Answer:
[tex]\boxed {\frac{dy}{dx}= 2x}[/tex]
Step-by-step explanation:
Solving :
⇒ log y = log (x²)
⇒ log y = 2 log x
⇒ [tex]\mathsf {\frac{1}{y} \frac{dy}{dx} = \frac{1}{x} \times 2}[/tex]
⇒ [tex]\mathsf {\frac{dy}{dx}= 2x}[/tex]
Answer:
y’ = 2x
Step-by-step explanation:
Let y = f (x), take the natural logarithm of both sides ln (y) = ln (f (x))
ln (y) = ln (x²)
Differentiate the expression using the chain rule, keeping in mind that y is a function of x.
Differentiate the left hand side ln (y) using the chain rule.
y’/y = 2 In (x)
Differentiate the right hand side.
Differentiate 2 ln (x)
y’/y = d/dx = [ 2 In (x) ]
Since 2 is constant with respect to xx, the derivative of 2 ln (x) with respect to x is 2 d/dx [ln (x)]
y’/y = 2 d/dx [In (x)]
The derivative of ln (x) with respect to x is 1/x.
y’/y = 2 1/x
Combine 2 and 1/x
y’/y = 2/x
Isolate y' and substitute the original function for y in the right hand side.
y’ = [tex]\frac{2}{x}[/tex] x²
Factor x out of x².
y’ = [tex]\frac{2}{x}[/tex] (x * x)
Cancel the common factor.
y’ = [tex]\frac{2}{x}[/tex] (x * x) (The x that is under 2 and the other x that I have underlined are the ones that cancel out)
Rewrite the expression.
y’ = 2x
So therefore, the answer would be 2x.
Could someone explain why the value of x is (-2).
Answer:
x = -2
Step-by-step explanation:
2x^3 +16 = 0
We are solving for x
Subtract 16 from each side
2x^3 +16- 16 = 0-16
2x^3 = -16
Divide each side by 2
(2x^3 ) /2 = -16/2
x^3 = -8
Take the cube root of each side
[tex]\sqrt[3]{x^3} = \sqrt[3]{-8}[/tex]
x = -2
Answer:
[tex]x=-2[/tex]
Step-by-step explanation:
Given equation:
[tex]2x^3+16=0[/tex]
To solve for the unknown variable x, apply arithmetic operations to isolate the variable.
Subtract 16 from both sides:
[tex]\implies 2x^3+16-16=0-16[/tex]
[tex]\implies 2x^3=-16[/tex]
Divide both sides by 2:
[tex]\implies \dfrac{2x^3}{2}=\dfrac{-16}{2}[/tex]
[tex]\implies x^3=-8[/tex]
Take the cube root of both sides:
[tex]\implies \sqrt[3]{x^3}=\sqrt[3]{-8}[/tex]
[tex]\implies x=-2[/tex]
When taking the cube root of a negative number, the result will be negative. To understand why, examine what happens when we cube a negative number.
When a number is cubed, it is multiplied by itself, then by itself again.
When multiplying a negative number by another negative number, the result is always positive.
When multiplying a positive number by a negative number, the result is always negative.
Therefore:
[tex]\begin{aligned}\implies (-2)^3 &= -2 \cdot -2 \cdot -2\\ & = 4 \cdot -2\\& = -8\end{aligned}[/tex]
So if -8 is cube rooted, the result is -2.
What is the solution of the inequality sh
below?
9+b≤2
Answer:
b≤-7Step-by-step explanation:
b+9≤2
Subtract 9 from both sides:
b+9-9≤2-9b≤-79+b≤2
subtract 9 on each side which will be -7
the answer will be b≤-7
A. Yes, the points shown on the line would be part of y=0.5x
B. Yes, all proportions can be shown on a graph of this line
C. No, the points shown would not be part of y=0.5x
D. No, the proportions cannot be represented on a graph
Answer:
B
Step-by-step explanation:
given the graph of y = 0.5x
Then any point on the line satisfies y = 0.5x
consider the points given
x = 1 then y = 0.5 × 1 = 0.5 ⇒ (1, 0.5 ) is on the line
x = 4 then y = 0.5 × 4 = 2 ⇒ (4, 2 ) is on the line
Which expression has a value of 16 when n = 5?
StartFraction 25 Over n EndFraction + 7
30 minus 3 n
7 + StartFraction 45 Over n EndFraction
n cubed minus 114
Answer:
7+45/n or 5
Step-by-step explanation:
45 divided by n or technically 5 is 9 plus 7 is 16 :)
Answer:7+45/n or 5
Step-by-step explanation:
What expression below is equivalent to -71n -
29?
A) -29 - 71n
B) 71n + (-29)
C) 29 + 71n
D) -29 - (-71n)
Answer:
A
Step-by-step explanation:
both numbers are negative, you can rearrange them in any order provided you keep the sign
Reason:
An expression in the form -a-b is the same as -a + (-b)
Recall we can add two numbers in any order we want. For instance, 2+3 = 3+2 since both sides are equal to 5. This is the commutative property of addition.
Therefore, the -a + (-b) is the same as -b + (-a) which in turn becomes -b - a
In short: -a - b = -b - a
This rule lets us say that -71n - 29 = -29 - 71n
\In a mouse population, some mice have thicker fur than others, but there are more mice with thinner fur. The climate in which these mice live has slowly gotten colder.
How will this change in the environment most likely affect the mouse population?
The adaptation of thinner fur will be selected over thicker fur, and the population will remain unchanged.
The adaptation of thicker fur will be selected over thinner fur, and the population will remain unchanged.
The adaptation of thinner fur will be selected over thicker fur, and the population will evolve.
The adaptation of thicker fur will be selected over thinner fur, and the population will evolve.
The adjustment of thicker fur will be selected over thinner fur, and the population will develop. Then the correct option is D.
What is decision-making?Determining the proper option, acquiring evidence, and exploring various options are all steps in the decision-making process.
In a mouse population, some mice have thicker fur than others, but there are more mice with thinner fur.
The climate in which these mice live has slowly gotten colder.
The change in the environment most likely affect the mouse population will be
The adjustment of thicker fur will be selected over thinner fur, and the population will develop.
Then the correct option is D.
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Marcela was assigned some math problems for
homework. She answered half of them in study hall.
After school she completed seven more. If she still
has 11 problems to do, how many problems were
assigned?
Answer:
36 problems
Step-by-step explanation:
so let's say x = how many problems Marcela was assigned
she completed half of them so the amount of problems left is x/2
She then completed 7 more so now we can subtract 7. We then set this to 11 to solve for x
x/2 - 7 = 11
x/2 = 18
x = 36
The steps to simplify the expression -100 ÷ 3 x (-0.6) are shown.
Answer:
That is the answer for you step 2
The top of a tree makes angles s and t with points K and L on the ground, respectively, such that the angles are complementary. Point K is x meters and point L is y meters from the base of the tree.
1. In terms of x and y, find the height of the tree. Include your calculations.
2. If m∠t = 40° and y = 8 meters, calculate the height of the tree, rounded to two decimal places.
1. In terms of x and y, the height of the tree is [tex]h = y \times tan \ t^\circ[/tex] and[tex]h = x \times tan \ s^\circ[/tex]
2. The height of the tree is 6.71 meters
Trigonometry
From the question, we are to determine the height of the tree in terms of x and y
Using SOH CAH TOA, we can write that
[tex]tan \ t^\circ =\frac{h}{y}[/tex]
∴ [tex]h = y \times tan \ t^\circ[/tex]
Also,
[tex]tan \ s^\circ =\frac{h}{x}[/tex]
∴ [tex]h = x \times tan \ s^\circ[/tex]
2. If m∠t = 40° and y = 8 meters,
Then, the height of the tree
h = 8 × tan40°
h = 6.71 meters
Hence, the height of the tree is 6.71 meters
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Genesis is going to see a movie and is taking her 3 kids. Each movie ticket costs $15 and there are an assortment of snacks available to purchase for $3 each. How much total money would Genesis have to pay for her family if she were to buy 6 snacks for everybody to share? How much would Genesis have to pay if she bought x snacks for everybody to share?
Total cost with 6 snacks:
Total cost with x snacks:.
need answer quickly as possibly!
Answer #1:
The total cost with 6 snacks is $78.
Answer #2:
The total cost with x snacks is represented by the expression 60 + 3x.
Step-by-step explanation for answer #1:
Starting off with what we know:
Genesis has 3 kids, so there are 4 people total including Genesis.Each movie ticket is $15.Each snack is $3.To answer the question of how much money Genesis would have to pay for her family if she bought 6 snacks for everybody, first consider the total cost of the movie tickets.
Multiply the total number of people by the cost of each movie ticket:
[tex]4* 15=60[/tex]
Genesis will be paying $60 total only to see the movie with her 3 kids. To find the total cost of both seeing the movie and buying 6 snacks, simply multiply the number of snacks by the cost of each snack:
[tex]6 * 3=18[/tex]
Then, add the total cost of the movie tickets by the total cost of the snacks to achieve your first answer:
[tex]60+18=78[/tex]
The total cost with 6 snacks is $78.
Step-by-step explanation for answer #2:
To find the total cost of "x" snacks (x is being used as a variable term to represent a quantity subject to change), we can create an algebraic expression.
Let "t" represent the total cost.
Since we've established that finding the total cost is just a matter of adding the total cost of movie tickets (60) and the total cost of snacks (3 multiplied by the number of snacks), let "x" represent the number of snacks in the equation to find the total cost with x snacks:
[tex]60+3x=t[/tex]
The total cost with x snacks is represented by the expression 60 + 3x.
Based on the given data, the total cost of 6 snacks: $78. total cost with x snacks: $60 + (3x).
Use the concept of multiplication defined as:
To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.
Given that,
Genesis is going to see a movie with her 3 kids.
Each movie ticket costs $15.
There are snacks available for purchase at $3 each.
Genesis wants to buy 6 snacks for everybody to share.
To calculate the total cost for Genesis and her 3 kids,
Consider the movie tickets and the snacks.
The cost of the movie tickets can be calculated by multiplying the number of people (4 in this case) by the cost of each ticket ($15).
So, Genesis would have to pay 4 x $15 = $60 for the movie tickets.
Now, let's calculate the total cost if Genesis were to buy 6 snacks for everybody to share.
Each snack costs $3, so 6 snacks would cost 6 x $3 = $18.
Therefore, the total cost for Genesis would be,
$60 (movie tickets) + $18 (snacks) = $78.
If we consider x snacks for everybody to share,
The total cost for snacks would be x $3.
So, the final cost would be $60 (movie tickets) + (3x) (snacks).
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Two gymnasts are running toward each other in a floor routine, and they plan to precisely time a flip to stay synchronized for the audience. The path of the gymnasts is parabolic and modeled by the following equations, where y is the height of the flip and x is the time in seconds:
Answer:
4 secs
Step-by-step explanation:
Since they have to be synchronized, their parabolic equations must be equal to one another
3(t^2 + 9 - 6t) - (t^2 + 25 - 10t) - t + 2 = 0
2t^2 - 9t + 4
(2t - 1)(t - 4) = 0
t = 1/2, 4
From the options, the answer is 4
The gymnasts will be at the same height during their flips at two different times: 1/2 seconds and 4 seconds.
so, correct option is: C.
Here, we have,
To determine when the gymnasts will be at the same height during their flips, we need to find the time (x) at which the equations for y are equal.
The given equations are:
y = –(x – 5)² + 3
y = –3(x – 3)² + x + 1
Setting the two equations equal, we have:
–(x – 5)² + 3 = –3(x – 3)² + x + 1
Expanding the squared terms:
–(x² – 10x + 25) + 3 = –3(x² – 6x + 9) + x + 1
Simplifying the equation:
–x² + 10x – 25 + 3 = –3x² + 18x – 27 + x + 1
Combining like terms:
–x² + 10x – 22 = –3x² + 19x – 26
Rearranging the equation:
2x² - 9x + 4 = 0
To solve this quadratic equation, we can factor it:
(2x - 1)(x - 4) = 0
Setting each factor equal to zero:
2x - 1 = 0 or x - 4 = 0
Solving for x:
2x = 1 or x = 4
Dividing by 2:
x = 1/2 or x = 4
Therefore, the gymnasts will be at the same height during their flips at two different times: 1/2 seconds and 4 seconds.
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2. The smallest number from the given numbers 6895, 4875 and 5689 will have its smallest digits at place. (A)
The smallest digit of the smallest number is at the thousands place.
What is Number System ?The system of representing numbers is called Number System.
The given numbers are 6895 , 4875 , 5689
The smallest number among these is 4875
The smallest digit in this number is 4
4 is at thousands place
Therefore , The smallest digit of the smallest number is at the thousands place.
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write the faction in the simplest form 28/36
Answer:
7/9
Step-by-step explanation:
Divide the numerator and denominator by 4.
28/36 = 7/9
Answer:
7/9
Step-by-step explanation:
28 and 36 are both divisible by 4.
28÷4 is 7 and
36÷4 is 9
28/36 = 7/9
You can walk over the mountains to
your campsite or take a safer, easier trip around the mountains. You
are on the ground at P(0, 0) and the campsite is at C(-2, 7). You
travel around the mountain by going to A(2, 5). The coordinate
system is measured in meters. Draw a diagram of the situation.
Find PA and AC.
Answer:
PA = [tex]\sqrt{29}[/tex]
AC =[tex]\sqrt[2]{5}[/tex]
Step-by-step explanation:
distance formula : if [tex](x_{1},y_{1}) and (x_{2},y_{2})[/tex] are two points on a line segment then distance between both the points is given by distance formula.
[tex]d = \sqrt{[(x_{2} - x_{1} )^2 +(y_{2} - y_{1})^2]}[/tex]
therefore ,
given points are , A(2, 5), P(0, 0), C(-2, 7).
so PA = [tex]\sqrt{[({2} - {0} )^2 +({5} - {0})^2]} = \sqrt{({2} )^2 +({5} )^2}[/tex]
PA = [tex]\sqrt{\229}[/tex]
similarly,
AC = [tex]\sqrt{[({-2} - {2} )^2 +({7} - {5})^2]} = \sqrt{({4} )^2 +({2} )^2}[/tex]
AC = [tex]\sqrt{20} = \sqrt[2]{5}[/tex]
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perpendicular to the line y =6 - 3x; passes through (-3, -1)
Answer:
[tex]y=\frac{1}{3}x[/tex]
Step-by-step explanation:
Perpendicular lines
Perpendicular lines have slopes whose product is -1. In other words, to find a line that is perpendicular to a given line, the new line must have a that is the opposite reciprocal of the original line.
Passing through a point
Additionally, if a line contains a point, then the point must be a solution to the equation (meaning, when you plug in the "x" and "y", the equation must be true).
Finding our perpendicular line
Work on slope first
To find the slope of the original line, rewrite the original line in slope-intercept form: [tex]y=mx+b[/tex]
[tex]y=6-3x\\y=6+(-3x)\\y=(-3x)+6\\y=-3x+6\\y=-\frac{3}{1}x+6[/tex]
In this form, the slope is the number multiplied to the "x", so the original slope is -3/1.
Thus, the slope of the new perpendicular line will be the opposite reciprocal of -3/1 ... which is 1/3.
Writing what we do know about the new perpendicular line, we have [tex]y=\frac{1}{3}x+b[/tex]
Making sure it passes through the point
This new line is also supposed to contain the point (-3,-1), so (-3,-1) must be a solution to the equation, for whatever constant-value the "b" is. To find out the "b", substitute the known quantities, and solve:
[tex]-1=\frac{1}{3} (-3)+b\\-1=-1+b\\(-1)+1=(-1+b)+1\\0=b[/tex]
Substituting this into our equation:
[tex]y=\frac{1}{3}x+(0)\\y=\frac{1}{3}x[/tex]
Conclusion
So, the equation for a line that is perpendicular to the original line [tex]y=6-3x[/tex], and that also passes through (-3,-1) is [tex]y=\frac{1}{3}x[/tex]
Solve for x.
log x = 3
-------------------------------------------------------------------------------------------------------------
Answer: [tex]\textsf{x = 1000}[/tex]
-------------------------------------------------------------------------------------------------------------
Given: [tex]\textsf{log(x) = 3}[/tex]
Find: [tex]\textsf{The value of x}[/tex]
Solution: In order to solve for x we need to get rid of the log which means that we need to put it to the power of the same number as the base. When a base isn't displayed it defaults to 10 so we raise everything to the base of 10.
Get rid of log
[tex]10^{\textsf{log}_{10}\textsf{(x)}}} = 10^{\textsf{3}}[/tex][tex]x = 10^{\textsf{3}}[/tex][tex]x = 1000[/tex]Therefore, the final answer would be that x is equal to 1000.
Gerald purchased a rectangular plot of land. the length of the plot is 20 feet more than the width. the cost of the land was $12 per square foot. gerald also had a fence put around the entire perimeter of the plot, at a cost of s8 per linear foot. the total amount he spent on both the land and the fence was $10,560. part a write an equation in two variables for the perimeter, p, and an equation in two variables for the area, a, of the plot of land where x is the width of the plot in feet. provide evidence to support your equations
Answer:
P =2(x+ y) [tex]ft[/tex].
A =xy [tex]ft^{2}[/tex]
Step-by-step explanation:
perimeter of rectangle (P)= 2(L+B
Area of rectangle (A) = L×B
Given,
width = 'x'
length = y = x+20
therefore using above formula
P = 2(x+20 + x) = 2(2x+20)
A = (x+20)(x)
cost of land per square foot = $12
cost of land per linear foot = $8
total cost of land = $10560
now,
P×8 + A×12 = 10560
2(2x+20)×8 + (x+20)(x)×12 = 10560
32x + 320 + 12x²+ 240x = 10560
12x² + 272x = 10240
12x² + 272x - 10240 = 0
solving this quadratic equation using Discriminant method
x = [-b ±√(b² - 4ac)]/2
where b is the coefficient of x
a is the coefficient of x²
and c is the constant term
by replacing values in the above formula
x = 19.5 , x = -42.24
as the area cannot be negative therefore -42.24 is neglected
hence x = 19.5ft
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In the triangle above, what is the measure of
A.
M
B.
M
C.
M
D
, cannot be determined
Use the drawing tool(s) to form the correct answer on the provided graph.
Graph the solution to the following linear inequality in the coordinate plane.
5x - y > -3
27
Drawing Tools
Select
Line
Dashed Line
Shaded Region
Click on a tool to begin drawing.
10
8
6
The solution to the inequality 5x - y > -3 is shown in the graph.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
An inequality shows the non equal comparison of two or more variables and numbers.
The solution to the inequality 5x - y > -3 is shown in the graph.
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Someone please help with this!!
Answer:
A
Step-by-step explanation:
Comment (and answer)
Both start with the plus x axis as one of the arms. The other arm of 110o goes anti clockwise until it is 30 degrees into the second quadrant.
The answer can be found by using 110 - 360 = negative angle.
So 110 - 360 = - 250 which goes clockwise from the plus x axis. The answer must be A.
Make sure you understand the concept of clockwise and anti clockwise. If you don't know, ask your teacher, write it down, and try not to forget it. The distinction comes up a lot.
Which of the following are solutions to the equation below?
Check all that apply.
(3x - 5)^2 = 19
Answer:
To solve the answer, you can do it in 2 ways:
1) you could take the square root on both sides and solve for x
OR
2)you could multiply 3x-5 times itself and subtract 19 on both sides and use the grouping or quadratic formula method to find x.
Hope it helps.
The graph of the function f(x) = –(x + 3)(x – 1) is shown below.
On a coordinate plane, a parabola opens down. It goes through (negative 3, 0), has a vertex at (negative 1, 4), and goes through (1, 0).
Which statement about the function is true?
The function is positive for all real values of x where
x < –1.
The function is negative for all real values of x where
x < –3 and where x > 1.
The function is positive for all real values of x where
x > 0.
The function is negative for all real values of x where
x < –3 or x > –1.
Answer:
The second statement is true.
Step-by-step explanation:
Draw the corresponding parabola and use it to verify all statements.
Answer:
The function is negative for all real values of x where
x < –3 and where x > 1.
Step-by-step explanation:
The answer above is correct.
what is
f(-1)=-3x^{2}+7x+4
Step-by-step explanation:
please mark me as brainlest
Fill in the table with the appropriate value to describe the amount of hot dogs eaten at a hot dog eating contest.
Which value should be placed in the empty space in the table?
A. 1/6
B. 1/4
C. 1/12
D. 1/8
20 hot dogs was eaten by the 10 people who attended the hot dog eating contest.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let us assume that there were 10 people at the hot dog eating contest. If each person eats 2 hot dog, hence:
Number of hot dogs eaten = 2 hot dog per person * 10 people = 20 hot dogs.
20 hot dogs was eaten by the 10 people who attended the hot dog eating contest.
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The length of a rectangle is 7 inches and the width is 42 inches. What is the ratio,
using whole numbers, of the length to the width?
Answer:
6 : 1
Step-by-step explanation:
The length is the longer side compared to the width.
Hence, the length is 42 inches and width is 7 inches.
Taking the ratio :
Length : Width42 : 77 x 6 : 76 : 1Answer:
1 : 6
Step-by-step explanation:
42 /7 = 6
We will have to simply the ratios in order to get the answer.
area = ft
help me please ty:)
find the length using the Pythagorean theorem.
Length = sqrt(20^2 - 12^2)
Length = 16 feet
area = 12 x 16 = 192 square feet
Find (f • g) (x) Assume x>0
Answer:
[tex]\textsf{B.} \quad (f \cdot g)(x)=10x[/tex]
Step-by-step explanation:
Given:
[tex]\begin{cases}f(x)=\sqrt{50x}\\g(x)=\sqrt{2x}\end{cases}[/tex]
[tex]\begin{aligned}\textsf{As }(f \cdot g)(x) & = f(x) \cdot g(x)\\\implies (f \cdot g)(x)& = \sqrt{50x} \cdot \sqrt{2x}\end{aligned}[/tex]
[tex]\textsf{Apply radical rule} \quad \sqrt{a}\sqrt{b}=\sqrt{ab}:[/tex]
[tex]\begin{aligned}\implies (f \cdot g)(x) &= \sqrt{50x2x}\\& = \sqrt{100x^2}\end{aligned}[/tex]
Rewrite 100 as 10²:
[tex]\implies (f \cdot g)(x)=\sqrt{10^2x^2}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^bc^b=(ac)^b:[/tex]
[tex]\implies (f \cdot g)(x)= \sqrt{(10x)^2}[/tex]
[tex]\textsf{Apply radical rule} \quad \sqrt{a^2}=a, \quad a \geq 0:[/tex]
[tex]\implies (f \cdot g)(x)=10x[/tex]
Is in between 2 number on the number line
Answer:
3 and 4
Step-by-step explanation:
well you can approximate by squaring values
2^2 = 4
3^2 = 9
4^2 = 16
the value should fall somewhere between 3 and 4 since the square root of 9 is 3 and the square root of 16 is 4
simplify (3+√5)(3+√2)
When we simplify (3 + √5)(3 + √2), the result obtained is:
9 + 3√2 + 3√5 + √10
Surd operationa√b × c√d = (a × c)√(b × d)
How to simplify (3 + √5)(3 + √2)(3 + √5)(3 + √2)
Expand by clearing the bracket
3(3 + √2) + √5(3 + √2)
9 + 3√2 + 3√5 + √10
Thus,
(3 + √5)(3 + √2) = 9 + 3√2 + 3√5 + √10
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Mr wong predicted that he would sell 45 automobiles , he actually sold 48
Answer:
well done mr wong lol
Step-by-step explanation:
what's the question is it the percent he beat his prediction?
if it is then its 48 divided by 45 times 100
this equals 106.67%
then take this away from his prediction percent originally which is 100%
106.67-100= 6.67%
