total number of boys in the sample:
Relative frequency = Number of boys who want to go to the water park / Total number of boys
Assuming you have the necessary data, you can compute this value by dividing the number of boys who want to go to the water park by the total number of boys and rounding the result to the nearest hundredth.
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Aiden gave each member of his family a playlist of random songs to listen to and asked them to rate each song between 0 and 10. He compared his family’s ratings with the release year of each song and created the following scatterplot:
What would the linear equation be?
The linear equation from the given scatterplot will be y = -0.1x + 9.
On the given scatterplot we have the song released details on the x-axis and the average rating of the songs by the family members on the y-axis.
To get the linear equation from the given scatterplot we have to find the y-intercept of the equation.
The general form of the equation is y = mx + c
here, m is the slope and c is the y-intercept.
By, the given graph we can say that y is intercepting at the value '9'. So, the y-intercept is 9.
To find the slope we have to take two points,
Let's take two points as (1970, 7) and (1990, 5).
From the points slope = (5-7)/(1990-1970)
= -2/20 = -1/10 = -0.1
So, the equation from the given scatterplot is y = mx+c
So, y = -0.1x + 9.
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PLEASE HELP ASAP (BRAINLIEST) 39 POINTS
Answer:
3x + 5 = 4x - 40
x = 45
5y - 50 = y
4y = 50
y = 25/2
In ΔMNO, n = 88 inches, m = 60 inches and ∠M=38°. Find all possible values of ∠N, to the nearest 10th of a degree.
answer is 64. 6 and 115. 4 delta
In ΔMNO, possible values of ∠N are 64.6° and 115.4°.
To find the possible values of ∠N, follow these steps:
1. Since the sum of angles in a triangle is 180°, we first find ∠O by subtracting ∠M from 180°: 180° - 38° = 142°.
2. Next, we use the Law of Sines to find the sine of ∠N: sin(∠N) = (n * sin(∠O)) / m = (88 * sin(142°)) / 60.
3. Solve for sin(∠N), which gives us two possible values: sin(∠N) ≈ 0.8988 and sin(∠N) ≈ -0.8988.
4. Find the inverse sine (arcsin) of both values to get the possible angles for ∠N: arcsin(0.8988) ≈ 64.6° and arcsin(-0.8988) ≈ 115.4° (adding 180° to the negative result).
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Casey is going to wear a gray sportscoat and is trying to decide what tie he should wear to work. In his closet, he has 22 ties, 13 of which he feels go well with the sportscoat. If Casey selects one tie at random, determine the probability and the odds of the tie going well or not going well with the sportscoat.
The probability the tie goes well with the jacket is?
The probability the tie will not go well with the jacket is?
The odds against the tie going well with the jacket is?
(Simplify your answer. Type an integer or a fraction.)
The odds in favor of the tie going well with the jacket is?
(Simplify your answer. Type a ratio using a colon.)
Martin collected data from students about whether they played a musical Instrument. The table shows his
results.
Instrument
No Instrument TOTAL
Boys
42
70
112
Girls
48
88
TOTAL
90
110
200
Of the students surveyed, how many played an instrument?
The number of students surveyed who played an instrument is
Out of the students surveyed, 90 played a musical instrument.
To find the total number of students who played a musical instrument, we need to look at the table provided and sum up the number of boys and girls who played an instrument.
From the table, we can see the following:
- Boys who played an instrument: 42
- Girls who played an instrument: 48
To find the total number of students who played a musical instrument, simply add the number of boys and girls together:
Total students who played an instrument = (Number of boys who played an instrument) + (Number of girls who played an instrument)
Total students who played an instrument = 42 (boys) + 48 (girls)
Total students who played an instrument = 90
So, of the students surveyed, 90 played a musical instrument.
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Answer and solution please (Quickly)
Answer:
p=3
Step-by-step explanation:
A linear function that represents the number of animals adopted from the shelter is compared to a different linear function that represents the hours volunteers work at the shelter each week. describe the key features of the functions that are needed to determine if these lines intersect.
please help i don't understand >.
To determine if two lines intersect, compare their slopes and y-intercepts, and solve their equations simultaneously.
How to determine if two lines intersect?To determine if two lines intersect, you need to compare their key features, such as their slope and y-intercept.
If the slopes of the two lines are different, then they will intersect at some point.
If the slopes are the same, then the lines may or may not intersect, depending on whether or not their y-intercepts are also the same.
To find the point of intersection, you can set the two linear functions equal to each other and solve for the variable. The resulting value will give you the x-coordinate of the intersection point, which can then be substituted back into either equation to find the corresponding y-coordinate.
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Pls help me it’s due tmrw
Answer:
Problem 8:
Option C) DE = √21
Problem 9:
Option C) Perimeter = 30 cm
Step-by-step explanation:
1. Problem 8
Draw a line segment connecting the center of the circle, X, to A. This is the radius of the circle
The points ABX form a right triangle with AX as the hypotenuse and AB, BX as the legs of the right triangle
By the Pythagorean theorem
hypotenuse² = sum of the squares of the two legs
Plugging in line segment references
AX² = AB² + BX²
We are given AC = 8, BX = 3
Since the segment BX intersects AC at right angles, AB = BC = AC/2
So AB = 8/2= 4
Plugging these values into the Pythagorean formula
AX² = AB² + BX²
AX² = 4² +3²
AX² = 16 + 9
AX² = 25
Draw a line connecting X and D
The points DEX form a right triangle with DX as the hypotenuse and EX and DE as the legs
Again, by the Pythagorean theorem
DX² = DE² + EX²
But DX is the radius of the circle so it must be the same as AX
Substitute for DX in terms of AX
DX² = DE² + EX²
=> AX² = DE² + EX²
But AX² = 25 and EX = 2 giving EX² = 4
Therefore
AX² = DE² + EX² becomes
25 = DE² + 4
DE² = 25 - 4
DE² = 21
DE = √21
This is choice C
Problem 9
To find the perimeter, just add up the individual side lengths:
Perimeter = 10 + 8 + 7 + 5 = 30 cm
Option C
The table shows the number of beads used to make a necklace.
Ginger wants to make a smaller necklace using the same ratio of pink to white beads.
How many different necklaces could Ginger make?
How do you know?
Answer:
The Answer is 5
Step-by-step explanation:
Pink beads : White beads
30 : 35
( 30 / 5 ) : ( 35 / 5 )
6 : 7
To find the number of beads, You divide the number of pink beads available by the number of pink beads in the ratio:
= 30 / 6
= 5 necklaces
hope this helps :)
Draw a rectangle that is 4 squares long and
1/2 of a square wide.
then add up the partial squares to find the area.
multiply to check your answer
The exact area of our rectangle is 2 square units
The total area of the rectangle can be found by adding up the area of each square. We have four unit squares and one half-square, which can be represented as 4 + 1/2. To add these two values, we need to find a common denominator, which is 2.
Thus, we can represent the area of the rectangle as
=> 8/2 + 1/2 = 9/2 square units.
We know that the area of a rectangle is calculated by multiplying its length by its width. Thus, we can represent the area of our rectangle as follows:
A = lw
where A is the area, l is the length, and w is the width.
Now, we need to differentiate this equation with respect to the width w. This means we are finding the rate of change of the area with respect to the width. Using the product rule of differentiation, we get:
dA/dw = l * dw/dw + w * dl/dw
Since the width is constant (it does not change), the second term on the right-hand side of the equation is zero. Thus, we are left with:
dA/dw = l
To calculate the exact area of our rectangle, we can use the concept of limits. We can start by approximating the area of our rectangle with a width of 1 square unit. In this case, the area of the rectangle would be 4 square units.
We can then approach the width of 1/2 of a square unit by taking the limit as the width approaches 1/2. Using the derivative we calculated earlier, we can represent this limit as follows:
lim(w→1/2) A = lim(w→1/2) lw = l * lim(w→1/2) w = 4 * (1/2) = 2 square units
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Find the most important variable in the problem.
If a company hired an additional 12 employees, and every employee needed a
phone, it would require 8 more phones. How many phones does the company
have available now?
OA. the number of employees hired
OB. the money required to purchase phones
OC. the number of phones available
Find a rule for the following table of value
Answer:
y-3x-2
Step-by-step explanation:
take 2 points (1,1), (3,7)
find the slope: 7-1/3-1=3
plug into y=2x+b (used pt (1,1) )
1=3(1)+b
1=3+b
b=-2
y=3x-2
What adds to +10 and multiplys to 290
You are at the beach with your friends. You have brought some supplies to make sand castles. These supplies include a pail that has a base with a circumference of 87 inches, is 12 inches tall, and has an opening on top that is twice the diameter of the base. You also have a plastic pyramid mold that has a square base with an edge that measures 6 inches and is 7 inches tall, and an empty soup can with a diameter of 5. 25 inches and is 6. 5 inches tall
The opening of the pail has a diameter of approximately 55.4 inches (since it's twice the diameter of the base).
Based on the information you provided, it sounds like you have some great supplies for making sand castles at the beach with your friends.
Firstly, let's take a look at the pail. You mentioned that it has a base with a circumference of 87 inches, which means that the diameter of the base is approximately 27.7 inches (since circumference = pi x diameter). The pail is also 12 inches tall and has an opening on top that is twice the diameter of the base. Therefore, the opening has a diameter of approximately 55.4 inches (since it's twice the diameter of the base). With these measurements, you can use the pail to make some pretty big sand castles!
Next, you mentioned a plastic pyramid mold that has a square base with an edge that measures 6 inches and is 7 inches tall. This mold should be perfect for making pyramid-shaped sand castles. Just fill it with sand, pack it down, and carefully remove the mold to reveal your pyramid!
Finally, you mentioned an empty soup can with a diameter of 5.25 inches and is 6.5 inches tall. This can could be used to make cylindrical shapes in your sand castles. Simply pack sand around the can, press down firmly, and carefully remove the can to reveal your cylinder.
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Three-fifths of seventh graders have a cell phone. in a seventh grade class of 450, how many students would you predict to have a cell phone
270 students in a seventh-grade class of 450 would have a cell phone which denotes three-fifths of seventh graders using fractions.
Total number of students = 450
Percent of students who have cell phones = 3/5 th
In a class, if there are three-fifths of students have cell phones, that means we need to calculate the remaining percent of students who did not have cell phones.
Students without cell phones = 1 - 3/5 = 2/5
The total number of students with cell phones = (3/5) x 450
The total number of students with cell phones = 270
Therefore, we can conlcude that 270 students in a seventh-grade class of 450 would have a cell phone.
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Find the line of tangency to the circle defined by (x-3)^2 + (y-7)^2 = 169 at the point (15,2).
first off, let's look at the equation of the circle
[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{}{h}~~,~~\underset{}{k})}\qquad \stackrel{radius}{\underset{}{r}} \\\\[-0.35em] ~\dotfill\\\\ (x-\stackrel{h}{3})^2+(y-\stackrel{k}{7})=169\implies (x-\stackrel{h}{3})^2+(y-\stackrel{k}{7})=\stackrel{ r }{13^2}[/tex]
so we have a circle centered at (3 , 7) with a radius of 13, Check the picture below.
so the line we want is the line in purple, which is tangential to the circle and therefore perpendicular to the blue line.
keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the blue line
[tex](\stackrel{x_1}{3}~,~\stackrel{y_1}{7})\qquad (\stackrel{x_2}{15}~,~\stackrel{y_2}{2}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{2}-\stackrel{y1}{7}}}{\underset{\textit{\large run}} {\underset{x_2}{15}-\underset{x_1}{3}}} \implies \cfrac{ -5 }{ 12 } \implies - \cfrac{5 }{ 12 } \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{-5}{12}} ~\hfill \stackrel{reciprocal}{\cfrac{12}{-5}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{12}{-5} \implies \cfrac{12}{ 5 }}}[/tex]
so we're really looking for the equation of a line whose slope is 12/5 and it passes through (15 , 2)
[tex](\stackrel{x_1}{15}~,~\stackrel{y_1}{2})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{12}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{ \cfrac{12}{5}}(x-\stackrel{x_1}{15}) \\\\\\ y-2=\cfrac{12}{5}x-36\implies {\Large \begin{array}{llll} y=\cfrac{12}{5}x-34 \end{array}}[/tex]
Michael is using a rotating
sprinkler to water his lawn. The
sprinkler rotates in a complete
circle. It sprays water at most 8
feet. Find the area of the lawn
that is watered. Use 3. 14 for π. Show Your Work
The area of the lawn that is watered by the sprinkler is approximately 200.96 square feet.
The area of the field that's doused by the sprinkler is a sector of a circle with a compass of 8 bases and a central angle of 360 degrees.
To find the area of the sector, we can use the formula
Area = ( θ/ 360) x πr2
where θ is the central angle in degrees,
r is the radius of the circle,
and π is the constant pi.
Substituting the given values,
we get Area = (360/360) x3.14 x 82
Area = 3.14 x 64 Area = 200.96
Thus, the area of the field that's doused by the sprinkler is roughly 200.96 square feet.
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The complete question is as follows:
Michael is using a rotating sprinkler to water his lawn. The sprinkler rotates in a complete circle. It sprays water at most 8 feet. Find the area of the lawn that is watered.
PLEASE HELP!!
Line A has a slope of -1/3 and passes through the point (1, 10 1/3). Line B has a slope of 1/3 and passes through the point (-34, -2). Find the point where line A intersects like B.
The point where line A intersects line B is [2, 10].
How to determine an equation of this line?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:
x and y represent the data points.m represent the slope.At data point (1, 10 1/3) and a slope of -1/3, a linear equation for this line can be calculated or determined by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 10 1/3 = -1/3(x - 1)
y - 31/3 = -x/3 + 1/3
For Line B, we have:
y - y₁ = m(x - x₁)
y - (-2) = 1/3(x - (-34))
y + 2 = x/3 + 34/3
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The monthly demand function for a product sold by a monopoly is p = 2104 – (1/3)x^2 dollars, and the average cost is C = 1000 + 22x + x2 dollars. Production is limited to 1000 units and x is in hundreds of units. (a) Find the quantity (in hundreds of units) that will give maximum profit. (b) Find the maximum profit. (Round your answer to the nearest cent.)
a) The quantity (in hundreds of units) that will give maximum profit: [tex]x^{2}[/tex] + 6x - 3022 = 0
b) The maximum profit is approximately $202,573.42.
What is Equation?An equation is a mathematical statement that shows that two expressions are equal. It typically contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
(a) To find the quantity that will give maximum profit, we need to find the level of output at which marginal revenue equals marginal cost.
The total revenue function for the monopoly is TR = px, where p is the price and x is the quantity sold. The marginal revenue is the derivative of the total revenue with respect to x, which is MR = d(TR)÷dx = p + x(dp÷dx).
To find the marginal revenue function, we differentiate the demand function with respect to x:
dp÷dx = -(2÷3)x
So, the marginal revenue function is:
MR = (2104 - (1÷3)[tex]x^{2}[/tex]) - (2÷3)[tex]x^{2}[/tex]
To find the marginal cost function, we differentiate the average cost function with respect to x:
dC÷dx = 22 + 2x
So, the marginal cost function is:
MC = 22 + 2x
To find the level of output at which MR = MC, we set the two functions equal to each other:
(2104 - (1÷3)[tex]x^{2}[/tex]) - (2÷3)[tex]x^{2}[/tex] = 22 + 2x
Simplifying this equation, we get:
[tex]x^{2}[/tex] + 6x - 3022 = 0
(b) Using the quadratic formula, we find that:
x = (-6 ± √( 36- 4(1)(-3022))) / 2(1)
x = (-6 ± √(36444)) / 2
x = (-6 ± 190.81) ÷ 2
x ≈ -98.4 or x ≈ 92.4
Since we can't produce a negative quantity, we choose x ≈ 92.4 as the quantity that will give maximum profit.
Therefore, the level of output that will maximize profit is 924 units (in hundreds of units).
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Una carretera recta forma un angulo de 22° con la horizontal dde cierto punto Q en ella el angulo de elevacion del avion en el punto A 57 en el mismo instante dde otro punto Q a 100 m adelante del primero el angulo de elevacion 63 los puntos P Q A quedan en el plano vertical calcule la distancia de P al avion
The distance from P to the airplane is y = x + 100 ≈ 628.38 m.
What is the triangle?A triangle is a three-sided polygon with three angles. It is a fundamental geometric shape and is often used in geometry and trigonometry.
From a certain point Q on a straight road, which forms an angle of 22° with the horizontal, the angle of elevation of an airplane at point A is 57°. At the same instant from another point Q, 100 meters ahead of the first point, the angle of elevation of the airplane is 63°. The points P, Q, and A are in the same vertical plane. Find the distance from P to the airplane.
To solve the problem, we can use the concept of similar triangles. Let's call H the height of the airplane and x the distance from Q to the airplane. Then, the distance from P to the airplane is given by y = x + 100.
From triangle QA1H, we have:
tan(57°) = H / x
From triangle QA2H, we have:
tan(63°) = H / (x + 100)
Dividing these two equations, we get:
tan(57°) / tan(63°) = x / (x + 100)
Solving for x, we get:
x = 100 * tan(63°) / (tan(63°) - tan(57°)) ≈ 528.38 m
Therefore, the distance from P to the airplane is:
y = x + 100 ≈ 628.38 m.
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can someone please help will give brain
Answer:
y = 9
Step-by-step explanation:
• Parallel lines have equal slopes
calculate the slope m of TU using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = T(- 7, 4 ) and (x₂, y₂ ) = U(- 2, - 6 )
[tex]m_{TU}[/tex] = [tex]\frac{-6-4}{-2-(-7)}[/tex] = [tex]\frac{-10}{-2+7}[/tex] = [tex]\frac{-10}{5}[/tex] = - 2
now calculate the slope of VW and equate to - 2
with (x₁, y₁ ) = V(8, 7 ) and (x₂, y₂ ) = W(7, y )
[tex]m_{VW}[/tex] = [tex]\frac{y-7}{7-8}[/tex] = [tex]\frac{y-7}{-1}[/tex]
now equating gives
[tex]\frac{y-7}{-1}[/tex] = - 2 ( multiply both sides by - 1 to clear the fraction )
y - 7 = 2 ( add 7 to both sides )
y = 9
what dose y equal when the equation is negitive 5 y plus 4 is equal to negitive 11
Answer: y = 1.4
Step-by-step explanation:
if you write the equation it would be
-5y - 4 = -11
so first you would subtract 4 from -4 and -11 to cancel out the four.
so your equation would look like this -5y= -7
so now u would divide -5 by both sides to canceled out the -5
your equation should end up looking like
y=1.4
N equation for the depreciation of a car is given by y = A(1 – r)t , where y = current value of the car, A = original cost, r = rate of depreciation, and t = time, in years. The value of a car is half what it originally cost. The rate of depreciation is 10%. Approximately how old is the car?
3. 3 years
5. 0 years
5. 6 years
6. 6 years
the car is approximately 6.6 years old. The closest option provided is 6 years, so the answer is (C) 6 years.
A car's original value depreciates by 10% per year. If the current value of the car is half of its original value, approximately how old is the car?Given:
y = A(1 – r)t
The value of a car is half what it originally cost, which means:
y = 1/2 A
The rate of depreciation is 10%, which means:
r = 0.1
Substituting these values in the equation, we get:
1/2 A = A(1 – 0.1)t
Simplifying, we get:
1/2 = 0.9t
Solving for t, we get:
t = ln(1/2) / ln(0.9) ≈ 6.6 years
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What’s my gpa?
For school
Answer:
2.2
Step-by-step explanation:
To find your GPA, use the attached image:
After you've written down your numerical scores, divide it by the total classes you're taking, which is 7.
So, your GPA is 2.2
50 POINTS ASAP Triangle 1 and triangle 2 are similar right triangles formed from a ladder leaning against a building.
Triangle 1 Triangle 2
The distance, along the ground, from the bottom of the ladder to the building is 12 feet. The distance from the bottom of the building to the point where the ladder is touching the building is 18 feet. The distance, along the ground, from the bottom of the ladder to the building is 8 feet. The distance from the bottom of the building to the point where the ladder is touching the building is unknown.
Determine the distance from the bottom of the building to the point where the ladder is touching the building for triangle 2.
27 feet
18 feet
12 feet
5 feet
The distance where the ladder is touching the building for triangle 2 is 12 ft
Determining the distance from the bottom of the building to the pointFrom the question, we have the following parameters that can be used in our computation:
Ladder 1
Distance along the ground = 12 ft
Distance touching the ladder = 8 ft
Ladder 2
Distance along the ground = 18 ft
Distance touching the ladder = x
Using proportion of similar triangles, we have
x : 18 = 8 : 12
Express as fraction
x/18 = 8/12
So, we have
x = 18 * 8/12
Evaluate
x = 12
Hence, the distance where the ladder is touching the building for triangle 2 is 12 ft
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Answer:
12?
Step-by-step explanation:
Not too sure! I am in the middle of taking the test right now though
Solve for x: √8x + 4 = 6
The solution to the equation √8x + 4 = 6 is x = 0.5.
What is the value of x?An equation is simply a mathematical formula that expresses the equality of two expressions, using the equals sign as a connection between them.
Given the equation in the question:
√8x + 4 = 6
To solve for x in the equation, isolate the term containing the variable x.
Subtract 4 from both sides of the equation:
√8x + 4 - 4 = 6 - 4
√8x = 6 - 4
√8x = 2
Square both sides of the equation:
( √8x )² = 2²
8x = 4
Divide both sides of the equation by 8:
x = 4/8
x = 1/2
x = 0.5
Therefore, the value of x is 0.5.
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A sequence can be generated by using an=an−1+7, where a1=4 and n is a whole number greater than 1. What are the first 3 terms in the sequence? 7, 11, 15 7, 28, 112 4, 11, 18 4, 28, 196
To generate the sequence, we start with a1 = 4, and then use the formula an = an-1 + 7 for n > 1.
So, to find the first 3 terms of the sequence, we can use the formula:
a2 = a1 + 7 = 4 + 7 = 11
a3 = a2 + 7 = 11 + 7 = 18
The first 3 terms of the sequence are 4, 11, and 18.
So, the answer is 4, 11, 18, which corresponds to the third option.
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If a person drives his car at the speed of 50 miles per hour, how far can he cover in 2.5 hours?
The marginal cost function, in dollars per item, for producing the x th item of a certain brand of bar stool is given by MC(x)=20−0. 5 x , 0≤ x≤ 100. The fixed cost is $200. Estimating the total cost of producing 100 bars tools using the left-rectangle approximation with five rectangles, we conclude that the total cost is approximately $
The total cost of producing 100 bar stools using the left-rectangle approximation with five rectangles is approximately $200.
The marginal cost function for producing the x-th item of a certain brand of bar stool is given by MC(x) = 20 - 0.5x, where 0 ≤ x ≤ 100. To estimate the total cost of producing 100 bar stools using the left-rectangle approximation with five rectangles, we first need to calculate the width of each rectangle.
The width of each rectangle is (100 - 0) / 5 = 20.
Now, we will evaluate the marginal cost function at the left endpoints of each rectangle:
MC(0) = 20 - 0.5(0) = 20
MC(20) = 20 - 0.5(20) = 10
MC(40) = 20 - 0.5(40) = 0
MC(60) = 20 - 0.5(60) = -10
MC(80) = 20 - 0.5(80) = -20
Next, multiply each value by the width (20) and sum the results to approximate the variable cost:
Variable cost ≈ 20(20) + 10(20) + 0(20) + (-10)(20) + (-20)(20) = 400 + 200 - 200 - 400 = 0
Finally, add the fixed cost of $200 to the variable cost:
Total cost ≈ $200 + $0 = $200
So, the total cost of producing 100 bar stools using the left-rectangle approximation with five rectangles is approximately $200.
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