Answer:
The probability that atleast one ticket labeled 2-6 is drawn is 0.94
Numbered ticket = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Number of draws = 4
Required picks = {2, 3, 4, 5, 6}
Recall :
Probability = required outcome / Total possible outcomes
Probability of choosing a required ticket :
5/10 = 1/2
Therefore, the probability that none of the required tickets is chosen :
(1/2 × 1/2 × 1/2 × 1/2) = 1/16
The probability that atleasr one ticket labeled 2-6 is drawn is :
1 - P(none is chosen) = 1 - 1/16 = 15/16 = 0.9375
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Finding the mean of data
In a triathlon, Farhan swims a distance of 1.5 km at an average speed of 2.5 km/h,
hours.
cycles 40 km in 12 hours and runs at an average speed of 9 km/h for 1
Find his
average speed for the entire competition.
Answer:
We can use the formula:
average speed = total distance / total time
To find the total distance, we add the distances for each event:
total distance = 1.5 km (swimming) + 40 km (cycling) + 1 km (running) = 42.5 km
To find the total time, we add the times for each event:
time for swimming = distance / speed = 1.5 km / 2.5 km/h = 0.6 hours
time for cycling = 40 km / 12 km/h = 3.33 hours
time for running = 1 km / 9 km/h = 0.11 hours
total time = 0.6 hours + 3.33 hours + 0.11 hours = 4.04 hours
Now we can calculate the average speed:
average speed = total distance / total time = 42.5 km / 4.04 hours ≈ 10.52 km/h
Therefore, Farhan's average speed for the entire competition is approximately 10.52 km/h.
I will give brainliest and ratings if you get this correct
Answer:
below
Step-by-step explanation:
To show that (AB)^-1 = B^-1A^-1, we need to show that:
(AB)(B^-1A^-1) = (B^-1A^-1)(AB) = I
where I is the identity matrix.
Using the associative property of matrix multiplication, we can simplify the left-hand side of the equation as:
(AB)(B^-1A^-1) = A(BB^-1)A^-1 = AIA^-1 = AA^-1 = I
Similarly, using the associative property of matrix multiplication, we can simplify the right-hand side of the equation as:
(B^-1A^-1)(AB) = B^-1(A^-1A)B = B^-1IAB = BB^-1 = I
Therefore, we have shown that (AB)^-1 = B^-1A^-1.
To show this identity, we need to demonstrate that:
(AB)^-1 = B^-1 A^-1
To do this, we start by multiplying both sides of the equation by (AB):
(AB)^-1 (AB) = B^-1 A^-1 (AB)
On the left side, we know that (AB)^-1 is the inverse of AB. So we can simplify to:
I = B^-1 A^-1 (AB)
Where I is the identity matrix. We can now simplify the right side by rearranging the terms:
B^-1 A^-1 (AB) = B^-1 (A^-1 A) B
Since A^-1 A = I, we can simplify further to:
B^-1 (A^-1 A) B = B^-1 IB = B^-1 B = I
Therefore, we have shown that:
(AB)^-1 = B^-1 A^-1
Q.E.D.
If cos a = 0.5, then sin a could equal:
Answer:
Step-by-step explanation:
If cos a = 0.5, then sin a could equal √(1 - cos^2 a) = √(1 - 0.5^2) = √0.75 ≈ 0.866.
Adjacent angles .help me solve
BGC or AGF
Adjacent means next to or adjoining something else and both of the mentioned angles are next to angle AGB.
please answer question on picture
The probability that Mason does not arrive at work by 9 a.m. is given as follows:
p = 17/50.
How to calculate a probability?A probability is calculated dividing the desired number of outcomes by the total number of outcomes.
The outcomes relating to the bus not arriving on time are divided as follows:
1 - 3/4 = 1/4 of 4/5. -> bus arrives on time.1 - 3/10 = 7/10 of 1 - 4/5 = 1/5. -> bus does not arrive on time.Hence the probability is given as follows:
p = 1/4 x 4/5 + 7/10 x 1/5
p = 1/5 + 7/50
p = 10/50 + 7/50
p = 17/50.
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What are the correct answers for this.
Answer:
True, False, True, False, False (answered top to bottom)
Step-by-step explanation:
Can anyone help with the geometry notes
Based on the Tangent Chord Theorem:
m∠1 = ¹/₂ ABm∠1 = ¹/₂ ACBm∠DEG = 154°m∠DBC = 84°m XY = 56°What is the tangent chord theorem?The Tangent Chord Theorem states that if a chord and a tangent intersect at the point of tangency, then the measure of each angle formed is equal to half the measure of its intercepted arc.
Considering the given angles;
m∠1 = ¹/₂ AB
m∠1 = ¹/₂ ACB
m∠DEG = ¹/₂ * 308°
m∠DEG = 154°
m∠DBC = ¹/₂ * (360 - 192)
m∠DBC = 84°
m XY = 360 - (151 * 2)
m XY = 56°
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A new hair cream was just given to a random sample of
500 people that are either bald, or currently losing their hair. The results showed that
175 of people either started growing back hair or stopped losing their hair, and the results had a margin of error of 0.065
.
If the new hair cream will be administered to 7810
people, how many are expected to see improvement?
Between [DROPDOWN1] and [DROPDOWN2] are predicted to see an improvement in their baldness.
Between 2226 and 3241 people are predicted to see an improvement in their baldness.
How to obtain the amounts?The expected amount of people expected to see improvement is obtained applying the proportions in the context of the problem.
The sample proportion is given as follows:
175/500 = 0.35.
Considering the margin of error, the bounds are given as follows:
0.35 - 0.065 = 0.285.0.35 + 0.065 = 0.415.Hence the amounts out of 7810 people are obtained as follows:
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Write an equation for the function graphed below
The given graph represents the function with the equation f(x) = 12(x - 1)/[(x + 2)(x - 3)].
To write an equation for the function, we first observe that the vertical asymptotes occur at x = -2 and x = 3, indicating that the denominator (Q(X)) has roots at x = -2 and x = 3. Hence, the equation of the denominator is:
Q(X) = (x + 2)(x - 3)
Furthermore, from the graph, we can see that the zero of the function occurs at x = 3, indicating that the numerator (P(X)) has the factor (x - 3). Let a be the leading coefficient of P(X), so we can write:
P(X) = a(x - 1)(x - 3)
To find the value of a, we use the fact that the y-intercept of the function is (0, -2). Substituting x = 0 and f(0) = -2 into the equation of the function, we get:
-2 = a(-1)/[(2)(-3)]
Simplifying this equation gives us:
a = -2/(2*3) = -1/3
Substituting this value of a into the equation for P(X), we get:
P(X) = (-1/3)(x - 1)(x - 3)
Substituting the equations for P(X) and Q(X) into the formula for f(x), we get:
f(x) = (-4)(x - 1)/[(x + 2)(x - 3)]
Simplifying this equation gives us:
f(x) = -4(x - 1)/[(x + 2)(x - 3)]
Therefore, the equation of the function graphed in the figure is f(x) = -4(x - 1)/[(x + 2)(x - 3)].
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At the beginning of an environmental study, a forest covered an area of 1500 km^2. Since then, this area has decreased by 7.75% each year.
Let t be the number of years since the start of the study. Let y be the area that the forest covers in km^2.
Write an exponential function showing the relationship between y and t.
If "t" number of years the study started and "y" denote the forest cover area, then the exponential-function representing the relation between "y" and "t" is y = 1500 × [tex](0.9225)^{t}[/tex].
An "Exponential-Function" is a mathematical expression in which a constant (called the base) is raised to a variable power.
In this case, we want to find an exponential function that relates the area of a forest (y) to the number of years since the start of an environmental study (t).
We know that the forest is decreasing by 7.75% each year, which means that the area of the forest after "t years" will be some percentage of its original area.
The formula for an exponential function is y = a × [tex]b^{t}[/tex], where "a" = initial value and "b" = growth or decay factor.
In this case, the initial area of the forest is = 1500 km², and
The forest is decreasing by 7.75% each year.
So, the "decay-factor" is = 1 - 0.0775 = 0.9225,
The exponential function that relates the area of the forest (y) to the number of years since the start of the study (t) is:
⇒ y = 1500 × [tex](0.9225)^{t}[/tex],
Therefore, the required "exponential-function" is y = 1500 × [tex](0.9225)^{t}[/tex].
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How likely is it that
at least > or them are vowe tiles?
) Which simulation could be used to fairly represent the situation?
There is a probability of 0.117 for at least 2 of the tiles to be vowel tiles.
Given that,
Probability that the tiles getting is a vowel tile is 30%.
P(V) = 30% = 0.3
That is there will be only 3 tiles out of 10 tiles which are vowels.
Out of 8, there will be 8 × 0.3 = 2.4 tiles which are vowels.
There would be either 2 or 3 vowels.
Probability that at least 2 of them is vowel tiles is,
Probability = (0.3)² + (0.3)³
= 0.117
Here,
The simulation which can be used to fairly represent the situation is,
Use a computer to randomly generate 8 numbers from 1 to 10. Each time 1, 2 or 3 appears, it represents a vowel tile.
Hence, the required probability is 0.117.
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Select any of these that are LINEAR. (Select MORE THAN ONE answer !)
Responses
A
A
B
B
C
C
D
D
The linear equations from the options given are expressed as: y = 5 and y = -2/5x + 4.
How to Determine an Equation that is Linear or Non-linear?An equation is linear can be expressed in the following forms:
y = mx + b
x = b
y = b
In the above, y is the dependent variable and x is the independent variable, m = slope, and b = y-intercept.
Note that in a linear equation the exponent of the variable is always 1. On the other hand, a nonlinear equation has an exponent that is not equal to 1 and can be expressed as:
y = x²
Therefore, the equations that are linear are:
y = 5 and y = -2/5x + 4.
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14. As shown in the figure, the area of square ABCD is known to be 1320 square centimeters, E is the midpoint of AB, F is a quarter point of BC close to point B, and the intersection points of CE, DB and DF are P and Q , find the area of the quadrilateral BPQF.
The coordinates of all four vertices of quadrilateral BPQF, we can use the shoelace formula to find its area. The shoelace formula states that the area of a polygon with vertices (x1, y1), (x2,
What is the triangle?A triangle is a closed, two-dimensional geometric shape that has three sides and three angles. It is one of the basic shapes in geometry and can be formed by connecting three non-collinear points. Triangles can be classified based on their side lengths and angles.
To solve this problem, we can use the fact that the area of a triangle is equal to half the product of its base and height. Let's first find the length of the sides of the square.
Since the area of the square is 1320 square centimeters, we can find its side length by taking the square root of 1320:
√(1320) ≈ 36.32
So the side length of the square is approximately 36.32 cm.
Next, let's find the coordinates of points E and F. Since E is the midpoint of AB, its coordinates are:
E = ((0+36)/2, (36+0)/2) = (18, 18)
Similarly, since F is a quarter point of BC close to point B, its coordinates are:
F = (36, (3/4)36) = (36, 27)
Now let's find the equation of line CE. Since we know two points on the line (C and E), we can use the point-slope form of a linear equation to find its equation:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is one of the points on the line. We can find the slope of the line by taking the difference in y-coordinates and dividing by the difference in x-coordinates:
m = (0 - 36)/(36 - 18) = -2
Using the point-slope form with point C, we get:
y - 0 = -2(x - 36)
y = -2x + 72
Similarly, the equation of line DB can be found using points D and B:
m = (0 - 36)/(0 - 18) = 2
y - 0 = 2(x - 0)
y = 2x
Finally, we can find the coordinates of point P by solving the system of equations:
y = -2x + 72
y = 2x
Setting the two expressions for y equal to each other, we get:
-2x + 72 = 2x
4x = 72
x = 18
Substituting x = 18 into either equation, we get:
y = 2x = 36
So the coordinates of point P are (18, 36). To find the coordinates of point Q, we can use the fact that it lies on line DF. Since we know two points on the line (D and F), we can again use the point-slope form of a linear equation:
m = (27 - 0)/(36 - 0) = 3/4
y - 0 = (3/4)(x - 0)
y = (3/4)x
To find the intersection point of lines DF and CE, we need to solve the system of equations:
y = (3/4)x
y = -2x + 72
Setting the two expressions for y equal to each other, we get:
(3/4)x = -2x + 72
(11/4)x = 72
x = 64/11
Substituting x = 64/11 into either equation, we get:
y = (3/4)(64/11) ≈ 14.73
So the coordinates of point Q are (64/11, 14.73).
Hence, Now that we have the coordinates of all four vertices of quadrilateral BPQF, we can use the shoelace formula to find its area. The shoelace formula states that the area of a polygon with vertices (x1, y1), (x2,
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Adult men have heights with a mean of 69.0 inches and a standard deviation of 2.8 inches. Find the z-score of a man who is 66.7 inches tall. (to 4 decimal places)
Answer:
!
Step-by-step explanation:
a local bakery was trying to determine what kind of desserts their customers would like to see offered in the bakery they conducted several random surveys over a period of two weeks to gather data on customers preferred desserts the results of the surveys are shown in the table?
Therefore, the closest option to the calculated percentage is option (d) 44%.
What is percent?Percent, denoted by the symbol "%", is a way of representing a fraction or proportion out of 100. It is used to express a part of a whole in terms of hundredths. For example, 50% is the same as 50 out of 100, which simplifies to 1/2 or 0.5 as a decimal. Percentages are commonly used in many areas such as finance, statistics, and everyday life to express changes, rates, or comparisons.
Here,
To calculate the percentage of customers who prefer cookies, we need to add up the number of customers who prefer cookies from each survey and divide it by the total number of customers who took the survey.
Number of customers who prefer cookies: 25 + 9 + 28 + 11 = 73
Total number of customers who took the survey: 17 + 11 + 20 + 13 + 25 + 8 + 9 + 30 + 26 + 11 = 170
Percentage of customers who prefer cookies: (73/170) x 100% ≈ 44%
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The country of Scotstats requires the people in their country to have license tags on their
car such that the first 3 characters are English letters but no letter may repeat. The last 3
characters must each be a number 0 - 9 and again no numbers can be repeated. How
many license tags are possible?
The licenses have space for 6 characters.
We need to note that there are 26 alphabets and 10 numbers to pick from.
So, for the first character, any of the 26 alphabets can take this spot.
For the second character, 25 alphabets are now available for that space. (Since repetition is not allowed)
For the third character, 24 alphabets are available for that.
For the fourth character, any of the 10 numbers can take up that spot.
For the fifth character, only 9 numbers can take this spot now. (No repetition rule too)
For the sixth character, 8 numbers can take that spot.
So, mathematically, the number of license tags possible will be
[tex]\bold{26 \times 25 \times 24 \times 10 \times 9 \times 8 = \underline{11,232,000} \ possible \ license \ tags}[/tex]
Answer:
11,232,000
Step-by-step explanation:
26 choices for the first letter.
25 choices left for the second letter (can't repeat letters)
24 choices left for the third letter
26 x 25 x 24 = 15,600 = possible first 3 characters.
10 choices for the first number
9 choices left for the second number (can't repeat numbers)
8 choices left for the third number
10 x 9 x 8 = 720 = possible last 3 characters.
chatgpt
find the discriminant x^2-5x+22=7x+2
[tex]x^2-5x+22=7x+2\implies x^2-12x+20=0 \\\\[-0.35em] ~\dotfill\\\\ \qquad \qquad \qquad \textit{discriminant of a quadratic} \\\\\\ \stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{-12}x\stackrel{\stackrel{c}{\downarrow }}{+20}=0 ~~~~~~~~ \stackrel{discriminant}{b^2-4ac}= \begin{cases} 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution} \end{cases} \\\\\\ (-12)^2-4(1)(20)\implies 144-80\implies 64[/tex]
Find the standard form of the equation of the hyperbola with the given characteristics.
Vertices:
(1, −2), (5, −2);
passes through the point
(−3, 6)
The standard form of the hyperbola is:
(x-3)²/4 - (y+2)²/(64/35) = 1
How to solve
Given vertices (1, -2) and (5, -2), the center is (3, -2) and the semi-major axis, a = 2.
The hyperbola is horizontal.
Using point (-3, 6), we find the semi-minor axis, b² = 64/35.
The standard form of the hyperbola is:
(x-3)²/4 - (y+2)²/(64/35) = 1
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The owner of a new pizzeria in town wants to study the relationship between weekly revenue and advertisi dollars. The summary output for the regression model is given below.
R2, since the model only has one independent variable.
How to solveFind R2:
R^{2} = [tex]\frac{SSR}{SST}[/tex]
R^{2} = [tex]\frac{18.26347892}{26.24578040}[/tex]
R^{2} = 0.6958634
[tex]R^{2} = 0.6959}}[/tex]
Part 2 of 3:
Find Ra2:
[tex]R^{2}_{a}= 1-\frac{(1-R^{2}) \\times (n-1)}{n-k-1}[/tex]
where
n = 9
since dftotal = n -1 = 8
thus n = 8+1=9
and k = number of independent variables = 1
Thus
[tex]R^{2}_{a}= 1-\frac{(1-0.6958634 ) \\times (9-1)}{9-1-1}[/tex]
[tex]R^{2}_{a}= 1-\frac{0.3041366 \\times 8}{7}[/tex]
[tex]R^{2}_{a}= 1-0.3475846[/tex]
[tex]R^{2}_{a}= 0.6524154[/tex]
[tex]R^{2}_{a}= 0.6524}}[/tex]
Part 3 of 3:
R2, since the model only has one independent variable.
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Which scenario has the ratio 1/2?
Triangles to total shapes because triangles= 5 total, and total shapes = 10. 5/10 simplifies to 1/2
Determine the number of triangles ABC possible with given parts
The number of triangles formed by triangle ABC is 1
How to determine the number of angles
In order to calculate the possible number of triangles ABC that can be made with the specified parts, it is essential to employ conditions for triangle congruence, which includes the Triangle Inequality Theorem.
If sides a, b, and angle A can create a valid triangle, then they must follow these requirements:
a + b > c
a + c > b
b + c > a
By replacing given values with assigned variables, we get:
38 + 77 > c,
38 + c > 77,
77 + c > 38.
After solving these inequalities, we determine that c should be
39 < c < 115.However, if the Triangle Inequality Theorem is not observed, an unfeasible configuration results. Therefore, it would be impossible to generate a triangle using the following: AB = 38, AC = 77, and angle BAC = 74 degrees.
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3. Ronald Kivetsky bought a new car and received these price quotes
from his insurance company.
a. What is the annual premium?
b. What is the semiannual premium?
c. How much less would Ronald's semiannual payments be if he
dropped the optional collision insurance?
Answer:
A.the annual premium can be found by multiplying the monthly premium by 12:
Annual premium = Monthly premium x 12
Annual premium = $65 x 12 = $780
b. The semiannual premium can be found by dividing the annual premium by 2:
Semiannual premium = Annual premium / 2
Semiannual premium = $780 / 2 = $390
c. To calculate how much less Ronald's semiannual payments would be if he dropped the optional collision insurance, we need to subtract the cost of the optional insurance from the semiannual premium:
Semiannual payment with optional insurance = Semiannual premium - $120
Semiannual payment with optional insurance = $390 - $120 = $270
Semiannual payment without optional insurance = Semiannual premium - $120
Semiannual payment without optional insurance = $390 - $120 = $270
Therefore, Ronald's semiannual payments would be $270 if he dropped the optional collision insurance.
Select the correct answer.
What transformations are applied to the graph of the function f(x) = 10 to produce the graph of the function g (a) = 3(10) 2?
O A. a vertical dilation by a factor of 3 and a horizontal shift to the right 2 units
OB.
a vertical dilation by a factor of 3 and a vertical shift down 2 units
O C.
a vertical dilation by a factor of
and a vertical shift down 2 units
OD.
a vertical dilation by a factor of
and a horizontal shift to the right 2 units
The transformations that are applied to the graph of the function [tex]f(x) = 10^x[/tex] to produce the graph of the function [tex]g(x) = 3(10)^x - 2[/tex] is: C. a vertical dilation by a factor of 3 and a vertical shift down 2 units.
What is a transformation?In Mathematics and Geometry, a transformation can be defined as the movement of a point from its initial position to a new location. This ultimately implies that, when a function or object is transformed, all of its points would also be transformed.
What is a dilation?In Mathematics and Geometry, a dilation refers to a type of transformation which typically changes the size of a geometric object, but not its shape.
Based on the parent function (exponential), we can logically deduce that the exponential function g(x) can be produced by vertically stretching the parent function by a scale factor of 3, followed by a vertical shift down 2 units;
[tex]f(x) = 10^x[/tex]
[tex]g(x) = 3(10)^x - 2[/tex]
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Which graph represents the function f(x) = 2^x-1 + 2?
The graph that represents the exponential function f(x) = 2^(x - 1) + 2 is given as follows:
Graph A.
How to define an exponential function?An exponential function has the definition presented as follows:
y = ab^x.
In which the parameters are given as follows:
a is the value of y when x = 0.b is the rate of change.The function for this problem is defined as follows:
f(x) = 2^(x - 1) + 2
It has an horizontal asymptote at y = 2, which removes the option C.
The parameter b is of b = 2, meaning that the function is increasing, which removes option D.
The numeric value of the function when x = 0 is given as follows:
f(0) = 2^(0 - 1) + 2
f(0) = 1/2 + 2
f(0) = 5/2
f(0) = 2.5.
Hence graph A is correct.
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Gary, the technology coordinator of a high school, is planning to purchase one memory stick for each of the teachers. There are 86 teachers at the school.
Answer: 86 memory sticks should be purchased
Step-by-step explanation: If he needs to be a stick for each teacher, and there are 86 teachers, then he needs to be 86 sticks.
Question 8 (Essay Worth 10 points)
(07.04 MC)
Given the functions f(x)=x²+x²-3x+4 and g(x)=2"-4, what type of functions are f(x) and g(x)? Justify your answer. What key feature(s) do f(x) and g(x) have in common?
(Consider domain, range, x-intercepts, and y-intercepts.)
The prominent similar feature between f(x) and g(x) is the present y-intercept.
How to explain the informationSince the discriminant is negative, there are no possible solutions to the given equation, which implies that f(x) has no x-intercepts.
It should be noted that to identify the y-intercept, we set x = 0 and evaluate f(x):
f(0) = 2(0)² - 3(0) + 4
f(0) = 4,
The y-intercept of f(x) can be determined as (0, 4).
Moreover, the function g(x) = 2x-4 is a power function having a negative exponent, which makes it a reciprocal function. It's domain comprises all real numbers but 0, while its range includes all real numbers excluding 0.
In order to obtain the x-intercept of g(x), we set g(x) equal to zero and solve for x:
0 = 2x-4
2x = 4
x = 2
Consequently, the x-intercept of g(x) is (2, 0), together with the y-intercept (0, -4).
The prominent similar feature between f(x) and g(x) is the present y-intercept. On the other hand, f(x) does not contain any x-intercepts, whereas g(x) has a single x-intercept. Also, f(x) is categorized as a second-degree polynomial function, whereas g(x) is classified as a reciprocal function.
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Name an acute angle and give its measure.
Angle: ______ Measure: _____
Name an obtuse angle and give its measure. (2 points)
Angle: _____ Measure: _____
Name one right angle. (1 point
Name one straight angle. (1 point)
11. Angle KEF = 40°, Angle KEJ = 50°
12. Angle HEF = 115°, Angle KED = 140°
13. Angle JEF = 90°, Angle KEF = 90°
14. Angle DEF = 180°
Define the term geometry?The study of points, lines, and shapes in two and three dimensions, as well as their properties and relationships, is the focus of the mathematical discipline known as geometry.
11. Acute angles: Those angles are less than 90 degree angles.
Angle KEF = 40°
Angle KEJ = 50°
Angle KEH = 75°
Angle JEH = 25°
Angle HED = 65°
12. Obtuse angles: Those angles are more than 90 degree angles.
Angle HEF = 115°
Angle KED = 140°
13. Right angle: Those angles are 90 degree angle.
Angle JEF = 90°
Angle KEF = 90°
14. Straight angle: Those angles make 180 degree angle.
Angle DEF = 180°
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The table below shows the costs for
different numbers of folders.
Folder Costs:
•Number of Folders
1 = $0.17
3 = $0.51
5 = $0.85
7 = $1.19
Based on this information, what is the constant of proportionality?
A0.15
B 0.17
C 0.51
D 0.85
39. Firefighters must reach a fire anywhere
in the city within 7 minutes. They are
currently reaching fires in 3.5 minutes
less than the maximum time allotted.
Which equation shows how much time
it takes them to reach a fire?
A. t + 3.5=7
B. t - 3.57
C. t - 7 = 3.5
D. t + 7 = 3.5