The errors of quadratic equation is [tex]x=4 \pm \sqrt{24} &[/tex].
The given quadratic equation is [tex]8x^{2}-x-1=0[/tex]
The polynomial equation whose highest degree is two is called a quadratic equation or sometimes just quadratics. It is expressed in the form of:
[tex]\ ax^{2} + bx + c = 0[/tex]where x is the unknown variable and a, b and c are the constant terms.
Standard Form of Quadratic Equation is [tex]ax^{2} + bx + c = 0[/tex] where a,b,c are real number and a≠0
Therefore the roots of the given equation can be found by:
[tex]x=\frac{-b\±\sqrt{b^{2}-4ac } }{2a}[/tex]
The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.
⇒[tex]$$8 x^2-x-1=0$$[/tex]
Multiply both sides by 8.
⇒[tex]8\left(8 x^2-x=1\right)[/tex]
[tex]x^2-8 x=8 & \\x^2-8 x+16=8+16 & \\(x-4)^2=24 & \\x-4=\pm \sqrt{24} & \\x=4 \pm \sqrt{24} &[/tex]
Therefore, the x is [tex]x=4 \pm \sqrt{24} &[/tex].
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Select the statement that is false.
Select one:
a. If 4 is a prime number, then 6 is a prime number.
b. If 4 is a prime number, then 5 is a prime number.
c. If 3 is a prime number, then 5 is a prime number.
d. If 3 is a prime number, then 6 is a prime number
Out of the given statements, statement a. If 4 is a prime number, then 6 is a prime number is the false one. A prime number is a positive integer that has exactly two distinct natural number divisors: 1 and itself.
For example, the number 2 is a prime number because it can only be divided evenly by 1 and 2. 4 is not a prime number because it can be divided evenly by 1, 2, and 4. 6 can be divided evenly by 1, 2, 3, and 6, so it is not a prime number. Therefore, statement a is false because if 4 is a prime number, 6 is not a prime number.
When determining whether a number is a prime number, it is important to remember that 1 is not a prime number. This is because 1 only has one divisor - itself. Any other number that can be divided evenly by more than two distinct natural numbers is not a prime number.
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A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options.
m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠5 + m∠6 =180° m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 = 180°
The statements which are always true in the triangle with interior angles, ∠2, ∠3, and m∠5, and exterior angles, ∠1, ∠4, and ∠5 are;
m∠5 + m∠6 = 180°
m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 180°
What is an exterior angle of a triangle?An exterior angle is an angle that is formed by the extension of a side of the triangle and the side adjacent to the extended side.
The interior angles of the triangle are; angle ∠2, angle ∠3, and angle ∠5
The exterior angles at the interior angles of the triangle ∠2, ∠3, and ∠5 are angles ∠1, angle ∠4, and angle ∠6 respectively.
Therefore, according to the exterior angle theorem, we get;
m∠1 = m∠3 + m∠5
m∠4 = m∠2 + m∠5
m∠6 = m∠2 + m∠3
The exterior angle and the adjacent internal angle are supplementary angles, therefore;
m∠2 + m∠1 = 180°
m∠3 + m∠4 = 180°
m∠5 + m∠6 = 180°
The angle sum property of a triangle indicates that the sum of the interior angles of a triangle is 180°, therefore;
m∠2 + m∠3 + m∠5 = 180°
The statements that are always true are therefore;
m∠5 + m∠6 = 180°
m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 = 180°
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Math help linear function
Answer:
Choice B
Step-by-step explanation:
The y increases by 3 every time that the x increases by 1.
Hope this helped! :)
Which of the following grids correctly graphs the points A (3, 1), B (0, 3), and C (1, 3)?
A grid that correctly graphs the points A (3, 1), B (0, 3), and C (1, 3) include the following: B. A coordinate plane has an x-axis from 0 to 5 in increments of 1 and a y-axis from 0 to 5 in increments of 1. Point A is plotted 3 units to the right and 1 unit above the origin. Point B is plotted 3 units above the origin. Point C is plotted 1 unit to the right and 3 units above the origin.
What is a graph?In Mathematics, a graph simply refers to a type of chart that is typically used for the graphical representation of ordered pairs (data points) on both the horizontal and vertical lines of a cartesian coordinate, which indicates the x-axis (x-coordinate) and y-axis (y-coordinate) respectively.
Next, we would use an online graphing calculator to plot the above set of data points with a point of intersection at (2, 3) as shown in the graph attached below.
By critically observing graph B of the given set of data points, we can logically deduce that it correctly graphs the points A(3, 1), B(0, 3), and C (1, 3) as shown in the image attached below.
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Complete Question:
Which of the following grids correctly graphs the points A (3, 1), B (0, 3), and C (1, 3)?
A. A coordinate plane has an x-axis from 0 to 5 in increments of 1 and a y-axis from 0 to 5 in increments of 1. Point A is plotted 3 units to the right and 1 unit above the origin. Point B is plotted 3 units to the right of the origin. Point C is plotted 1 unit to the right and 3 units above the origin.
B. A coordinate plane has an x-axis from 0 to 5 in increments of 1 and a y-axis from 0 to 5 in increments of 1. Point A is plotted 3 units to the right and 1 unit above the origin. Point B is plotted 3 units above the origin. Point C is plotted 1 unit to the right and 3 units above the origin.
C. A coordinate plane has an x-axis from 0 to 5 in increments of 1 and a y-axis from 0 to 5 in increments of 1. Point A is plotted 1 unit to the right and 3 units above the origin. Point B is plotted 3 units above the origin. Point C is plotted 3 units to the right and 1 unit above the origin.
D. A coordinate plane has an x-axis from 0 to 5 in increments of 1 and a y-axis from 0 to 5 in increments of 1. Point A is plotted 1 unit to the right and 3 units above the origin. Point B is plotted 3 units to the right of the origin. Point C is plotted 3 units to the right and 1 unit above the origin.
A company sells snow globes made out of glass that are in the shape of hollow
spheres. Nolan measures the outer diameter of a snow globe to be 30 cm. If
the glass has a thickness of 0.5 cm, find the total volume of glass that makes
up the globe. Round your answer to the nearest hundredth if necessary.
(Note: diagram is not drawn to scale.)
The total volume of glass that makes up the globe to the nearest hundredth is equal to 12,771.71 cubic centimeters.
How to calculate the volume of a sphere?Mathematically, the volume of a sphere can be calculated by using this mathematical expression:
Volume = 4/3 × πr³
Where:
r represents the radius.
Based on the information provided about the globe, the diameter of the hollow sphere is given by;
Diameter of sphere = 30 cm
Radius = diameter/2 = 30/2 = 15 cm.
Actual radius = 15 cm - 0.5 cm = 14.5 cm.
Substituting the given points into the volume of a sphere formula, we have the following;
Volume of sphere = 4/3 × 3.142 × 14.5³
Volume of sphere = 12,771.71 cubic centimeters.
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The figure shows intersecting lines k and m. What is the measure of angle a?
The figure shows intersecting lines k and m, the measure of angle a is 94 degrees.
What are parallel lines ?
parallel lines can be defined as the lines which are equi distant from each other and which never meet or intersect each other.
Given,
In the figure the lines k and m are intersecting.
so,
by adding 86 degrees with a it should result into 180 degrees.
86+a = 180
a = 180 - 86
a = 94
Hence, The figure shows intersecting lines k and m, the measure of angle a is 94 degrees.
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Question 7 of 10
Which of the following presents the correct order of buttons to press on a
scientific calculator to enter 2.8 x 107?
OA. [2] [] [8] [EE] [7]
OB. [EE] [7] [2] [.] [8]
OC. [2] [] [8] [7] [EE]
O D. [EE] [7] [2] [.] [8]
The required order of buttons to press on a scientific calculator to enter 2.8 x 10⁷ is [2] [.] [8] [EE] [7] . Option A is correct.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
While using the calculator, for the expression 2.8 × 10⁷,
We have to tap on tabs as, [2] [.] [8] [EE] [7] where [.] is the multiplication operator and {EE} is the exponential operator.
Thus, the required order of buttons to press on a scientific calculator to enter 2.8 x 10⁷ is [2] [.] [8] [EE] [7] . Option A is correct.
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Damian has a balance of $10,200 on his credit card. He threw the card away so he can never use it again. He has 3 1/2 years to pay off the balance. The interest rate on his card is 21%.
a. At the end of the 3 1/2 years, how much interest has he paid? ________
b. When Damian pays off his credit card, how much will he have paid in all? _____
a. At the end of the 3 1/2 years, the amount of interest Damian has paid is $4,288.73.
b) When Damian pays off his credit card, the total amount (principal and interest) he has paid will be $14,488.73.
How is the interest determined?The interest is computed using an online finance calculator, setting the following parameters, based on the compounding interest system.
The calculator also shows the total payments made for the duration of the credit balance.
N (# of periods) = 42 months (3.5 years x 12)
I/Y (Interest per year) = 21%
PV (Present Value) = $10,200
FV (Future Value) = $0
Results:
PMT = $344.97
Sum of all periodic payments = $14,488.73
Total Interest = $4,288.73
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Assuming the workers worked a five day week and worked 50 out of the 52 weeks each year, How many stones did they set each day.
If there were 50 workers, they would set 2,500 stones each day.
What is total number?Total number is a term used to describe the sum of all the values or items within a set. It is usually used to calculate an aggregate or grand total, such as the sum of all the numbers in a list or the total number of people in a population. Total number can also be used to describe the absolute value or quantity of something, such as the total number of days in a year.
Assuming each worker set the same number of stones each day, the total number of stones set each day would be 50 times the number of workers. For example, if there were 20 workers, they would set 1,000 stones each day.
The number of stones set each day could also vary depending on the number of workers and the number of days they worked in a week. For example, if the workers worked a six day week and worked 50 out of the 52 weeks each year, the total number of stones set each day would be 50 times the number of workers multiplied by six. For example, if there were 20 workers, they would set 12,000 stones each day. If there were 50 workers, they would set 30,000 stones each day.
The number of stones set each day can also be affected by the production rate of the workers, the quality of the stones, and the amount of time required to set each stone. If the production rate was low, the number of stones set each day would also be lower. Similarly, if the quality of the stones was poor, the workers would likely take longer to set each stone, thus reducing the number of stones set each day.
Overall, the exact number of stones set each day depends on the number of workers, the number of days they work each week, their production rate, and the quality of the stones.
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Select the correct answer. What is the equation of a line passing through (-6, 5) and having a slope of 1 3 ? A. y = 1 3 x − 23 3 B. y = 1 3 x − 3 C. y = 1 3 x + 7 D. y = 3x + 7 Reset Next
Answer:
C
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
given m = [tex]\frac{1}{3}[/tex] , then
y = [tex]\frac{1}{3}[/tex] x + c ← is the partial equation
to find c substitute (- 6, 5 ) into the partial equation
5 = [tex]\frac{1}{3}[/tex] (- 6) + c = - 2 + c ( add 2 to both sides )
7 = c
y = [tex]\frac{1}{3}[/tex] x + 7 ← equation of line
A point P is on the terminal side of angle θ. Evaluate six trigonometric functions for θ.
P(2, -7)
The exact values of the trigonometric functions for an angle are:
Sine: - 7√53 / 53 Cosine: 2√53 / 53 Tangent: - 7 / 2 Cotangent: - 2 / 7 Secant: √53 / 2 Cosecant: - √53 / 7How to derive the six trigonometric functions of the terminal side of an angle
The direction of an angle (θ), in degrees, with respect to origin is determine by coordinates on the terminal side. For a point in rectangular format we can determine the exact values of two fundamental trigonometric functions (sine, cosine) and four derivate trigonometric functions (tangent, cotangent, secant, cosecant:
For P(x, y) = (x, y)
Sine
sin θ = y / √(x² + y²)
Cosine
cos θ = x / √(x² + y²)
Tangent
tan θ = sin θ / cos θ
Cotangent
cot θ = 1 / tan θ
Secant
sec θ = 1 / cos θ
Cosecant
csc θ = 1 / sin θ
If we know that x = 2 and y = - 7, then exact values of trigonometric functions are, respectively:
Sine
sin θ = - 7 / √[2² + (- 7)²]
sin θ = - 7 / √53
sin θ = - 7√53 / 53
Cosine
cos θ = 2 / [2² + (- 7)²]
cos θ = 2 / √53
cos θ = 2√53 / 53
Tangent
tan θ = [- 7√53 / 53] / [2√53 / 53]
tan θ = - 7 / 2
Cotangent
cot θ = 1 / (- 7 / 2)
cot θ = - 2 / 7
Secant
sec θ = 1 / (2 / √53)
sec θ = √53 / 2
Cosecant
csc θ = 1 / (- 7 / √53)
csc θ = - √53 / 7
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y=sqrt(x), y=0, x=0, x=2
Cross sections are equilateral triangles and are perpendicular to the x-axis.
The police department in your city was asked by the mayor’s office to estimate the cost of crime. The police began their study with burglary records, taking a random sample of 500 files. If the average dollar loss in a burglary, for this sample size 500 is $678, with a standard deviation of $560, construct the 95% confidence interval for the true average dollar loss in burglaries rounded to the nearest dollar amount.
The 95% confidence interval for the true average dollar loss in burglaries rounded to the nearest dollar amount is given as follows:
($629, $727).
What is a z-distribution confidence interval?The bounds of the confidence interval are given as follows:
[tex]\overline{x} \pm z\frac{\sigma}{\sqrt{n}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.z is the critical value.n is the sample size.[tex]\sigma[/tex] is the standard deviation for the population.The confidence level is of 95%, hence the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.
The parameters for this problem are of:
[tex]\overline{x} = 678, \sigma = 560, n = 500[/tex]
Hence the lower bound of the interval is of:
[tex]678 - 1.96\frac{560}{\sqrt{500}} = 629[/tex]
The upper bound of the interval is of:
[tex]678 + 1.96\frac{560}{\sqrt{500}} = 727[/tex]
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What is the value of k in the function f(x)=2-k/5x-k if it passes through the point (3,-2/19)
The value of k from the equation is k = 4
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
f ( x ) = ( 2 - k ) / ( 5x + k ) be equation ( 1 )
Now , the equation passes through the point P ( 3 , -2/19 )
On simplifying the equation , we get
Substituting the values of x and y in the equation , we get
y = ( 2 - k ) / ( 5 ( 3 ) + k ) )
-2/19 = ( 2 - k ) / ( 15 + k )
Multiply by ( 15 + k ) on both sides of the equation , we get
-2 ( 15 + k ) / 19 = 2 - k
Multiply by 19 on both sides of the equation , we get
-30 - 2k = 38 - 19k
Adding 2k on both sides of the equation , we get
38 - 17k = -30
Subtracting 38 on both sides of the equation , we get
17k = 68
Divide by 17 on both sides of the equation , we get
k = 4
Hence , the value of k = 4
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How would I simplify [tex]\frac{2}{1+\sqrt{5} }[/tex] (in simplified radical form) without just turning it into its reciprocal?
Answer:
[tex]\displaystyle \frac{\sqrt{5}-1}{2}[/tex]
Step-by-step explanation:
[tex]\displaystyle \frac{2}{1+\sqrt{5}}\\\\=\frac{2}{1+\sqrt{5}}*\frac{1-\sqrt{5}}{1-\sqrt{5}}\\ \\=\frac{2(1-\sqrt{5})}{1-5}\\ \\=\frac{2(1-\sqrt{5})}{-4}\\\\=\frac{\sqrt{5}-1}{2}[/tex]
now complete your visual overview by identifying the known variables and the variables you must find. assume charge 1 is located at the origin of the x axis and the positive x axis points to the right. let x1 , x2 , and x3 denote the positions of charge 1, charge 2, and charge 3, respectively. determine which of the following quantities are known and which are unknown.
The known quantities are then identified using the visual overview as a guide, and they are q1 q2 q3 x1 and x2. The unknown quantity is then x3
Based on the details provided;
e charge 1 is located at the origin of the x-axis and the positive x-axis points to the right. Let x1, x2, and x3 denote the positions of charge 1, charge 2, and charge 3, respectively
Considering that charge 1 is thought to be at the source;
The known quantities are then identified using the visual overview as a guide, and they are:
q1 q2 q3 x1 and x2
However, given that we are unsure of its precise position x3,
The unknown quantity is then x3
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Solve for z. (10+z)-4= 18 Select the number from the drop down menu to correctly complete the statement. The value of z is 4 6 12 32
Answer:HEYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY TOMATOE TOMATOE TOMATOE BOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
Step-by-step explanation:
divide using long division
(3r³ +11r²-6r-18)÷(r + 4)
Answer:
To divide using long division, we first divide the leading term of the dividend (3r³) by the leading term of the divisor (r) to get the first term of the quotient (3r²). Then, we multiply the divisor (r + 4) by the quotient term (3r²) and subtract it from the dividend. We then bring down the next term of the dividend (11r²) and repeat the process. This will give us the quotient and the remainder.
The long division process is as follows:
(r + 4) | (3r³ +11r²-6r-18)
| 3r² (3r²)
|__
| -9r² +11r²-6r-18
|____
| -2r-18
|____
| +12r + 72
|____
| -36
|____
So the quotient is 3r² - 2r + 12 and the remainder is -36
And we can write the division as (3r³ +11r²-6r-18)÷(r + 4) = 3r² - 2r + 12 +(-36)/(r + 4)
Suppose f(x) = x2. What is the graph of g(x) = f(2x)?
The required function g(x) = 4x² and the related graph is a parabola that is opened upwards on the x-axis
Graph of a function?The collection of all points in the plane with the form (x, f(x)) that make up a function f's graph. This means that when we replace x - coordinate with a numerical value in the expression, we get the y-value for the given function f(x).
Here we have
f(x) = x²
Given that g(x) = f(2x)
To find the g(x) take x = 2x
=> f(2x) = (2x)²
=> g(x) = 4x²
Therefore,
The required function g(x) = 4x² and the related graph is a parabola that is opened upwards on the x-axis
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please help will mark BRAINLIEST
Consider two completely different data sets: price per gallon of gas in Fort Pierce and SAT scores of students
at a certain high school.
The price per gallon of gas data set has a mean of $2.05 and a standard deviation of $1.04.
The high school SAT scores data set has a mean of 1014 and a standard deviation of 160.
Calculate the Coefficient of Variation for both data sets. Round solutions to one decimal place, if necessary.
Coefficient of Variation for the price per gallon of gas data set:
Coefficient of Variation for high school SAT scores data set:
Which data set has greater variability? Price per gallon of gas
Thus, we see that when comparing two data sets, the data set with the larger standard deviation
necessarily have greater variabilty.
does not
CV stands for standard deviation divided by sample mean multiplied by 100.
How are CV and SD calculated?By applying the formula, she assesses: Example calculation: CV = standard deviation / sample mean x 100CV stands for standard deviation divided by sample mean multiplied by 100. CV also stands for volatility divided by expected return multiplied by 100.Price Variation (CV): The CV is the difference between the budgeted amount and the earned value of the task that was completed (Actual Cost). EV-AC = CV. - Schedule Variance (SV): The SV is the discrepancy between the earned value of work that has been completed and the planned value of work that has been scheduled. SV = EV – PV.CV = (std. deviation*100)/mean = 41.2 %
for SAT score:
CV = (std. deviation*100)/mean = 17.55%
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George said if 2 and 4 are factors of a number, then 8 is a factor of the number. Is he correct?Explain...
Answer:
yes
Step-by-step explanation:
the mid point of AB is M(-4,-3). If the coordinates of A are (-1,-2), what are the coordinates of B?
If the mid point of AB is M (-4,-3). and the coordinates of A are (-1,-2), the coordinates of B is solved to be (-7, -4)
What is midpoint of a line?The midpoint of a line segment is the centroid of the segment. It is equally distant from both of the ends and hence cuts the section in half.
How to find the second point of the line segmentThe formula to determine the center point of a straight line using the coordinates of its endpoints is known as the midpoint formula in coordinate geometry.
The formula is of the form
X= (x₂ + x₁)/2
Y = (y₂ + y₁)/2
From the formula the second end point is calculated
-4= (-1 + x₁)/2
-8 = -1 + x₁
x₁ = -7
-3 = (-2 + y₁)/2
-6 = -2 + y₁
y₁ = -4
The coordinates of B are (-7, -4)
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The parallelogram shown on the grid represents a mirror. The mirror hangs on a
wall that is covered with square tiles. Each grid square represents ane tile The side
length of each square tile is 4 in. What is the area of the mirror? Show your work
The area of the mirror is 480 square inches
How to determine the area of the mirrorFrom the question, we have the following parameters that can be used in our computation:
A mirror that has the shape of a parallelogram (see attachment)
From the attachment, we have the following dimensions
Base length = 5 units
Height = 6 units
The area is the product of the dimensions
Since the side length of each square tile is 4 in, then we have
Area = 4 * 4 * 5 * 6
Evaluate
Area = 480
Hence, the area is 480 square inches
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Help help help help help help help
Answer: with ?
Step-by-step explanation:
The sum of two numbers is 8 more than four times the first number. Their difference is 10 less than three times the second number. Find each of the numbers.
Answer:
first number= -2 and the second number = 2
Step-by-step explanation:
x= one number and y= the other
your two equations are x+y=4x+8 and x-y=3y-10
by solving the first one for y, y=3x+8 so plug that in for y in the second equation
x-(3x+8)=3(3x+8)-10
x-3x-8=9x+24-10
-2x-8=9x=14
11x=-22 so x=-2
now plug that in and solve for y
-2+y=4(-2)+8
y=2
Solve the following question:
|p+2|=5
Write an equation of the ellipse with foci (0,+-4) and co-vertices at (±4,0).
When the ellipse has co-vertices at (4,0) and foci at (0,+-4), the equation is y2/32+x2/16=1.
what is equation ?The algebraic equation y=mx+b is known as a linear equation. M serves as the y-intercept, and B serves as the slope. The previous clause has two variables, y and x, and is sometimes referred to as a "linear equation with two variables". Bivariate linear equations are those with two independent variables. The following are a few examples of linear equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. When an equation takes the form y=mx+b, with m denoting the slope and b denoting the y-intercept, it is referred to as being linear. The term "linear" refers to an equation having the form y=mx+b, where m stands for the slope and b for the y-intercept.
given
The center is
=(0,0)
The equation is
y2/a2+x2/b2=1
b=4
c=4
a2=b2+c2
=16+16=32
When the ellipse has co-vertices at (4,0) and foci at (0,+-4), the equation is y2/32+x2/16=1.
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The circle shown below has a diameter of 12 centimeters. What is the
approximate area of the shaded sector?
310*
OA. 390 cm²
OB. 102 cm²
C. 97 cm²
OD. 226 cm2
The approximate area of the shaded sector is 97 cm^2
What is area of circle ?
area of circle can be defined as the product of pi and square of radius ,the value of pi is 22/7 or 3.14 .
Given , diameter = 12
We need to convert diameter to r
d =2r
d/2 =r
12/2=r
6 =r
The radius is 6
A= pi *6^2
A = 36 *(3.14)
A = 113.04 cm^2
Now the fraction of the circle that is shaded is 301degrees out of 360
(A circle is 360 degrees)
=310/360 * area of a circle
=310/360 * 113.04
= 97.34
Hence , The approximate area of the shaded sector is 97 cm^2 .
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Find the slope of the line between the following points. (a) x = 3 and x = 3.1 (b) x = 3 and x = 3.01 (c) x = 3 and x = 3.001 Now use algebra to find a simplified rational expression for the slope of the line between (3, f(3)) and (3 + h, f(3 +h)), (h 0). (Simplify your answer completely.) What do these slopes tend toward as h is made closer and closer to 0?
Draw the graph of the function y = f(x) = 2/x between x = 1/2 and x = 4.
The slope is (2/3 - 2/(3+h))/ h and as h is made closer and closer to 0, the slopes of the lines tend toward the derivative of the function f(x) at x = 3.
a) To find the slope of the line between the points (3, f(3)) and (3.1, f(3.1)), we use the formula m = (f(3.1) - f(3)) / (3.1 - 3) = (f(3.1) - f(3)) / 0.1
b) To find the slope of the line between the points (3, f(3)) and (3.01, f(3.01)), we use the formula m = (f(3.01) - f(3)) / (3.01 - 3) = (f(3.01) - f(3)) / 0.01
c) To find the slope of the line between the points (3, f(3)) and (3.001, f(3.001)), we use the formula m = (f(3.001) - f(3)) / (3.001 - 3) = (f(3.001) - f(3)) / 0.001
As h is made closer and closer to 0, the slopes of the lines tend toward the derivative of the function f(x) at x = 3.
Now to find a simplified rational expression for the slope of the line between (3, f(3)) and (3 + h, f(3 +h)), (h 0) :
The slope is equal to (f(3+h) - f(3)) / h
With f(x) = 2/x, we substitute it into the equation above and we get (2/(3+h) - 2/3) / h
The simplified rational expression for the slope is (2/3 - 2/(3+h))/ h
The graph of y = f(x) = 2/x between x = 1/2 and x = 4 :
This is the graph of a hyperbola that is symmetric about the y-axis and passes through the origin (0,0) and as x increases from 1/2 to 4, y decreases from 4 to 1/2.
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