Answer:
190Step-by-step explanation:
This is the combination of 2 out of 20:
20C2 = 20!/18!2! = 20*19/2 = 190 waysWe have to find,
The number of ways in which he can choose two students.
Now the 2 out of 20 combination is,
→ 20C2
→ 20!/18!2!
→ 20 × 19/2
→ 10 × 19
→ 190 ways
Hence, there are 190 ways.
If a and b are positive numbers, find the maximum value of f(x) = x^a(2 − x)^b on the interval 0 ≤ x ≤ 2.
Answer:
The maximum value of f(x) occurs at:
[tex]\displaystyle x = \frac{2a}{a+b}[/tex]
And is given by:
[tex]\displaystyle f_{\text{max}}(x) = \left(\frac{2a}{a+b}\right)^a\left(\frac{2b}{a+b}\right)^b[/tex]
Step-by-step explanation:
Answer:
Step-by-step explanation:
We are given the function:
[tex]\displaystyle f(x) = x^a (2-x)^b \text{ where } a, b >0[/tex]
And we want to find the maximum value of f(x) on the interval [0, 2].
First, let's evaluate the endpoints of the interval:
[tex]\displaystyle f(0) = (0)^a(2-(0))^b = 0[/tex]
And:
[tex]\displaystyle f(2) = (2)^a(2-(2))^b = 0[/tex]
Recall that extrema occurs at a function's critical points. The critical points of a function at the points where its derivative is either zero or undefined. Thus, find the derivative of the function:
[tex]\displaystyle f'(x) = \frac{d}{dx} \left[ x^a\left(2-x\right)^b\right][/tex]
By the Product Rule:
[tex]\displaystyle \begin{aligned} f'(x) &= \frac{d}{dx}\left[x^a\right] (2-x)^b + x^a\frac{d}{dx}\left[(2-x)^b\right]\\ \\ &=\left(ax^{a-1}\right)\left(2-x\right)^b + x^a\left(b(2-x)^{b-1}\cdot -1\right) \\ \\ &= x^a\left(2-x\right)^b \left[\frac{a}{x} - \frac{b}{2-x}\right] \end{aligned}[/tex]
Set the derivative equal to zero and solve for x:
[tex]\displaystyle 0= x^a\left(2-x\right)^b \left[\frac{a}{x} - \frac{b}{2-x}\right][/tex]
By the Zero Product Property:
[tex]\displaystyle x^a (2-x)^b = 0\text{ or } \frac{a}{x} - \frac{b}{2-x} = 0[/tex]
The solutions to the first equation are x = 0 and x = 2.
First, for the second equation, note that it is undefined when x = 0 and x = 2.
To solve for x, we can multiply both sides by the denominators.
[tex]\displaystyle\left( \frac{a}{x} - \frac{b}{2-x} \right)\left((x(2-x)\right) = 0(x(2-x))[/tex]
Simplify:
[tex]\displaystyle a(2-x) - b(x) = 0[/tex]
And solve for x:
[tex]\displaystyle \begin{aligned} 2a-ax-bx &= 0 \\ 2a &= ax+bx \\ 2a&= x(a+b) \\ \frac{2a}{a+b} &= x \end{aligned}[/tex]
So, our critical points are:
[tex]\displaystyle x = 0 , 2 , \text{ and } \frac{2a}{a+b}[/tex]
We already know that f(0) = f(2) = 0.
For the third point, we can see that:
[tex]\displaystyle f\left(\frac{2a}{a+b}\right) = \left(\frac{2a}{a+b}\right)^a\left(2- \frac{2a}{a+b}\right)^b[/tex]
This can be simplified to:
[tex]\displaystyle f\left(\frac{2a}{a+b}\right) = \left(\frac{2a}{a+b}\right)^a\left(\frac{2b}{a+b}\right)^b[/tex]
Since a and b > 0, both factors must be positive. Thus, f(2a / (a + b)) > 0. So, this must be the maximum value.
To confirm that this is indeed a maximum, we can select values to test. Let a = 2 and b = 3. Then:
[tex]\displaystyle f'(x) = x^2(2-x)^3\left(\frac{2}{x} - \frac{3}{2-x}\right)[/tex]
The critical point will be at:
[tex]\displaystyle x= \frac{2(2)}{(2)+(3)} = \frac{4}{5}=0.8[/tex]
Testing x = 0.5 and x = 1 yields that:
[tex]\displaystyle f'(0.5) >0\text{ and } f'(1) <0[/tex]
Since the derivative is positive and then negative, we can conclude that the point is indeed a maximum.
Therefore, the maximum value of f(x) occurs at:
[tex]\displaystyle x = \frac{2a}{a+b}[/tex]
And is given by:
[tex]\displaystyle f_{\text{max}}(x) = \left(\frac{2a}{a+b}\right)^a\left(\frac{2b}{a+b}\right)^b[/tex]
what is an example of a quintic bionomial?
What is the 21st term of the sequence with a1 = -6 and d = 4?
a) 70
b) 74
c) -40
d) 78
Answer:
74
Step-by-step explanation:
for this question you have to use the formula of the nth term which is Tn=a+(n-1)d,
T21=-6+(21-1)4
=-6+(20)4
=-6+80
=74
I hope this helps
Choose the expression that represents “divide 0.04 by n.”
Answer:
.04/n=4/100n or 4÷100n
Only the top runners in the world can finish a marathon in close to 2 hours. This graph
shows the pace that one such runner followed in a race. What is the unit rate?
Marathon Pace
N
28
24
20
Distance (mi)
10
12
B
8
A
4
0
0 20 40 60 80 100120140
Time (min)
Answer:
the unit rate is 4mi per minute.
Step-by-step explanation:
if you find the slope between point A and point B, you see that the line rises at 4 mi (distance) per minute (time). thus the unit rate is 4mi per minute. please correct me if I'm wrong, I hope this answer helps.
Scientists have steadily increased the amount of grain that farms can produce each year. The yield for farms in France is given by y=−2.73x2+11000x−11000000 where x is the year and y is the grain yield in kilograms per hectare (kg/ha).
What does the y-intercept of this function represent?
9514 1404 393
Answer:
the yield in year 0
Step-by-step explanation:
The y-value is the yield for farms in France in year x. The y-value when x=0 is the yield for farms in France in year 0.
_____
Additional comment
The reasonable domain for this function is approximately 1843 ≤ x ≤ 2186. The function is effectively undefined for values of x outside this domain, so the y-intercept is meaningless by itself.
Write the quadratic equation whose roots are 2 and -4 and whose leading coefficient is 2
Answer:
2x^2+4x-16
Step-by-step explanation:
The quadratic can be written as
f(x) = a(x-z1)(x-z2) where z1 and z2 are the roots
f(x) = a (x-2)(x- -4)
a is the leading coefficient
f(x) = 2(x-2)(x+4)
= 2(x^2 -2x+4x-8)
= 2(x^2 +2x-8)
= 2x^2 +4x-16
what’s the value of x? and what’s the measure of angel JHK?
Answer:
x = 14
JHK = 21
Step-by-step explanation:
The angles are vertical angles and vertical angles are equal
3x-21 = x+7
Subtract x from each side
3x-x -21 = x+7-x
2x-21 = 7
Add 21 to each side
2x-21+21 = 7+21
2x = 28
Divide by 2
2x/2 =28/2
x = 14
JHK = 3x-21 = 3(14) -21 = 42-21 = 21
Answer:
Because ∠GHI and ∠JHK are vertical angles, they're congruent. Therefore, set their angle measures equal to each other & solve for x.
[tex]x+7=3x-21\\x-3x=-7-21\\-2x=-28\\x=\frac{-28}{-2} =14\°[/tex]
Substitute in the value of x to find ∠JHK:
[tex]3x-21=3(14)-21=42-21=21\°[/tex]
3^x= 3*2^x
solve this equation
Answer:
[tex] {3}^{x} = 3 \times {2}^{x} \\ x = \frac{ln(3)}{ln(3) -ln(2) } = \frac{1.09}{1.09 - 0.69} \\ x = 2.7[/tex]
I hope I helped you^_^
Three numbers are in the ratio of 1:2:4. If 3 is added to the first and 8 is subtracted from the third, the new numbers will be the first and third terms of an A.P., whose second term is the second number. Find the original numbers.
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Answer:
5, 10, 20
Step-by-step explanation:
Suppose the three numbers are x, 2x, and 4x. Then they have the required ratios. After the transformation, we have ...
((x+3) +(4x -8))/2 = 2x . . . . . 2nd term is average of 1st and 3rd
5x -5 = 4x ⇒ x = 5
The original numbers are 5, 10, 20.
_____
After the adjustment, the arithmetic sequence is 8, 10, 12.
could anyone help me solve this? I’ve had several questions like this and I don’t understand how to solve it. I’ll give brainliest:)
Answer:
-2, - 1, - 2 and - 3
Step-by-step explanation:
As the graph depicts an odd function, it will follow the rule f(-x) = - f(x)
Is interquartile range a measure of center or a measure of variation?
Answer:
The interquartile range is the middle half of the data that is in between the upper and lower quartiles. ... The interquartile range is a robust measure of variability in a similar manner that the median is a robust measure of central tendency.
an expression equivalent to 12+ 21 is
Answer:
33
Step-by-step explanation:
The answer is 1628. Because 16 x 21 = 28 x 12 = 336
Hope this helps
I need to know the answer please
Focusing on the center point of f(x) (0,0), we can see that it has moved to the left 4 units and up 3 units.
g(x) = [tex](\sqrt[3]{x + 4}) + 3[/tex]
Option C
Hope this helps!
Oscar has 1/5 of a jar of mustard. He puts equal amounts of the mustard onto 7 sandwiches and uses all of the mustard. What fraction of a jar of mustard does each sandwich have?
Answer:
1/35 jar of mustard yuh yuh
Will give brainliest answer
Find TAN
Instructions: Find the value of the trigonometric ratio. Make
sure to simplify the fraction if needed.
please mark this answer as brainlist
The number of patients treated at Dr. Frank's dentist office each day was recorded for nine days: 18, 19, 19, 4, 14, 8, 22, 3, 1. Using the given data, find the mean for this sample.
Answer:
12
Step-by-step explanation:
To find the mean, add up all the numbers
18+ 19+ 19+ 4+14+ 8+ 22+ 3+ 1
108
Then divide by the number of terms
108/9 =
12
Help me with this question plz
9514 1404 393
Answer:
17
Step-by-step explanation:
The points at the ends of the interval are ...
(0, f(0)) = (0, 0)
(7, f(7)) = (7, 119)
The average rate of change is given by the slope formula:
m = (y2 -y1)/(x2 -x1)
m = (119 -0)/(7 -0) = 119/7 = 17
Help me out!! Anyone
Answer:
4:10
Step-by-step explanation:
if they have to wait for plane B and it arrives every 10 mins then 4:10 is the anser
Been stuck on this since yesterday !!?!?
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
I have uploaded a graph for you. The x axis is the number of years. The y axis is the salary multiplied by 1000. I should have made the multiplication factor 10000 but 1000 will do.
The 5 given points are plotted in red. The blue line is the function.
The function is y = 5x + 35. That means for every year you add 5 times the year onto the salary.
No years is 35000
1 year is 1 * 5000 + 35000
2 years is 2 * 5000 + 35000 = 45000
6 years is 5 * 5000 * 35000 = 65000
and so on.
The point you want is x = 12
12 years is 12 * 5000 + 35000 = 95000
Forgot the graph
4
5
7
11
19
?
a. 41
b. 35
c. 23
d. 29
Answer:
35
Step-by-step explanation:
The pattern is adding powers of 2.
4+1=5 (exception)
5+2=7
7+4=11
11+8=19
19+16=35
Answer:
35
Step-by-step explanation:
4 + 1 = 5
5 + (1 × 2) = 5 +2 =7
7 + (2×2) = 7 + 4 = 11
11 +(4×2) = 11 + 8 = 19
19 + (8×2) = 19 + 16 = 35
A triangle has base of 7 1/8 feet and height 6 1/4 feet. Find the area of a triangle as a mixed number.
Answer: The area is 22 17/64.
Step-by-step explanation:
base = 7 1/8 = 57/8
height = 6 1/4 = 25/4
area = 1/2*b*h
= 1/2*57/8*25/4
= 1425/64
= 22 17/64
Is u=−12 a solution of 8u−1=6u?
Answer:
No, -12 is not a solution.
Step-by-step explanation:
8u-1=6u
8(-12)-1=6(-12)
-96-1=-72
-97=-72
Untrue, to it’s not a solution
Evaluate the expression (2²x² over xy³ )² for x = 4 and y == 2.
Answer:
4
Step-by-step explanation:
State the domain and range of the following function:
{(- 3,4), (0,6), (2, - 2), (1, – 3), (6, - 7), (3, 2)}
Answer:
[tex]Domain = \{-3,0,2,1,6,3\}[/tex]
[tex]Range = \{4,6,-2,-3,-7,2\}[/tex]
Step-by-step explanation:
Given
[tex]\{(- 3,4), (0,6), (2, - 2), (1, - 3), (6, - 7), (3, 2)\}[/tex]
Required
The domain and range
A function is represented as:
[tex]Function = \{(x_1,y_1),(x_2,y_2),....(x_n,y_n)\}[/tex]
Where
[tex]x \to[/tex] domain
[tex]y \to[/tex] range
So, we have:
[tex]Domain = \{-3,0,2,1,6,3\}[/tex]
[tex]Range = \{4,6,-2,-3,-7,2\}[/tex]
Solve the following formula for a.
Answer:
B is correct .trust me
Step-by-step explanation:
Steve thinks he can drive legally 30 minutes after he drinks 5 beers. The legal limit is BAC = 0.08. Give a 90% prediction interval for Steve’s BAC. Can he be confident he won’t be arrested if he drives and is stopped?
Answer: Hello your question has some missing data attached below is the missing data
How well does the number of beers a student drinks predict his or her blood alcohol content (BAC)? Sixteen student volunteers at Ohio State University drank a randomly assigned number of 12-ounce cans of beer. Thirty minutes later, a police officer measured their BAC. Here are the data. The students were equally divided between men and women and differed in weight and usual drinking habits. Because of this variation, many students don’t believe that number of drinks predicts BAC well.
answer:
prediction interval : (0.040 , 0.114)
Step-by-step explanation:
Given data:
Confidence level = 90%
Legal limit ( BAC ) = 0.08
solution
sample size = 16
Degree of freedom ( df ) = 14
critical t value = 1.761
X = 4.81
Σ(x-x)² (Sx) = 72.44
also standard error of estimates = 0.0204
Y= -0.01270 + 0.01796 * 5 = 0.077
given that ; the predicted value of Y at x = 5
Considering individual response Y
standard error = 0.0211
margin of error = 1.761 * 0.021 = 0.0371
Hence the limits of the prediction interval is :
Lower limit = 0.077 - 0.037 = 0.040.
Upper limit = 0.077 + 0.037 = 0.114
Finally
90% prediction interval = (0.040 , 0.114)
What is the area of the right triangle with sides 10,26 and 24
Answer:
[tex]\boxed {\boxed {\sf 120 \ units^2}}[/tex]
Step-by-step explanation:
We are asked to find the area of a triangle. The formula for calculating this is:
[tex]a= \frac{1}{2} bh[/tex]
This is a right triangle, so the base and height are the legs of the triangle. The 2 smallest sides are the legs because the longest side is the hypotenuse. Since the side lengths are 10, 26, and 24, the base and height must be 10 units and 24 units.
b= 10 unitsh= 24 unitsSubstitute these values into the formula.
[tex]a= \frac{ 1}{2} ( 10 \ units)(24 \ units)[/tex]
Multiply the numbers in parentheses.
[tex]a= \frac{1}{2}(240 \ units^2)[/tex]
Multiply by 1/2 or divide by 2.
[tex]a= 120 \ units^2[/tex]
The area of the triangle is 120 units squared.
HELPSSS PLSSSS I need help!!
Step-by-step explanation:
The perimeter of the rectangle is
[tex]P = 2(4x + 2x) = 12x[/tex]
The perimeter of the octagon is
[tex]P = 8(1.5x) = 12x[/tex]
So for x = 1, the perimeter of the rectangle, as well as the octagon, is 12 cm. For x = 2, its 24 cm. For x = 3, it's 36 and so on. So the pattern here is with each integer increase in x, the perimeter increases by 12 cm. Also that the perimeters of both shapes are equal.
