The function is differentiable at these two x-values, the x-values at which the function is differentiable is (-1, 1/3).
The x-values at which the function is differentiable can be found by taking of the function and setting it equal to 0. The derivative of the function y = x3 + 2x2 - 5x + 8 is 3x2 + 4x - 5. Setting this equal to 0 and solving for x yields x = -1 and x = 1/3. So, the x-values at which the function is differentiable is (-1, 1/3).
In order to find the derivative of the function, the power rule was used. The power rule states that the derivative of a function f(x) = xn is f'(x) = nxn-1. Applying this to the function y = x3 + 2x2 - 5x + 8, the derivative is 3x2 + 4x - 5. Next, setting this equal to 0 and solving for x yields two solutions: x = -1 and x = 1/3. Since the function is differentiable at these two x-values, the x-values at which the function is differentiable is (-1, 1/3).
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Describe the mistake that was made in solving the following equation:
The mistake is in the second step, where the sign of the "4" is changed. The real solution is x = -5
Which one is the mistake in the solution of the equation?Here we have the solution of the following equation:
4x + 6 = 2x - 4
First, they subtract 2x from both sides, then we will get:
4x + 6 - 2x = 2x - 4 - 2x
2x + 6 = -4
While notice that in the image, in this step they wrote:
2x + 6 = 4
They changed the sign of the 4, there is the mistake.
Now we can keep solving our equation:
2x + 6 = -4
2x = -4 - 6
2x = -10
x = -10/2
x = -5
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The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space be positive, real numbers. Define the events A and B as follows: A = { x | x < 72.5] and B = { x | x > 52.5]. Describe each of the following events: a. A' b. B' c. A ∩ B d. A ∪ B
the events A and B as follows: A = { x | x < 72.5] and B = { x | x > 52.5].
From the question given above, the following data were obtained:
Angle θ = 71°
Hypothenus = 25
Adjacent = x
Thus, we can obtain the x component of the vector by using the cosine ratio as illustrated below:
Cos θ = Adjacent /Hypothenus
Cos 71 = x/25
Cross multiply for events
x = 25 × Cos 71
x = 25 × 0.3256
x = 8.1
Therefore, the x component of the vector is 8.1
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5(z+6) − 10=10z How do I do this qustion??
Answer: z=4
Step-by-step explanation:
5(z+6) - 10 = 10z
The first step is to distribute the 5 to the inside of the parenthesis. So,
5*z = 5z , 5*6 = 30 .
Now the equation has become 5z+30-10=10z. Next, the like terms can be reduced. So,
30-10= 20
The equation is: 5z+20=10z, 5z will be subtracted on both sides, canceling one side out.
5z-5z=0 , 10z-5z=5z
: 20=5z
The last step is to isolate the variable by dividing 5 on both sides.
5/5 = 0 , 20/5 = 4
Therefore, z=4
Suppose f ( x ) = 3 x 2 + 7 x + 5 . Compute the following:
The values of the function are
a) f(-1)+ f(2) = 32
b) f(-1) - f(2) = -30.
What is Function?In mathematics, a function is represented as a rule that produces a distinct result for each input x. The collection of all the values that the function may input while it is defined is known as the domain. The entire set of values that the function's output can produce is referred to as the range. The set of values that could be a function's outputs is known as the co-domain.
Given:
f( x ) = 3x² + 7x + 5
a) f(-1)+ f(2)
= 3(-1)² + 7(-1) + 5 + 3(2)² + 7(2) + 5
= 3 -7 + 5 +12 + 14 + 5
= 32
So, the value of f(-1)+ f(2)= 32
b) f(-1) - f(2)
= 3(-1)² + 7(-1) + 5 -{ 3(2)² + 7(2) + 5}
= 3 -7 + 5 -{12 + 14 + 5}
=-30
So, the value of f(-1) - f(2)= -30.
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if you are using a simple-moving average and you want your forecast to ignore occasional noise in the data, which of the following is true? a. You want a stable forecast b. You want a responsible forecast c. You use more periods of data. d. You use fewer periods of data e.Only a and c f.Only b and d
if you are using a simple-moving average and you want your forecast to ignore occasional noise in the data, Only A & C are of the following is true .
What is the meaning of the simple moving average?
The average of a range of prices is determined by the number of periods that fall within the range using simple moving averages.
A technical indicator called a simple moving average can help forecast whether a price trend—be it bullish or bearish—will continue or reverse.
What is not necessary for simpler forecasting techniques?
various types and amounts of data are needed for various forecasting methodologies. Less information is necessary for simpler techniques, such as the time series techniques outlined below.
Complex models require a lot of data. It might be expensive to create precise forecasts.
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HELP NOW PLS!! 100PTS!
Marcus loves baseball and wants to create a home plate for his house. Marcus needs to calculate the area of the home plate at the ball field so he can reconstruct it when he gets home. Calculate the area of the polygon.
30 in2
31.5 in2
36.5 in2
39 in2
Answer: 39 in^2
Step-by-step explanation: To find the area of this polygon, we can break it down into two triangles and one rectangle. The following formulas are to solve for the area of each polygon:
Area of a Triangle: A = b x h / 2
Area of a Rectangle: A = l x w
We can plug in the values by what we are given. The triangle to the top has a base of 6 in and a height of 5 in. We can multiply these to get 30 in but cut it in half to get 15 in. The second triangle has a base of 3 in and a height of 4 in, so when we multiply, we get 12 in / 2 to get 6 in. The rectangle's length is 6 in and the width is 3 in. So, 6 x 3 is 18. Add all these values together, and you get 39 in^2.
Let me know if this is right! :)
Question
Which models represent the sum?
1.2 + 0.3
Select each correct answer.
Responses
Ten by 10 grid of 100 squares. The first column and 2 squares of the second column are shaded one color. The next three columns are shaded another color.
Image with alt text: Ten by 10 grid of 100 squares. The first column and 2 squares of the second column are shaded one color. The next three columns are shaded another color.
One column, divided into 10 small squares. Two small squares. Plus sign. Three columns, each divided into 10 small squares.
Image with alt text: One column, divided into 10 small squares. Two small squares. Plus sign. Three columns, each divided into 10 small squares.
Two 10 by 10 grids of 100 squares. All 10 columns of first square and 2 columns of the second square shaded one color. Three columns of the second square shaded a different color.
Image with alt text: Two 10 by 10 grids of 100 squares. All 10 columns of first square and 2 columns of the second square shaded one color. Three columns of the second square shaded a different color.
Large square divided into a 10 by 10 grid of 100 small squares. Two columns, each divided into 10 small squares. Plus sign. Three columns, each divided into 10 small squares.
Image with alt text: Large square divided into a 10 by 10 grid of 100 small squares. Two columns, each divided into 10 small squares. Plus sign. Three columns, each divided into 10 small squares.
Number line from 0 to 4 by tenths. An arrow shows a jump starting at 0 and ending 2 marks past 1. Another arrow shows a jump starting at 2 marks past 1 and ending at 5 marks past 1.
The models that represent the sum 1.2 + 0.3 are
Option 3 and Option 5
Why are they the models?For Option 3:
The first square represents 1.2, with all 10 columns shaded in one color. The next two columns represent 0.3, shaded in a different color. The sum of 1.2 and 0.3 is 1.5, which is represented by the combination of the two squares.
For Option 5:
On the number line, the first arrow represents 1.2, starting from 0 and ending 2 marks past 1, which is equal to 1.2. The second arrow represents 0.3, starting from 2 marks past 1 and ending at 5 marks past 1, which is equal to 1.5. The sum of 1.2 and 0.3 is 1.5, which is represented by the end point of the second arrow on the number line.
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Pizza is available with three kinds of cheese and four toppings. How many different ways can you order your pizza with one kind of cheese and two toppings?
Answer:
Therefore you will be able to get 6 * 3 different pizzas = 18 pizza combinations.
Answer: 12
Step-by-step explanation:
Graph the inequality. y is less than negative one fourth times x minus 3 The graph shows a solid line passing through the point negative 12 comma 0 and 0 comma negative 3, with shading above the line. The graph shows a dashed line passing through the point negative 12 comma 0 and 0 comma negative, 3 with shading above the line. The graph shows a solid line passing through the point negative 12 comma 0 and 0 comma negative 3, with shading below the line. The graph shows a dashed line passing through the point negative 12 comma 0 and 0 comma negative 3, with shading below the line.
A graph of the inequality "y is less than negative one fourth times x minus 3" is: D. The graph shows a dashed line passing through the point negative 12 comma 0 and 0 comma negative 3, with shading below the line.
How to determine the required ordered pair?In order to determine the ordered pair that is included in the solution to this linear inequality, we would first of all translate the word problem into an algebraic equation as follows;
y is less than negative one-fourth times x minus 3 ⇒ y < -x/4 - 3.
Next, we would use an online graphing calculator to graphically solve the linear inequality. Therefore, the required solution for the given linear inequality is the shaded area below the dashed line, which is given by the ordered pairs (−12, 0) and (0, -3).
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Answer:
it’s D dotted line with shading below
Step-by-step explanation:
i got it on my quiz
components a, b, c, and d function properly with probabilities 0.6, 0.7, 0.8, and 0.9 respectively. calculate the reliability of the system show in the figure. assume the components are independent
(a) The reliability of this system is 0.42
(b) The reliability of this system is 0.8376
As per the given data that the probabilities that components A, B, C, and D function properly are 0.6, 0.7, 0.8, and 0.9 respectively.
Let the name of the components itself denote the event that the corresponding component works properly.
So P(A) = 0.6, P(B) = 0.7, P(C) = 0.8, P(D) = 0.9.
We have to find the reliability of the given structure of the combination of the above-said components.
The diagrammatic representation(attached at the end of solution(a)) of the system is as follows.
The above system consists of components A and B connected in the sequel. So the reliability of the system is the probability that both A and B function properly.
This probability is given by P(A ∩ B) = P(A)P(B) as the events are independent. Substitute the probabilities in the above formula.
P(A ∩ B) = (0.6)(0.7)
P(A ∩ B) = 0.42
The diagrammatic representation(attached at the end of solution(b)) of the system is as follows.
In the above system, the components A and B and components C and D are connected the in the sequel. The entire system is connected in parallel with A, B and C, D. The reliability of this system is given by the probability.
P(A ∩ B) ∪ P(C ∩ D) = P(A ∩ B) + P(C ∩ D) - P(A ∩ B) ∩ P(C ∩ D)
= P(A)P(B) + P(C)P(D) - P(A)P(B)P(C)P(D)
= (0.6)(0.7) + (0.8)(0.9) - (0.6)(0.7)(0.8)(0.9)
= 0.42 + 0.72 - 0.3024
= 1.14 - 0.3024
= 0.8376
Therefore answer is 0.8376
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My bill at the hair salon came up to 110$ if I tip my stylist 20% what will my total bill be
Answer:
Step-by-step explanation:
110 x 0.20 = 22
your total is 132
A bike tire just ran over a nail, and it is losing pressure at a rate of 10% every minute. The tire pressure is currently 1,060 kilopascals. What will it be in 4 minutes?
Answer:
699.626 kilopascals after 4 minutes
Step-by-step explanation:
We can use the formula for simple interest to calculate the tire pressure after 4 minutes. Let's call the initial pressure "P", the rate of loss "r", and the time elapsed "t".
P = P * (1 - r)^t
Where r is the rate of loss in decimal form and t is the number of minutes elapsed.
Substituting the given values, we have:
P = 1,060 * (1 - 0.10)^4
P = 1,060 * (0.9)^4
P = 1,060 * 0.6561
P = 699.626 kPa
Therefore, the tire pressure will be 699.626 kilopascals after 4 minutes.
Barry took out a 20-year loan for $55,000 at an APR of 6.8%, compounded
monthly, and he is making monthly payments of $419.84. Assuming that his
balance is $31,019.97 with 8 years left on the loan, how much would he save
by paying off the loan 8 years early?
The solution is, A = $94193.421., would he save by paying off the loan 8 years early.
What is interest?Interest is the price you pay to borrow money or the cost you charge to lend money. Interest is most often reflected as an annual percentage of the amount of a loan. This percentage is known as the interest rate on the loan.
here, we have,
We must determine the amount owed for the first 8 years.
The monthly compound interest formula is:
A = P(1+r/12)^12t - Xnt
given,
P = $55,000
r = 6.8%
t = 8
Xnt = $419.84
solving we get,
A = 94193.421
Hence, The solution is, A = $94193.421.
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Find the correct volume and round to the nearest tenth. Please I need help!!! Answer both questions
The volume of the given cylinder would be = 1.33mm³
How to calculate the volume of cylinder?The formula that can be used to calculate the volume of the cylinder = v = πr²h
Where,
π = 3.14
r= 0.5mm
h= 1.7 mm
V = 3.14×0.5²×1.7
V= 3.14× 0.25×1.7
V= 1.3345
V= 1.33mm³ ( to the nearest tenth)
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What is the next fraction in this sequence? Simplify your answer.
2 4
8 16
27.
3, 9, 27, 81, ...
Submit
Answer: 32/243
Step-by-step Explanation: multiply 16/81 by 2/3
The expression 7n - 19 represents the perimeter of the triangle.
a. What is the length of the third side?
b. What are the lengths of the three sides of the triangle when n = 5? Is the triangle a right triangle? Explain.
c. What are the lengths of the three sides of the triangle when n = 7? Is the triangle a right triangle? Explain.
a) The length of the third side of the triangle is; 6n - 9
b) The lengths of the three sides of the triangle when n = 5 are;
8, 7 and 22.
c) The lengths of the three sides of the triangle when n = 7 are;
12, 13 and 33.
How to find the Perimeter of a Triangle?The perimeter of a triangle is defined as the total length of the boundary of the triangle.
The parameters are;
Perimeter of triangle = 7n - 19
Length of side 1 = 2n - 2
Length of side 2 = 3n - 8
a) Thus;
Length of third side = 7n - 19 - (2n - 2 + 3n - 8)
= 7n - 19 - n + 10
= 6n - 9
b) When n = 5, the lengths of the three sides are;
Length of side 1 = 2(5) - 2 = 8
Length of side 2 = 3(5) - 8 = 7
Length of third side = 6(5) - 9 = 22
c) When n = 7, the lengths of the three sides are;
Length of side 1 = 2(7) - 2 = 12
Length of side 2 = 3(7) - 8 = 13
Length of third side = 6(7) - 9 = 33
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in each of problems 1 through 4: a. draw a direction field and sketch a few trajectories. b. express the general solution of the given system of equations
The general solution is an equation that describes all possible solutions of the system. Finally, we can use the general solution to find the specific solutions to the system.
1. [tex]\frac{dy}{dt} = t - y, \frac{dx}{dt} = y - t$a.[/tex]
b. [tex]$y^2 - 2ty + x = c$[/tex]
2.[tex]$\frac{dy}{dt} = y - t^2, \frac{dx}{dt} = t^2 - x$a.[/tex]
b. [tex]$y^2 - 2tx + x^2 - t^2 = c$[/tex]
3. [tex]$\frac{dy}{dt} = ty - x, \frac{dx}{dt} = y - tx$[/tex]
a.
b. [tex]$y^2 - 2tx + x^2 = c$[/tex]
4.[tex]$\frac{dy}{dt} = xy, \frac{dx}{dt} = x$[/tex]
a.
b. [tex]$y = cxe^{x^2/2}$[/tex]
In each of the four problems, we are given a system of equations. To solve the system, we first need to draw a direction field and sketch a few trajectories. This gives us a visual representation of the system, which helps us understand it better. Next, we need to find the general solution of the system. To do this, we need to solve the system of equations algebraically. The general solution is an equation that describes all possible solutions of the system. Finally, we can use the general solution to find the specific solutions to the system.
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(100 points) Corey spies a bald eagle in a tall tree. He estimates the height of the tree to be 80 feet and the angle of elevation to the bird from where he stands to be 65°. The leaves on the tree make it difficult for Corey to watch the bird, so he moves away from the tree 53 feet to get a better view.
What is his new angle of elevation to the bird? Round your answer to the nearest hundredth of a degree.
Answer:
The answer should be should be 51.45°
Find the median and mean of the data set below: 21 , 40 , 48 , 26 , 32 , 31
What does the problem asks for:
--> mean and median
Let's calculate the mean:
[tex]\text{Mean}=\dfrac{\text{sum of all the numbers}}{\text{number of numbers given}}[/tex]
--> the sum of all the numbers = 21 + 40 + 48 + 26+ 32 + 31 = 198
--> number of numbers given: 6
==> let's calculate
[tex]\text{Mean}=\dfrac{198}{6} =33[/tex]
Let's calculate the median
--> Median definition: middle number in an ordered sequence
==> let's order the sequence: 21, 26, 31, 32, 40, 48
--> since there are an even number of numbers, there isn't an exact
middle number
--> thus, we must find the mean of the two numbers closest to the
middle
[tex]\text{Median} = \dfrac{31+32}{2} =\dfrac{63}{2} =31.5[/tex]
Answer: 31.5
Step-by-step explanation:
21, 26, 31, 32, 40, 48
Median
31+32= 63
63÷2= 31.5
Mean
21+26+31+32+40+48=198
198÷6=33
The value of an electronic music device decreases by 11% each year. The device was purchased for $125. What will its value be after 8 years?
The cost of an electronic music device after 8 years is, $59.83
What is Simple interest?A quick and easy method of calculating the interest charge on a loan is called a Simple interest.
We have to given that;
The value of an electronic music device decreases by 11% each year.
And, The device was purchased for $125.
We know that;
The formula as the value of the machine depreciates every year by 11%,
⇒ A = P (1 - r)ⁿ
Here, P = $125
r = 11% = 0.11
n = 8 years
Substitute all the values, we get;
⇒ A = P (1 - r)ⁿ
⇒ A = 125 (1 - 0.11)⁸
⇒ A = 125 (0.89)⁸
⇒ A = 125 × 0.3936
⇒ A = $59.83
Thus, The cost of an electronic music device after 8 years is, $59.83
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a rectangular field is to be enclosed by 400 feet of fence. one side of the field is a building, so fencing is not required on that side. rectangular field with one side against a building. if x denotes the length of one side of the rectangle perpendicular to the building, determine the function in the variable x giving the area (in square feet) of the fenced-in region. area, as a function of x
The function A(x) = x(400 - x) gives the area (in square feet) of the fenced-in region as a function of the length of the side of the rectangle perpendicular to the building (x).
The area of a rectangular field enclosed by 400 feet of fence can be calculated using the formula Area = Length x Width. Since one side of the field is against a building, fencing is not required on that side. Thus, the length of the rectangle perpendicular to the building (x) is one of the sides of the rectangle. The other side is calculated by subtracting x from 400 feet. The area (in square feet) of the fenced-in region is then given by the function A(x) = x(400 - x).
To illustrate, if the length of the rectangle perpendicular to the building (x) is 200 feet, then the width is calculated by subtracting 200 from 400, giving a width of 200 feet. The area of the fenced-in region is then given by A(x) = 200 x (400 - 200) = 200 x 200 = 40,000 square feet.
Therefore, the function A(x) = x(400 - x) gives the area (in square feet) of the fenced-in region as a function of the length of the side of the rectangle perpendicular to the building (x).
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solve for 1/x(x-3)=2/x-3+3/x
To solve for 1/x(x-3) = 2/(x-3) + 3/x, we can simplify and rearrange the terms on the right side:
1/x(x-3) = 2/(x-3) + 3/x
Multiplying both sides by x(x-3):
1 = 2x + 3x(x-3)/x
Expanding the right side:
1 = 2x + 3x^2 - 9x/x
Cancelling x from the right side:
1 = 2x + 3x^2 - 9
Dividing both sides by x^2:
1/x^2 = 2/x + 3 - 9/x^2
Rearranging the terms:
1/x^2 - 9/x^2 = 2/x + 3
Isolating the x terms:
-8/x^2 = 2/x + 3
Multiplying both sides by x^2:
-8 = 2x + 3x^2
Expanding the right side:
-8 = 2x + 3x^2
Rearranging the terms:
3x^2 + 2x + 8 = 0
This is a quadratic equation and can be solved using the quadratic formula or factoring. The solutions are the values of x that make the equation equal to zero.
true/false. an experiment was run to compare four groups. the standard deviations were 4974, 6585, 3288, 5488. in this case, it is reasonable to assume all groups have equal variances.
False. It is not reasonable to assume all groups have equal variances in this case.
The standard deviation is a measure of variability or spread of the data. If the standard deviations of different groups are significantly different, then it suggests that the spread of the data within each group is also different. When comparing multiple groups, equal variances are typically assumed so that a valid statistical test can be performed.
If the variances are unequal, then a different type of test, such as Welch's ANOVA, should be used. In this case, the large difference in the standard deviations of the four groups (4974, 6585, 3288, 5488) suggests that it is not reasonable to assume equal variances and a different test should be used.
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Shawn would like to accumulate $460,000 for his retirement in 11 years. If he is promised a rate of 4.59% compounded semi-annually by his local bank, how much should he invest today?
The amount he should invest today is $279,229.23.
What is compound interest?It is the interest we earned on the interest.
The formula for the amount earned with compound interest after n years is given as:
A = P [tex](1 + r/n)^{nt}[/tex]
P = principal
R = rate
t = time in years
n = number of times compounded in a year.
We have,
A = $460,000
r = 4.59%
t = 11 years
Semi-annually.
So,
n = 2
Now,
A = P [tex](1 + r/n)^{nt}[/tex]
460,000 = P [tex](1 + 0.0459/2)^{22}[/tex]
P = 460,000 / [tex]0.02295^{22}[/tex]
P = $279,229.23
Thus,
He should invest $279,229.23 today.
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Answer the question that follows based on the graph to the right 
1) The independent variable is ''Day'' shown in x - axis.
2) Days shows on x - axis are incorrectly arranged.
What is Coordinates?A pair of numbers which describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines is called the coordinates.
We have to given that;
The graph is shown in graph.
On the graph;
Day is shown in x - axis.
And, Earnings shown in y - axis.
Now, We know that;
The independent variable is shown in x - axis.
Hence, The independent variable is Day.
And, One wrong thing in graph is,
Days shows on x - axis are incorrectly arranged.
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HELP ASAP ON HW PLEASE I BEEGGGGGGGGGFF PLEASEEE
There are 1049 children went to movie.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Let the number of children are x and number of adults were y.
Tickets are $5.50 for children and $7 for adults.
1250 are the total number of people attended the movie.
x+y=1250
5.50x+7y=7979
x=1250-y
Now plug in x value in the second equation.
5.5(1250-y)+7y=7979
6875-5.5y=7979
Subtract 6875 on both sides
-5.5y=7979-6875
-5.5y=1104
y=-200.7
Now plug in y value
x+(-200.7)=1250
x=1049
Hence, there are 1049 children went to movie.
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What are the lengths of the sides of the trapezoid shown at the right if the perimeter of the trapezoid is 17 cm?
The side lengths of Trapezoid are 4 cm, 5 cm, 4 cm and 4 cm.
What is Perimeter?The perimeter of a form is the space surrounding its edge.
A closed path that covers, encircles, or outlines a one-dimensional length or a two-dimensional shape is called a perimeter.
Given:
Perimeter of Trapezoid = 17 cm
So, the Perimeter of Trapezoid
17 = Sum of parallel sides + Sum of non parallel side
17 = x + x+ 1 + 2√x + 2√x
17 = 2x + 4√x + 1
2x + 4√x = 16
2x + 4√x - 16 = 0
x + 2√x = 8
2√x = 8- x
Now, Squaring both side
4x = 64 + x² - 16x
x² - 16x -4x + 64 = 0
x² - 20x + 64 = 0
x² - 16x - 4x + 64 = 0
x(x-16) -4( x- 16)= 0
x= 16, 4
So, the sides lengths are
x= 4 cm, x+1 = 5 cm, 2√x = 4 cm, 2√x = 4 cm.
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Noah found a shirt at the nearby clothing store that was 20 percent off. The total price after the discount is $58.20, which includes $3.80 in taxes.
The original price of the shirt was $68
How to find the original price?We know that the price after a discount of the 20% and an added $3.80 in taxes is $58.20
Then we can write:
$58.20 = P*(1 - 0.2) + $3.80
Where the factor (1- 0.2) corresponds to a discount of 20%.
Solving that for P we will get:
P = ($58.20 - $3.80)/0.8
P = $68
That is the original price of the shirt.
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(3x)^-3 show work pls
Answer: -81x
Step-by-step explanation:
3x times -3= -9x
-9x times -3= 27x
27x times -3= -81x
Select all of the linear transformations from ℝ3 to ℝ3 that are invertible. There may be more than one correct answer.A. Identity transformationB. Projection onto the xz-planeC. Reflection in the y-axisD. Rotation about the x-axisE. Dilation by a factor of 6F. Projection onto the z-axis
A linear transformation is a function from one vector space to another that preserves each vector space's underlying (linear) structure. A linear transformation can also be referred to as a linear operator or map.
The correct answers are A, D, and E.
A linear transformation from ℝ3 to ℝ3 is invertible if and only if it is bijective, meaning it is both one-to-one and onto.
The following linear transformations are invertible:
A. Identity transformation: The identity transformation maps every vector to itself and is one-to-one, so it is invertible.
D. Rotation about the x-axis: A rotation about the x-axis is a bijective linear transformation that maps each point in ℝ3 to another point in ℝ3 in a one-to-one manner, so it is invertible.
E. Dilation by a factor of 6: A dilation by a factor of 6 scales every vector by 6, and the scaling factor is nonzero, so this transformation is bijective and invertible.
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