The coordinates of the focus of the parabola y = -(1/8)x² - 2x - 4 is (-8, 2).
What are the coordinates of the focus of the parabola?y = -(1/8)x² - 2x - 4
Given by the equation y = -(1/8)x² - 2x - 4,
To find the focus of the parabola we first need to put the equation in standard form:
y = a(x - h)² + k
Where (h, k) is the vertex of the parabola, and a is a constant that determines the shape and size of the parabola.
Completing the square on the x terms, we get:
y = -(1/8)(x² + 16x + 64) - 4 + 8
= -(1/8)(x + 8)² + 4
Comparing this to the standard form, we see that the vertex is (-8, 4) and a = -1/8.
Since the coefficient of x² is negative, the parabola opens downwards.
The focus of the parabola is located at a distance of 1/(4a) units from the vertex, on the axis of symmetry, which is a vertical line passing through the vertex.
Substituting the value of a, we get:
= 1/(4a)
= 1/(4(-1/8))
= 2
Therefore, the focus of the parabola is located 2 units below the vertex, on the line x = -8.
So the focus has coordinates (-8, 2).
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Let v = - 8i+2j, and w=-i-5j. Find 6w + 9v.
6w +9v = (Type your answer in terms of i and j.)
..
The value of the given expression in terms of i and j is -78i-12j.
Given that, v = - 8i+2j and w = -i-5j.
We need to find the value of 6w+9v.
Substitute v = - 8i+2j and w = -i-5j in 6w+9v, we get
6(-i-5j)+9(-8i+2j)
= -6i-30j-72i+18j
= -78i-12j
Therefore, the value of the given expression in terms of i and j is -78i-12j.
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PLEASE PLEASE HELP DUE IN A FEW HOURS ILL GIVE U ANYTHING
What age represents the median of this data? (1 point)
4. What age(s) represent the mode? (1 point)
5. Beth is 63 years old. She loves to bike. She decides to look at other clubs to join. Why do you think she didn’t want to join this club? Explain your answer using the stem-and-leaf plot above. (1 point)
Are there any outliers in this data? If so, what age(s) are they? (Hint: you’ll need to find the quartiles first. Show your work.) (4 points)
Beth could not join the club because the highest age which is allowed here is only 55 years
We know that;
A statistical expression obtained from a list of data that refers an abnormal gap from other values.
And, The statistical rules that instruct us to divides the data or observation values into four parts.
After analyzing the stem leaf diagram, we noticed the youngest age allowed to join the club is 10 years and the club allow highest 55 years old to join.
the plot also defines that the number of members is declined with the age. There is only person in the club who is 55 years. There are few members with the age over 40 years.
hence, Beth did not join the club because her age was above the highest age that allowed in this club.
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if u are given the midpoint of a segment and one endpoint find the other endpoint (6,2) (1,3)
The other endpoint is (11, 1).
We have,
Midpoint = (6, 2) and endpoint = (1, 3).
and, ratio of m: n = 1 :1
Using section formula
6 = ( 1 + a) / (1+1)
6 = (1+a)/2
1 + a = 12
a = 11
and, 2 = (3 + b) / (1+1)
2 = (b + 3) /2
b +3 = 4
b = 1
Thus. the other endpoint is (11, 1).
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On a coordinate plane, a parabola opens down. It has an x-intercept at (negative 5, 0), a vertex at (negative 1, 16), a y-intercept at (0, 15), and an x-intercept at (3, 0).
The function f(x) = –x2 − 2x + 15 is shown on the graph. What are the domain and range of the function?
The domain is all real numbers. The range is {y|y < 16}.
The domain is all real numbers. The range is {y|y ≤ 16}.
The domain is {x|–5 < x < 3}. The range is {y|y < 16}.
The domain is {x|–5 ≤ x ≤ 3}. The range is {y|y ≤ 16}
The domain is all real numbers.
The range is {y|y ≤ 16}.
Option B is the correct answer.
We have,
The given parabola opens downwards and has a vertex at (-1,16).
So,
The parabola has a maximum value at y = 16.
Also, the y-intercept of the parabola is (0,15), which is below the vertex, and the parabola intersects the x-axis at (-5,0) and (3,0).
We can use the factored form of the equation of a parabola to find its equation in this case.
The factored form is:
y = a(x - h)² + k
where (h,k) is the vertex and "a" determines whether the parabola opens upwards or downwards.
Since the parabola opens downwards, "a" is negative.
Also, we know that the vertex is (-1,16), so we have:
y = a(x + 1)² + 16
To find "a", we can use one of the x-intercepts, say (-5,0).
Substituting these values into the equation gives:
0 = a(-5 + 1)² + 16
0 = 16a
a = 0
This indicates that the parabola is actually a line passing through (0,15) and (3,0). Therefore, the equation of the parabola is:
y = -x² - 2x + 15
The domain of this function is all real numbers because there are no restrictions on the values of x that can be plugged into the equation.
However, the range of the function is limited by the maximum value of y, which is 16.
Since the parabola opens downwards, all y values less than or equal to 16 are attainable.
Therefore, the domain is all real numbers, and the range is {y | y ≤ 16}.
Therefore,
The domain is all real numbers.
The range is {y|y ≤ 16}.
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The radius of a circle with area A is approximately $\sqrt{\frac{A}{3}}$
. The area of a circular mouse pad is 45 square inches. Estimate its radius to the nearest tenth.
in.
The radius of the circular mouse is 3.7 inches.
How to find the radius of a circle?The area of the circular mouse pad 45 square inches. Therefore, the radius of the circle can be found as follows:
area of the circle = πr²
where
r = radiusTherefore,
area of the circular mouse = 3.14 × r²
45 = 3.14r²
divide both sides by 3.14
r² = 45 / 3.14
r² = 14.3312101911
square root both sides
r = √14.331210911
r = 3.78565714243
r = 3.7 inches
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50 POINTS ANSWER FOR BRAINLIST SHOW YOUR WORK
Answer:
a) Two real solutions.
b) Solutions: y = -6, y = 12
Step-by-step explanation:
The discriminant is defined as the expression b² - 4ac, which appears under the square root sign in the quadratic formula.
The value of the discriminant determines the nature of the solutions to the quadratic equation:
[tex]\boxed{\begin{minipage}{12 cm}\underline{Discriminant}\\\\$b^2-4ac$ \quad when $ax^2+bx+c=0$\\\\when $b^2-4ac > 0 \implies$ two real solutions.\\when $b^2-4ac=0 \implies$ one real solution.\\when $b^2-4ac < 0 \implies$ no real solutions (two complex conjugate solutions).\\\end{minipage}}[/tex]
To find the number and type of solutions of the given quadratic equation, first rewrite the equation in the form ax² + bx + c = 0.
[tex]\begin{aligned}(y-3)^2-10&=71\\y^2-6y+9-10&=71\\y^2-6y-1&=71\\y^2-6y-72&=0\end{aligned}[/tex]
Compare the coefficients:
a = 1b = -6c = -72Substitute the values of a, b and c into the discriminant formula:
[tex]\begin{aligned}b^2-4ac&=(-6)^2-4(1)(-72)\\&=36-4(-72)\\&=36+288\\&=324\end{aligned}[/tex]
As the discriminant of the given equation is greater than zero, there are 2 real solutions.
To solve the equation, we can use the quadratic formula:
[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\quad\textsf{when}\;\;ax^2+bx+c=0[/tex]
As we have already calculated that the discriminant is 324, we can substitute this, along with the values of a and b, into the formula:
[tex]\implies y=\dfrac{-(-6) \pm \sqrt{324}}{2(1)}[/tex]
[tex]\implies y=\dfrac{6 \pm \sqrt{324}}{2}[/tex]
Solve for y:
[tex]\implies y=\dfrac{6 \pm 18}{2}[/tex]
[tex]\implies y=3 \pm 9[/tex]
[tex]\implies y=-6, 12[/tex]
Therefore, the solutions of the given equation are:
y = -6y = 12Fwam and Eva are both driving along the same highway in two different cars to a stadium in a distant city. At noon, Fwam is gar miles away from the stadium and Eva is 396 miles away from the stadium. Fwam is driving along the highway at a speed of 42 miles per hour and Eva is driving at speed of 65 miles per hour. Let F represent Fwam's distance, in miles, away from the stadium t hours after noon. Let E represent Eva's distance, in miles, away from the stadium t hours after noon. Write an equation for each situation, in terms of t, and determine how far both Fwam and Eva are from the stadium at the moment they are an equal distance from the stadium.
Fwam and Eva are an equal distance of 69 miles away from the stadium when the time t = 3 hours
Given data ,
The formula for the distance (F) of Fwam from the stadium is:
F(t) = 327 - 42t
Since Fwam is travelling at a 42 mph pace, his distance from the stadium is reducing at a 42 mph rate.
The formula for Eva's separation from the stadium is:
E(t) = 396 - 65t
Eva's distance from the stadium also shrinks at a pace of 65 miles per hour as she drives at a speed of 65 mph.
Set F(t) = E(t) and solve for t to get the time at which Fwam and Eva are equally far from the stadium
On simplifying the equations , we get
327 - 42t = 396 - 65t
Adding 65t to both sides:
65t + 327 - 42t = 396
23t + 327 = 396
Subtracting 327 from both sides:
23t = 396 - 327
23t = 69
Dividing both sides by 23:
t = 69 / 23
t = 3 hours
Hence , at t = 3 hours after noon, both Fwam and Eva are an equal distance of 69 miles away from the stadium.
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The complete question is attached below :
Fwam and Eva are both driving along the same highway in two different cars to a stadium in a distant city. At noon, Fwam is gar miles away from the stadium and Eva is 396 miles away from the stadium. Fwam is driving along the highway at a speed of 42 miles per hour and Eva is driving at speed of 65 miles per hour. Let F represent Fwam's distance, in miles, away from the stadium t hours after noon. Let E represent Eva's distance, in miles, away from the stadium t hours after noon. Write an equation for each situation, in terms of t, and determine how far both Fwam and Eva are from the stadium at the moment they are an equal distance from the stadium.
Construct a sinusoidal function that rises from a minimum point at (3,-2) to a maximum point at (7, 8)
The sinusoidal function that rises from a minimum point at (3,-2) to a maximum point at (7, 8) is y = 5 sin(π/2(x - 3)) + 3
We are given that;
Minimum point= (3,-2)
Maximum point= (7, 8)
Now,
To construct a sinusoidal function that rises from a minimum point at (3,-2) to a maximum point at (7, 8), we can use the following steps:
Find the amplitude A. The amplitude is the distance from the midline of the function to the maximum or minimum point. The midline is the average of the maximum and minimum values, which is (8 + (-2))/2 = 3. The distance from 3 to 8 or -2 is 5, so A = 5.
Find the period P. The period is the length of one cycle of the function, or the horizontal distance between two consecutive maximum or minimum points. In this case, the period is 7 - 3 = 4. The constant B is related to the period by the formula B = 2π/P, so B = 2π/4 = π/2.
Find the horizontal shift C. The horizontal shift is the amount that the function is shifted left or right from its standard position. In this case, we want the function to have a minimum point at x = 3, so we need to shift it right by 3 units. This means that C = 3.
Find the vertical shift D. The vertical shift is the amount that the function is shifted up or down from its standard position. In this case, we want the function to have a midline at y = 3, so we need to shift it up by 3 units. This means that D = 3.
Putting it all together, we get:
y = 5 sin(π/2(x - 3)) + 3
Therefore, by the function the answer will be y = 5 sin(π/2(x - 3)) + 3.
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What is the value of z in the triangle?
Enter your answer in the box. Round your final answer to the nearest hundredth.
Answer:
z = 12.52 in.
Step-by-step explanation:
Because this is a right triangle, we can solve for z using on the trigonometric ratios.
If we allow 37° to be our reference angle, the 10 in. side is the adjacent side and z (directly across from the right angle) is the hypotenuse.
Thus, we can use the cosine ratio which is
[tex]cos(angle)=\frac{adjacent}{hypotenuse}[/tex]
Now, we can plug everything into the formula and solve for z, the hypotenuse:
[tex]cos(37)=10/z\\z*cos(37)=10\\z=10/cos(37)\\z=12.52135658\\z=12.52[/tex]
Write an explicit formula for an, the nth term of the sequence 36, 32, 28, ....
WORTH 20
Answer:
an = 36 + -4(n - 1)
Step-by-step explanation:
Explicit formula:
[tex]a_n=a_1+(n-1)d[/tex]
Where:
an = nth terma1 = first termd = common differenceHere, we are given the following:
Sequence: 36, 32, 28The first term is 36
To get to the next term, 4 is subtracted from the previous term so the common difference is -4.
So we have:
First term(a1) = 36Common difference = -4By plugging this into [tex]a_n=a_1+(n-1)d[/tex]
We acquire : [tex]a_n=36+(n-1)(-4)[/tex]
i need the answer of this 9th grade question
The amount of fabric used to make the number cube is given as follows:
C. 294 cm².
How to obtain the amount of fabric?The amount of fabric used to make the number cube is represented by the surface area of the number cube.
The surface area of a cube of side length a is given by the equation presented as follows:
S = 6a².
The side length for this problem is given as follows:
a = 7 cm.
Hence the surface area of the cube is calculated as follows:
S = 6 x 7²
S = 294 cm².
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Select the correct answer. Which statement correctly describes this expression?\left|x^3\right|+5 A. the sum of the absolute value of three times a number and 5 B. 5 more than the absolute value of the cube of a number C. the cube of the sum of a number and 5 D. the absolute value of three times a number added to 5
The expression |x³| + 5 is 5 more than the absolute value of the cube of a number.
Option B is the correct answer.
We have,
The expression |x³| represents the absolute value of the cube of a number x.
This means that no matter what value x takes (positive, negative or zero), the result of the expression will always be positive.
For example, if x = 2, then |x³| = |2³| = |8| = 8.
If x = -2, then |x³| = |-2³| = |-8| = 8.
Adding 5 to the absolute value of the cube of x means adding 5 to the positive number that resulted from |x³|.
So, if we take the examples from above, if x = 2, then the expression becomes |2³| + 5 = 8 + 5 = 13.
If x = -2, then the expression becomes |-2³| + 5 = 8 + 5 = 13.
Therefore,
The expression |x³| + 5 is 5 more than the absolute value of the cube of a number.
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What is the answer to this question
The maximum profit that can be made is $1737.82
What is an equation?An equation is an expression showing the relationship between numbers and variables using mathematical operators.
The profit function is the difference between the revenue and cost of a commodity. Given that:
Revenue, R(x) = 700x - 11.3x²
Cost, C(x) = 8068 - 34.25x
Profit, P(x) = Revenue - Cost
P(x) = (700x - 11.3x²) - (8068 - 34.25x)
P(x) = 665.75x - 11.3x² - 8068
The maximum profit is at P'(x) = 0; hence:
P'(x) = 665.75 - 22.6x
665.75 - 22.6x = 0
22.6x = 665.75
x = 29.45
P(29.45) = 665.75(29.45) - 11.3(29.45)² - 8068
P(29.45) = 1737.82
The profit to be made is $1737.82
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What is the slope of the line that passes through the points (2,-4) and (5,2)
Answer:
2.
Step-by-step explanation:
To find the slope of the line that passes through the points (2,-4) and (5,2), we can use the slope formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) = (2,-4) and (x2, y2) = (5,2)
slope = (2 - (-4)) / (5 - 2)
slope = 6 / 3
slope = 2
Therefore, the slope of the line that passes through the points (2,-4) and (5,2) is 2.
What is the area of this figure?
17 m
15 m
4 m
6 m
5 m
Write your answer using decimals, if necessary.
5 m
7m
9 m
The calculated value of the area of the figure is 230.5 sq meters
What is the area of this figure?From the question, we have the following parameters that can be used in our computation:
The composite figure
The area is the sum of the individual areas
Using the above and the area formulas as a guide, we have the following:
Area = 5 * 5 + (6 + 5) * 4 + 1/2 * 17 * (15 + 4)
Evaluate
Area = 230.5
Hence, the area is 230.5 sq meters
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what is the average rate of change for the following intervals?
[-5,-4]
[-4,-3]
[-4,-1]
[-3,-1]
The average rate of change for the given intervals are:
(-5, -5) and (-4, 0): 0(-4, 0) and (-3, 3): 3(-4, 0) and (-1, 3): 1(-3, 0) and (-1, 0): 0How to find the average rate of changeTo compute the average rate of change for any given interval, determine the change between the function values at the endpoints and divide this value by the difference in independent variable values. it can be expressed as follows:
Average Rate of Change = (f(b) - f(a)) / (b - a)
The average rate of change for each point in the interval is determined form the graphs and used to solve for the average rate of change
(-5, -5) and (-4, 0):
Average Rate of Change = (0 - 0) / (-4 - (-5)) = 0 / 1 = 0
(-4, 0) and (-3, 3):
Average Rate of Change = (3 - 0) / (-3 - (-4)) = 3 / 1 = 3
(-4, 0) and (-1, 3):
Average Rate of Change = (3 - 0) / (-1 - (-4)) = 3 / 3 = 1
(-3, 0) and (-1, 0):
Average Rate of Change = (0 - 0) / (-1 - (-3)) = 0 / 2 = 0
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Triangle A and B are similar. Triangle A is shown below. The shortest side of triangle B is 10cm. What is the length of the other two sides?
Using the concept of similar triangles, the length of the other two sides of triangle B are: 20 cm and 25 cm
How to find the lengths of similar triangles?Similar triangles are defined as triangles that possess the same shape, but then their sizes may very well vary. This means that, if two triangles are similar, then we can say that their corresponding angles are congruent and corresponding sides are in equal proportion.
Since the shortest side of triangle B is 10cm, it means that it corresponds to the shortest side of triangle A which is 8 cm.
Thus:
Ratio of corresponding sides of B:A = 10/8
Thus:
Second side of triangle B: (10/8) = (x/16)
x = 160/8
x = 20 cm
Third side of triangle B: 10/8 = y/20
y = 200/8
y = 25 cm
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1. Joe has a chocolate box whose shape resembles a rectangular prism. Its length is 6 in, height is 2 in and width is 4 in. Find the volume of the box.
length= 5 in , width= 4 in, height= 3 in
Answer:
48 square inches
Step-by-step explanation:
lxwxh
6x4x2
Diego buys water bottles for his softball team, he buys 7 cases, each with t bottles. Diego buys a total of 168 bottles. Create an equation that models this situation, where t represents the number of bottles per case purchased
U={31,32,33……,50}
A={35,38,41,44,46,50}
B={32,36,40,44,48}
C= {31,32,41,42,48,50}
Find AorB
Find BandC
In the given sets, the values of A or B and B and C are:
A or B = {32, 35,36, 38, 40, 41, 44, 46, 48, 50}
B and C = {32, 48}
Set theory: Determining the intersect and union of A and BFrom the question, we are to determine the values of A or B and B and C in the given set.
From the question,
The universal set is
U = {31,32,33……,50}
and
A={35,38,41,44,46,50}
B={32,36,40,44,48}
C= {31,32,41,42,48,50}
A or B means we should find the union of A and B. That is, the elements present in A or B or both
A or B = {32, 35,36, 38, 40, 41, 44, 46, 48, 50}
B and C means we should find the intersect of B and C. That is, the elements that present both in B and C
B and C = {32, 48}
Hence,
A or B = {32, 35,36, 38, 40, 41, 44, 46, 48, 50}
B and C = {32, 48}
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Ellie is going to see a movie and is taking her 2 kids. Each movie ticket costs $13 and there are an assortment of snacks available to purchase for $5 each. How much total money would Ellie have to pay for her family if she were to buy 4 snacks for everybody to share? How much would Ellie have to pay if she bought x snacks for everybody to share?
Total cost with 4 snacks:
Total cost with x snacks:
The total cost that Ellie would pay with 4 snacks is $59.
The total cost that Ellie would pay with x snacks is $29 + $5x.
What is the total cost?The first step is to determine the total number of people going to the movies. There are 3 people going to the movies.
The total cost spent by Ellie is the sum of the tickets and the total cost of the snacks.
Total cost = total cost of the tickets + total cost of the snacks
= (cost of one ticket x number of people going to the movie) + (number of snacks bought x price of one snack)
= (13 x 3) + ($5 x 4)
$39 + $20 = $59
Total cost if x snacks is bought: (13 x 3) + ($5 x x) = $39 + $5x
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Write a polynomial function with the given zeros and their correspond possible answers. f(x) = Zeros 0 -6 -7 Mult. 1 3 1
[tex] - 13(t - 7) {6(t + 1) {8(t - 2)}^{?} }^{?} [/tex]
The value of a polynomial function with the given zeros would be,
⇒ f (x) = x (x + 7) (x + 6)³
What is mean by Function?A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given that;
Functions have zeroes 0, - 6, - 7 with multiplicity 1, 3, 1
Hence, We can formulate;
The value of a polynomial function with the given zeros would be,
⇒ f (x) = (x - 0)¹ (x - (- 6))³ (x - (- 7))¹
⇒ f (x) = x (x + 6)³ (x + 7)
⇒ f (x) = x (x + 7) (x + 6)³
Thus, The value of a polynomial function with the given zeros would be,
⇒ f (x) = x (x + 7) (x + 6)³
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The figure below is a right rectangular prism with
rectangle ABCD as its base.
What is the area of the base of the rectangular prism? 2, 9, 18 sq centimeters
What is the height of the rectangular prism? 2, 6, 9 centimeters
What is the volume of the rectangular prism? 17, 54, 108 cubic centimeter
Area of rectangular base is 18 square centimeters.
The height (given) is 6 centimeters.
The volume of the rectangular prism is 108 cubic centimeters.
For the area of the base, multiply the length of AD times length of CD
9cm × 2cm = 18cm² remember that is square unit measure:
cm² means square centimeters.
The height is the distance between the two bases. Given as AW = 6
To get Volume, multiply the area of one base by the height
18cm² × 6cm = 108cm³
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An electrician has 42.3 meters of wire to use on a job. On the first day, she uses 14.742 meters of the wire. How many meters of wire does she have remaining after the first day?
There are 27.558 meters of wire does she have remaining after the first day.
We have to given that;
An electrician has 42.3 meters of wire to use on a job.
And, On the first day, she uses 14.742 meters of the wire.
Hence, We get;
The remaining wire does she have remaining after the first day is,
⇒ 42.3 - 14.742
⇒ 27.558 meters
Thus, There are 27.558 meters of wire does she have remaining after the first day.
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(c) log, 12-( log, 9+ log 1 3 log, 8), write a single form equation
Help. Confused. Pic below
Answer:
Step-by-step explanation:
I agree it's convenience and biased.
It's the easiest way to convey a survey so it is convenient.
it is biased because not everyone has accessible to modern technology, or that particular social media app. Not all park users use social media. It is biased because it is targeting a younger demographic, but excludes the youngest demographic as well, that are not allowed ot have social media.
Answer:
I agree it's convenience and biased.
It's the easiest way to convey a survey so it is convenient.
it is biased because not everyone has accessible to modern technology, or that particular social media app. Not all park users use social media. It is biased because it is targeting a younger demographic, but excludes the youngest demographic as well, that are not allowed ot have social media.
Step-by-step explanation:
An hourglass is 6ft wide and 4 ft tall on top. On bottom the hourglass is 6ft wide and 2 feet tall. It can hold 2000 pounds. if the water takes exactly one year to pass through the hourglass, how much water pass through to the bottom each day? What is the linear function of the water passing through the hourglass?
The linear equation outlining the entire volume of water (y) which has coursed through the hourglass following x days. is
y = 5.48xThe volume of water passed each which is the slope is
5.48How to find the amount of water passing through the glass each dayTo ascertain the amount of water that is passing through the hourglass daily:
where:
Total water capacity: 2000 pounds
Time: 365 days
Water transpiring every day = Total water capacity / Time
Water pervading each day ≈ 200 / 365
Water pervading each day ≈ 5.48 pounds/day
the linear function of the water traveling through the hourglass.
Making x the number of days and y being the total quantity of water which has crossed through the hourglass after x days--as the fluid fluxes at a constant velocity--thus a linear relationship between x and y can be determined.
Every single day (x), the hourglass moves 5.48 pounds of water. This reveals the linear representation of:
y = 5.48x
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The graph shown corresponds to someone who makes
Total earnings
80
60
40
20
1
2
Hours worked
OA. $40 a day
B. $40 an hour
C. $20 an hour
D. $20 a day
The graph shown corresponds to someone who makes: C. $20 an hour.
What is a proportional relationship?In Mathematics and Geometry, a proportional relationship refers to a type of relationship that produces equivalent ratios and it can be modeled or represented by the following mathematical equation:
y = kx
Where:
y represents the total earnings in dollars.x represents the hours worked.k is the constant of proportionality.Next, we would determine the constant of proportionality (k) by using various data points as follows:
Constant of proportionality, k = y/x
Constant of proportionality, k = 20/1 = 40/2 = 60/3 = 80/4
Constant of proportionality, k = 20.
Therefore, the required linear equation or function is given by;
y = kx
y = 20x
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
6. Error Analysis Dakota said the third term of the expansion of (2g + 3h) is 36g2h². Explain Dakota's error. Then correct the error.
The binomial expansion is solved and the error in Dakota's statement is the incorrect substitution of 36g^2h^2 for the correct expression
Given data ,
Dakota made a mistake because the third term of the expansion of (2g + 3h) should have been 36g2h2. The binomial theorem asserts that the expansion of (2g + 3h) is as follows:
( x + y )ⁿ = ⁿCₐ ( x )ⁿ⁻ᵃ ( y )ᵃ
Here, x = 2g and y = 3h. Since term numbers begin at 0, since we are seeking for the third term, r = 2.
So , on simplifying the equation , we get
= nC2 * (2g)⁽ⁿ⁻²⁾ * (3h)²
= (n! / (2! * (n - 2)!)) * (2g)⁽ⁿ⁻²⁾ * (3h)²
= ((n * (n - 1)) / 2) * (2g)⁽ⁿ⁻²⁾ * (3h)²
Hence , the correct expression for the third term of the expansion of (2g + 3h) is ((n * (n - 1)) / 2) * (2g)^(n - 2) * (3h)², where n is the exponent in the binomial expansion.
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Write the inductive hypothesis for this statement
The inductive hypothesis is P₍ₐ₊₁₎ : 7 + 13 + 19 + ... + {6(a +1) +1} = (a + 1) {3(a + 1) + 4} is true for all (a+1) = n for the statement Pₙ : 7 + 13 + 19 + ... + (6n +1) = n(3n +4)
The statement is given as,
Pₙ : 7 + 13 + 19 + ... + (6n +1) = n(3n +4)
For n =1 we get,
P₁ : 6(1) + 1 =7
and, 1{3(1) + 4} = 7
Hence, 6n+ 1 = n(3n + 4) is true for n= 1.
Therefore assume that Pₐ is true for some a = n, we get,
Pₐ : 7 + 13 +19 +... + (6a +1 ) = a(3a + 4)
Let, n =a+1 , we get,
P₍ₐ₊₁₎ : 7 + 13 + 19 + ... + (6a +1) + {6(a +1) +1}
= Pₐ + (6a+ 7)
= a(3a + 4) + 6a + 7
= 3a² + 4a + 6a + 7
= 3a² +10a + 7
= 3a² + 7a + 3a + 7
= a( 3a + 7) + 1(3a + 7)
= (a + 1) ( 3a + 7)
= (a + 1) {3(a + 1) + 4}
Hence, (a +1) is true for all (a+1) = n by mathematical induction principle. Thus, the inductive hypothesis is true.
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