Answer:
Absolute minimum: x = -π / 6
Absolute maximum: x = π
Explanation:
The candidates for the absolute maximum and minimum are the endpoints and the critical points of the function.
First, we evaluate the function at the endpoints.
At x = -π, we have
[tex]f(-\pi)=3(-\pi)-6\cos (-\pi)[/tex][tex]\Rightarrow\boxed{f(-\pi)\approx-3.425}[/tex]At x = π, we have
[tex]f(\pi)=3(\pi)-6\cos (\pi)[/tex][tex]\Rightarrow\boxed{f(\pi)\approx15.425.}[/tex]Next, we find the critical points and evaluate the function at them.
The critical points = are points where the first derivative of the function are zero.
Taking the first derivative of the function gives
[tex]\frac{df(x)}{dx}=\frac{d}{dx}\lbrack3x-6\cos (x)\rbrack[/tex][tex]\Rightarrow\frac{df(x)}{dx}=3+6\sin (x)[/tex]Now the critical points are where df(x)/dx =0; therefore, we solve
[tex]3+6\sin (x)=0[/tex]solving for x gives
[tex]\begin{gathered} \sin (x)=-\frac{1}{2} \\ x=\sin ^{-1}(-\frac{1}{2}) \end{gathered}[/tex][tex]x=-\frac{\pi}{6},\; x=-\frac{5\pi}{6}[/tex]
on the interval [−π,π].
Now, we evaluate the function at the critical points.
At x = -π/ 6, we have
[tex]f(-\frac{\pi}{6})=3(-\frac{\pi}{6})-6\cos (-\frac{\pi}{6})[/tex][tex]\boxed{f(-\frac{\pi}{6})\approx-6.77.}[/tex]At x = -5π/6, we have
[tex]f(\frac{-5\pi}{6})=3(-\frac{5\pi}{6})-6\cos (-\frac{5\pi}{6})[/tex][tex]\Rightarrow\boxed{f(-\frac{5\pi}{6})\approx-2.66}[/tex]Hence, our candidates for absolute extrema are
[tex]\begin{gathered} f(-\pi)\approx-3.425 \\ f(\pi)\approx15.425 \\ f(-\frac{\pi}{6})\approx-6.77 \\ f(-\frac{5\pi}{6})\approx-2.66 \end{gathered}[/tex]Looking at the above we see that the absolute maximum occurs at x = π and the absolute minimum x = -π/6.
Hence,
Absolute maximum: x = π
Absolute minimum: x = -π / 6
J(-6-2)3-*NWMark this and return2--9-8-7-6-5-4-3-2-3₁ 1 2 3 4 5 61-5737-2-cd-6--7--8-2 do-9--10--11--12--13-8 9 10 11 xK(8,-9)What is the x-coordinate of the point that divides thedirected line segment from J to K into a ratio of 2:5?X == (m²²7 m )(x₂ − ×₁) + X₁m+n0 000-22Save and ExitNextSubmit
Use the given formula:
[tex]x=(\frac{m}{m+n})(x_2-x_1)+x_1[/tex]Being m: 2 and n: 5
x1: -6
x2: 8
[tex]\begin{gathered} x=(\frac{2}{2+5})(8-(-6))+(-6) \\ \\ x=\frac{2}{7}*(14)-6 \\ \\ x=4-6 \\ \\ x=-2 \end{gathered}[/tex]Then, the x-coordinate of th point that divides the directed line segment from J to K into a ratio 2:5 is -2Answer: -2Polynomial Functions:Find P(-1) and p(2) for each function.“P(x) = 4-3x”
P(-1):
[tex]\begin{gathered} P(-1)=4-3(-1) \\ P(-1)=4+3 \\ P(x)=7 \end{gathered}[/tex]P(2):
[tex]\begin{gathered} P(2)=4-3(2) \\ P(2)=4-6 \\ P(2)=-2 \end{gathered}[/tex]The fraction models below represent two fractions of the same whole: How much of the8음을16
So 4/5 times 5/8 is 1/2.
Which values are solutions to the inequality below?Check all that apply.√x ≤ 5A. 1B. 18C. -5D. 25E. 24F. 625
Given the inequality:
[tex]\sqrt[]{x}<5[/tex]We need to solve the inequality to get a range of values for x.
This we can do by finding the square of both sides:
[tex]\begin{gathered} (\sqrt[]{x})^2<5^2 \\ x<25 \end{gathered}[/tex]On checking the options given, we will pick the numbers that are strictly less than 25.
Therefore, the correct options are:
OPTION A
OPTION B
OPTION C
OPTION F
We have seen the isosceles triangles have two sides of equal length. The angles opposite these sides have the same measure. Use information to the right to help the measure of angles 1, 2, 3, 4, and 5.
In this case, we'll have to carry out several steps to find the solution.
Step 01:
data:
triangle diagram
Step 02:
angles:
we must analyze the diagram to find the solution
angle 1:
angle 1 = 180° - 155° = 25°
angle 2:
angle 2 = angle 1 = 25°
angle 3:
angle 3 = 180° - 25° - 25° = 130°
angle 4:
angle 4 = 155°
angle 5:
angle 5 = 180° - 25° = 155°
That is the full solution.
Send me Answers for Questions A, B, and C
Answer: A) 4 B) 30 C) 6
Step-by-step explanation:
For question A, you subtract the highest number and the lowest number (10-6)
For question B, you add all the frequency numbers together
For question C, you use your answer on B and divide it by 5
work out sues total pay
Sue's total pay for the year given the salary, bonus and share of profit is £38,110.
What is the total pay?Sue's total pay for the year is a function of the salary, the share of the profit that she earns and the bonus.
Salary for the year = monthly salary x number of months in a year
£1410 x 12 = £16,920
The next step is to determine the profit last year
Profit = total revenue - total cost
£549,000 - £473,500 = £75,500
Now determine the share of profit that Sue would earn.
Share of profit = 26% x £75,500
0.26 x £75,500 = £19,630
Now determine the total bonus she would earn : 4 x £390 = £1560
Total salary = £1560 + £19,630 + £16,920 = £38,110
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compare and contrast the graphs y=2x+1 with the domain {1,2,3,4} and y=2x+1 with the domain of all real numbers
Comparison of both the graphs y=2x+1 with domain {1,2,3,4} and set of all real numbers is :
Slope =2 , y-intercept =1 and x-intercept = -1/2 is same.
Contrast is range is different:
Range = { 3, 5, 7, 9} for domain {1,2,3,4}
Range = set of all real numbers for domain all real numbers.
As given in the question,
Given function for the graphs are:
y =2x+1
Different domains
Domain ={1,2,3,4}
Domain =All real numbers
Compare with y=mx +c
Slope m =2
For y-intercept put x=0
y=2(0) +1
=1
For x-intercept put y=0
0 =2x+1
⇒x=-1/2
Contrast:
For domain ={1,2,3,4}
Range is :
y = 2(1)+1
=3
y=2(2)+1
=5
y=2(3) +1
=7
y=2(4)+1
=9
Range ={ 3, 5, 7,9}
For domain= all real numbers
Range = set of all real numbers
Therefore, comparison of both the graphs y=2x+1 with domain {1,2,3,4} and set of all real numbers is :
Slope =2 , y-intercept =1 and x-intercept = -1/2 is same.
Contrast is range is different:
Range = { 3, 5, 7, 9} for domain {1,2,3,4}
Range = set of all real numbers for domain all real numbers.
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22The value of the hypotenuse in the right triangle shown isinches.14 in48 inFigure not drawn to scale
SOL
Step 1 :
In this question, we are meant to find the value of
the hypotenuse in the right angle below:
Before, we proceed, we still need to remind ourselves of Pythagoras' theorem,
Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“.
Step 2 :
From the above theorem, we can see that that the two adjacent sides are :
14 inches and 48 inches.
From the principle of Pythagoras' Theorem,
[tex]\begin{gathered} c^2=a^2+b^2 \\ \text{where a = 14 inches} \\ b\text{ = 48 inches} \\ c^2=14^2+48^2 \\ c^2\text{ = ( 14 x 14 ) + ( 48 x 48 )} \\ c^2\text{ = 196 + 2304} \\ c^2\text{ = 2500} \\ \text{square root both sides, we have that :} \\ c\text{ = 50 inches} \end{gathered}[/tex]
CONCLUSION :
The value of the hypotenuse in the Right angle, c = 50 inches.
In an arithmetic sequence with a1=-5 and d=-3, which term is -24?The term -24 is the ___th term of the sequence
Given:
[tex]\begin{gathered} a_1=-15 \\ d=-3 \\ a_n=-24 \end{gathered}[/tex]To find:
The value of n.
Explanation:
The nth term formula for the arithmetic sequence is,
[tex]a_n=a_1+(n-1)d[/tex]Substituting the given values we get,
[tex]\begin{gathered} -24=-15+(n-1)(-3) \\ -24=-15-3n+3 \\ -24=-3n-12 \\ -3n=-24+12 \\ -3n=-12 \\ n=4 \end{gathered}[/tex]Thus, -24 is the 4th term of the sequence.
Final answer:
The term -24 is the 4th term of the sequence.
Fill in the missing statements and reasons in each proof shown below. You must mark the diagram forcredit.15.Given: g | h and 21 22Prove: p | r3рStatementReason2.ghgh21 2 2321 2222 223pllr
To prove that p || r, we will complete the data in the given table
[tex]\begin{gathered} \text{Statement: g}\mleft\Vert h\mright? \\ R\text{eason : Given} \end{gathered}[/tex][tex]\begin{gathered} \text{Statement: <1}\cong<3 \\ \operatorname{Re}\text{ason: Corresponding angles} \end{gathered}[/tex][tex]\begin{gathered} \text{Statement: <1}\cong<2 \\ \operatorname{Re}\text{ason:Given} \end{gathered}[/tex][tex]\begin{gathered} Statement\colon\text{ <2}\cong<3 \\ Re\text{ason: Alternate exterior angle} \end{gathered}[/tex][tex]\begin{gathered} \text{Statement: P}\mleft\Vert r\mright? \\ \operatorname{Re}\text{ason:<2}\cong<3 \end{gathered}[/tex]Four points are labeled on the number line. M K L zo 0.5 1 Which point best represents 3? F. Point K G. H. Point 2 Point M Point N J.
The point that best represents 1/3 is point M .
The number line ranges from 0 to 0.5 with 10 divi
SOMEONE PLEASE HELP ME QUICKLY WITH THIS,ITS AN EMERGENCY!!!!! pls explain how you get the solution as well, sorry!
Thank you <3
The statement that reflects the running rates is Pepe ran 9/8 mile in 1/2 hour and Paul ran 19/24 mile in 1/3 of an hour.
What is the speed?Speed is the total distance run per time. It can be determined by dividing the total distance travelled by the total time.
Speed = distance / time
Speed if Paul ran 1/5 mile in 4/15 hour
Speed = 1/5 ÷ 4/15
1/5 x 15/4 = 3/4 miles per hour
Speed if Pepe ran 8/10 mile in 1/4 of an hour
Speed = 8/10 ÷ 1/4
8/10 x 4 = 16/5 = 3 1/5 mile per hour
Difference in speeds =
[tex]3\frac{1}{5}[/tex] - [tex]\frac{3}{4}[/tex]
[tex]3\frac{4 - 15}{20}[/tex] = [tex]2\frac{9}{20}[/tex]
Speed if Paul ran 4/15 mile in 1/5 hour
Speed = 4/15 ÷ 1/5
4/15 x 5 = 4/3 = 1 1/3 miles per hour
Speed if Pepe ran 1/4 mile in of 8/10 an hour
Speed = 1/4 ÷ 8/10
1/4 x 10/8 = 5 / 16
Difference in speeds = [tex]1\frac{1}{3} - \frac{5}{16}[/tex] = [tex]1\frac{1}{48}[/tex]
Speed if Paul ran 1/3 mile in 19/24 hour
Speed = 1/3 ÷ 19 / 24
1/3 x 24/19 = 8/19 miles per hour
Speed if Pepe ran 1/2 mile in of 9/8 an hour
Speed = 1/2 ÷ 9/8
1/2 x 8/9 = 4/9 mile per hour
Difference = 4/9 - 8/19 = 4/171
Speed if Pepe ran 9/8 mile in 1/2 hour
Speed = 9/8 ÷ 1/2
9/8 x 2 = 2 1/4 miles per hour
Speed if Paul ran 19 / 24 mile in of 1/3 an hour
Speed = 19 / 24 ÷ 1/3
19 / 24 x 3 = 2 3/8 miles per hour
Difference =
[tex]2\frac{3}{8} - 2\frac{1}{4}[/tex]
[tex]\frac{3 - 2}{8}[/tex] = [tex]\frac{1}{8}[/tex] miles per hour
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On a particular day, the amount of untreated water coming into the plant can be modeled by f(t) = 100 + 30cos(t/6) where t is in hours since midnight and f(t) represents thousands of gallons of water. The amount of treated water at any given time, t, can be modeled by g(t) = 30e^cos(t/2)a) Define a new function, a′(t), that would represent the amount of untreated water inside the plant, at any given time, t.b) Find a′ (t).c) Determine the critical values of this function over the interval [0, 24).
a)The amount of untreated water inside the plant will be the difference between the difference f(t) - g(t), then, a(t) can be defined as follows:
[tex]a(t)=100+30cos(\frac{t}{6})-30e^{cos(\frac{t}{2})}[/tex]b) the derivative of a(t) is the following:
[tex]a^{\prime}(t)=-5sin(\frac{t}{6})+15sin(\frac{t}{2})e^{cos(\frac{t}{2})}[/tex]c) the critical values of a(t) over the interval [0, 24) are:
[tex]\begin{gathered} t=0 \\ t=6\pi \end{gathered}[/tex]8. Anna withdrew $50 from her checking account. She spent $28 on a pair of shoes. What fraction of her money does Anna have left?
Explanation:
If she spent $28 of the $50 she withdrew, she now has:
[tex]50-28=22[/tex]$22
The fraction is:
[tex]\frac{22}{50}=\frac{11}{25}[/tex]Answer:
Anna has 11/25 of her money left.
what is the y intercept of y = 250 + 15x
The y-intercept of y = 250 + 15x is (0,250)
What is y-intercept?A line's y-intercept is the distance in y coordinates from the line's intersection with the y-axis at its origin. A location on the graph where x is 0 is known as the y-intercept. The y-intercept of a line that is perpendicular to the x-axis is undefined.
This is an illustration of a y-intercept. Think about the line y = x + 3. The point where this graph crosses the y-axis is (0,3). Therefore, the y-intercept of the line y = x+ 3 is (0,3).
Here to determine the y-intercept put x=0,
Given, y = 250 + 15x
Replacing x by 0 in the above equation we get,
y = 250 + 15×0
y=250 +0
y=250
Therefore, the y-intercept of y = 250 + 15x is (0,250)
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Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.Solve the inequality and describe the solution set.y-6 > 1232, Math symbolsRelations► Geometry► Groups► Trgonometry3 of 3 AnsweredType here to searcho66F Mosty clou
The problem gives the inequality:
[tex]y-6\ge12[/tex]solving for y we get:
[tex]\begin{gathered} y\ge12+6 \\ y\ge18 \end{gathered}[/tex]The solution set is all real numbers equal or greater than 18, i.e.,
[tex]\lbrack18,+\infty)[/tex]what property is used to solve this?
4x-3
x=2
4(2)-3
Given ΔABC with m∠B = 62°, a = 14, and c = 16, what is the measure of A?
1) Let's sketch this out to better grasp it
2) We can see that there are two legs and two angles (one of them is missing) so let's solve it using the Law of Sines:
[tex]undefined[/tex]Use the graph to answer the question.Which statement matches the vector operation shown on the coordinate grid?
We have the correct statement about the vectors in the graph.
We can already see that it is a sum of vectors like:
[tex]v+w=u[/tex]As v has starting point at (-1,0) and ending point at (3,3), we can describe the vector as:
[tex]v=(3-(-1))\hat{i}+(3-0)\hat{j}=4\hat{i}+3\hat{j}[/tex]As w starts at (3, 3) and ends on (5, 2), we can describe it as:w
[tex]w=(5-3)\hat{i}+(2-3)\hat{j}=2\hat{i}-1\hat{j}[/tex]Finally, u starts at (-1,0) and ends at (5,2), so it can be described as:
[tex]u=(5-(-1))\hat{i}+(2-0)\hat{j}=6\hat{i}+2\hat{j}[/tex]Answer: v + w = u for v = 4i + 3j, u = 2i - j and u = 6i + 2j [Option C].
Please help! I have 3 questions that I'm not sure what to do about :( if you could answer them that would be great
The resultant matrix for the given expressions are as follows -
[A] → 2B - C = [tex]\begin{pmatrix}7.5&2.4\\ -16.4&-8.8\end{pmatrix}[/tex]
[B] → C + 2A = [tex]\begin{pmatrix}18.5&-8.4\\ 16.4&6.8\end{pmatrix}[/tex]
[C] → 2B - 10C + A = [tex]\begin{pmatrix}12&-36\\ -31&25\end{pmatrix}[/tex]
What is a Matrix?Matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. A matrix has [n] rows and [m] columns. The order of a matrix is given by - [m] x [n].
Given are the three different matrices as -
A = [tex]\left[\begin{array}{ccc}9&-6\\7&5\\\end{array}\right][/tex]
B = [tex]\left[\begin{array}{ccc}4&3\\-7&-6\\\end{array}\right][/tex]
C = [tex]\left[\begin{array}{ccc}0.5&3.6\\2.4&-3.2\\\end{array}\right][/tex]
We can solve the given expressions as follows.
[A] → 2B - C = [tex]\begin{pmatrix}7.5&2.4\\ -16.4&-8.8\end{pmatrix}[/tex]
[B] → C + 2A = [tex]\begin{pmatrix}18.5&-8.4\\ 16.4&6.8\end{pmatrix}[/tex]
[C] → 2B - 10C + A = [tex]\begin{pmatrix}12&-36\\ -31&25\end{pmatrix}[/tex]
Therefore, the resultant matrix for the given expressions are as follows -
[A] → 2B - C = [tex]\begin{pmatrix}7.5&2.4\\ -16.4&-8.8\end{pmatrix}[/tex]
[B] → C + 2A = [tex]\begin{pmatrix}18.5&-8.4\\ 16.4&6.8\end{pmatrix}[/tex]
[C] → 2B - 10C + A = [tex]\begin{pmatrix}12&-36\\ -31&25\end{pmatrix}[/tex]
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A toy rocket is shot vertically into the air from a launching pad 5 feet above the ground with an initial velocity of 32 feet
per second. The height h, in feet, of the rocket above the ground at t seconds after launch is given by the function
h(t)=1612 +32t+5. How long will it take the rocket to reach its maximum height? What is the maximum height?
(a) Teresa owns 9 shares of a certain stock. Yesterday the total value of her shares went down by90 dollars. What was the change in value for each share?dollarsHelp ASAP PLEASE
we have that
Divide $90 by the number of shares
so
90/9=$10
the answer is $10
Hello! I need a little bit of help with this question please. (This information is not from an open test, it is a book as I'm studying for the ASVAB I am going to take later on.)
Given:
[tex]\sqrt{100}-\sqrt{64}[/tex]To find:
We need to solve this sum and find the final answer
Step-by-step solution:
To solve this problem, we need to know the square root of 100 and 64.
√100 = 10
√64 = 8
[tex]\begin{gathered} =\sqrt{100}-\sqrt{64} \\ \\ =10\text{ - 8} \\ \\ =2 \end{gathered}[/tex]Final answer:
Thus 2 (Option A) is the correct answer.
I need help with this question... it's about special triangles and I need to find y and z.. it should also not be a decimal.
To find z, consider the right-angled triangle at the botton in the diagram showm
[tex]\begin{gathered} \sin 45\text{ = }\frac{z}{20} \\ z\text{ = 20 }\sin 45 \\ z\text{ = }20\text{ }\times\frac{1}{\sqrt[]{2}} \\ z\text{ = }\frac{20}{\sqrt[]{2}} \\ z\text{ = }\frac{20}{\sqrt[]{2}}\times\frac{\sqrt[]{2}}{\sqrt[]{2}} \\ z\text{ = }\frac{20\sqrt[]{2}}{2} \\ z\text{ = 10}\sqrt[]{2} \end{gathered}[/tex]Let the common base of both triangles be m
[tex]\begin{gathered} \cos 45\text{ = }\frac{m}{20} \\ m\text{ = 20 }\cos 45 \\ m\text{ = }\frac{20}{\sqrt[]{2}} \\ m\text{ = }\frac{20}{\sqrt[]{2}}\times\frac{\sqrt[]{2}}{\sqrt[]{2}} \\ m\text{ = 10}\sqrt[]{2} \end{gathered}[/tex]To find y:
[tex]\begin{gathered} \tan 30\text{ = }\frac{y}{m} \\ \tan 30\text{ = }\frac{y}{10\sqrt[]{2}} \\ \frac{1}{\sqrt[]{3}}=\text{ }\frac{y}{10\sqrt[]{2}} \\ y\text{ = }\frac{10\sqrt[]{2}}{\sqrt[]{3}} \\ y\text{ = }\frac{10\sqrt[]{6}}{3} \end{gathered}[/tex]To find x:
[tex]\begin{gathered} \sin 30=\frac{y}{x} \\ \frac{1}{2}=\frac{10\sqrt[]{6}}{3}\div x \\ \frac{1}{2}=\frac{10\sqrt[]{6}}{3}\times\frac{1}{x} \\ x\text{ = }\frac{20\sqrt[]{6}}{3} \end{gathered}[/tex]I will show you a pic
GIven the table above :
We have that
x y
2 8
4 4
6 0
8 4
The table represents a Non - Linear Function
Reason: It is because there is no constant ratio or proportion between x and y.
I am struggling with this question. could you help me please??
Problem
Solution
Let x = age, W= weight the two variables of interest
We have the following probabilities given:
P(x<37) =0.142
P(W< 2500) = 0.051
P(x <37 AND W<2500)= 0.031
And we want the following probability and we can use the total probability rule:
P(x < 37 OR W< 2500) = P(x<37) +P(W< 2500) -P(x<37 AND W<2500)
If we replace we got:
P(x < 37 OR W< 2500)= 0.142+ 0.051- 0.031= 0.162
write an equation of each parabola in vertex form. Vertex (3,-2) Point (2,3)
The equation of Parabola in the vertex form with vertex (3,-2) and point(2,3) is y = 5(x-3)² - 2 .
The equation of parabola with vertex (h,k) is denoted by the equation
y = a(x-h)² + k
In the question ,
it is given that
the vertex of the Parabola is (3,-2) and the point is (2,3)
So, the equation of the parabola with vertex (3,-2) will be
y = a(x-3)² - 2
Since the point (2,3) lies on the parabola ,
So, 3 = a(2-3)² - 2
3 + 2 = a*(-1)²
5 = a
Substituting a in the equation y = a(x-3)² - 2 ,
we get
y = 5(x-3)² - 2
Therefore , The equation of Parabola in the vertex form with vertex (3,-2) and point(2,3) is y = 5(x-3)² - 2 .
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An envelope is 15 centimeters wide, and it measures 17 centimeters along the diagonal. The envelope is __ centimeters tall.
An envelope is rectangular in shape.
Given the width = 15cm, and diagonal = 17cm
Let h represent the tall length of the envelope
Applying Pythagoras theorem, we have
[tex]\begin{gathered} 17^2=15^2+h^2 \\ 289=225+h^2 \\ h^2=289-225 \\ h^2=64 \\ h=\sqrt[]{64} \\ h=8\operatorname{cm} \end{gathered}[/tex]The envelope is 8cm tall
Hello, I really need help on this assignment I don't understand what to do.
Answer:
-2 and -10
Explanation:
There are two numbers at a distance of 4 units from -6, the number that is 4 units to the right and the number that is 4 units to the left.
So, the number that is 4 units to the right is equal to
-6 + 4 = -2
And the number that is 4 units to the left is equal to
-6 - 4 = -10
Therefore, the numbers are -2 and -10 and they are represented as