Answer:
standard form, real nums
Step-by-step explanation:
The edge length of a cube is changing at a rate of 10 in/sec. At what rate is the cube's volume changing when the edge length is 3 inches?
Given:
The edge length of a cube is changing at a rate of 10 in/sec.
To find:
The rate by which cube's volume changing when the edge length is 3 inches.
Solution:
We have,
[tex]\dfrac{da}{dt}=10\text{ in/sec}[/tex]
We know that, volume of cube is
[tex]V=a^3[/tex]
Differentiate with respect to t.
[tex]\dfrac{dV}{dt}=3a^2\dfrac{da}{dt}[/tex]
Substituting [tex]\dfrac{da}{dt}=10[/tex] and a=3, we get
[tex]\dfrac{dV}{dt}_{a=3}=3(3)^2(10)[/tex]
[tex]\dfrac{dV}{dt}_{a=3}=3(9)(10)[/tex]
[tex]\dfrac{dV}{dt}_{a=3}=270[/tex]
Therefore, the volume increased by 270 cubic inches per sec.
In Exercises 11-18, use analytic methods to find the extreme values of the function on the interval and where they occur. Identify any critical points that are not stationary points.
15. f(x) = sin(x + π/4), 0 ≤ x ≤ 7π/4
Answer:
Absolute maximum of 1 at x = pi/4 ; [tex](\frac{\pi}{4}, \ 1)[/tex]
Absolute minimum of -1 at x = 5pi/4 ; [tex](\frac{5\pi}{4} , \ -1)[/tex]
Local maximum of √2/2 at x = 0 ; [tex](0, \ \frac{\sqrt{2} }{2} )[/tex]
Local minimum of 0 at x = 7pi/4 ; [tex](\frac{7\pi}{4}, \ 0)[/tex]
No critical points that are not stationary points.
Step-by-step explanation:
[tex]f(x)=sin(x+\frac{\pi}{4} ), \ 0 \leq x\leq \frac{7 \pi}{4}[/tex]
Take Derivative of f(x):Let's start by taking the derivative of the function.
Use the power rule and the chain rule to take the derivative of f(x).
[tex]f'(x)=\frac{d}{dx} [sin(x+\frac{\pi}{4})] \times \frac{d}{dx} (x+\frac{\pi}{4})[/tex]The derivative of sin(x) is cos(x), so we can write this as:
[tex]f'(x)=cos(x+\frac{\pi}{4})\times \frac{d}{dx} (x+\frac{\pi}{4})[/tex]Now, we can apply the power rule to x + pi/4.
[tex]f'(x)=cos(x+\frac{\pi}{4} ) \times 1[/tex] [tex]f'(x)=cos(x+\frac{\pi}{4} )[/tex]Critical Points: Set f'(x) = 0Now that we have the first derivative of [tex]f(x)=sin(x+\frac{\pi}{4})[/tex], let's set the first derivative to 0 to find the critical points of this function.
[tex]0=cos(x+\frac{\pi}{4})[/tex]Take the inverse cosine of both sides of the equation.
[tex]cos^-^1(0) = cos^-^1[cos(x+\frac{\pi}{4})][/tex]Inverse cosine and cosine cancel out, leaving us with x + pi/4. The inverse cosine of 0 is equal to 90 degrees, which is the same as pi/2.
[tex]\frac{\pi}{2} = x +\frac{\pi}{4}[/tex]Solve for x to find the critical points of f(x). Subtract pi/4 from both sides of the equation, and move x to the left using the symmetric property of equality.
[tex]x=\frac{\pi}{2}- \frac{\pi}{4}[/tex][tex]x=\frac{2 \pi}{4}-\frac{\pi}{4}[/tex] [tex]x=\frac{\pi}{4}[/tex]Since we are given the domain of the function, let's use the period of sin to find our other critical point: 5pi/4. This is equivalent to pi/4. Therefore, our critical points are:
[tex]\frac{\pi}{4}, \frac{5 \pi}{4}[/tex] Sign Chart(?):Since this is a sine graph, we don't need to create a sign chart to check if the critical values are, in fact, extreme values since there are many absolute maximums and absolute minimums on the sine graph.
There will always be either an absolute maximum or an absolute minimum at the critical values where the first derivative is equal to 0, because this is where the sine graph curves and forms these.
Therefore, we can plug the critical values into the original function f(x) in order to find the value at which these extreme values occur. We also need to plug in the endpoints of the function, which are the domain restrictions.
Let's plug in the critical point values and endpoint values into the function f(x) to find where the extreme values occur on the graph of this function.
Critical Point Values:[tex]f(\frac{\pi}{4} )=sin(\frac{\pi}{4} + \frac{\pi}{4} ) \\ f(\frac{\pi}{4} )=sin(\frac{2\pi}{4}) \\ f(\frac{\pi}{4} )=sin(\frac{\pi}{2}) \\ f(\frac{\pi}{4} )=1[/tex]There is a maximum value of 1 at x = pi/4.
[tex]f(\frac{5\pi}{4} )=sin(\frac{5\pi}{4} + \frac{\pi}{4} ) \\ f(\frac{5\pi}{4} )=sin(\frac{6\pi}{4}) \\ f(\frac{5\pi}{4}) = sin(\frac{3\pi}{2}) \\ f(\frac{5\pi}{4} )=-1[/tex]There is a minimum value of -1 at x = 5pi/4.
Endpoint Values:[tex]f(0) = sin((0) + \frac{\pi}{4}) \\ f(0) = sin(\frac{\pi}{4}) \\ f(0) = \frac{\sqrt{2} }{2}[/tex]There is a maximum value of √2/2 at x = 0.
[tex]f(\frac{7\pi}{4} ) =sin(\frac{7\pi}{4} +\frac{\pi}{4}) \\ f(\frac{7\pi}{4} ) =sin(\frac{8\pi}{4}) \\ f(\frac{7\pi}{4} ) =sin(2\pi) \\ f(\frac{7\pi}{4} ) =0[/tex]There is a minimum value of 0 at x = 7pi/4.
We need to first compare the critical point values and then compare the endpoint values to determine whether they are maximum or minimums.
Stationary Points:A critical point is called a stationary point if f'(x) = 0.
Since f'(x) is zero at both of the critical points, there are no critical points that are not stationary points.
x+y =783
10x+7y=6459
9514 1404 393
Answer:
(x, y) = (326, 457)
Step-by-step explanation:
Use the first equation to make an expression for y that can be substituted into the second.
y = 783 -x
10x +7(783 -x) = 6459
3x = 978 . . . . . . . . . . . . subtract 7(783) = 5481
x = 326 . . . . . . . . . . . . . divide by 3
y = 783 -326 = 457
The solution is (x, y) = (326, 457).
Christina is buying a new rug for her bedroom. The rug's length is 8 yards, 2 feet, and 4 inches long. The rug's width is 1/4 of its length. What is the perimeter of the rug?
Answer:
20 yards, 6 feet
Step-by-step explanation:
Width of the rug is [tex]\frac{1}{4}[/tex] of the length:
[tex]\frac{1}{4}[/tex] × 8 yards = 2 yards
[tex]\frac{1}{4}[/tex] × 2 feet = [tex]\frac{1}{2}[/tex] foot = 6 inches
[tex]\frac{1}{4}[/tex] × 4 inches = 1 inch
P = 2( l + w )
2 × l = 16 yards, 4 feet, 8 inches
2 × w = 4 yards, 1 foot, 2 inches
P = 20 yards, 5 feet, 12 inches = 20 yards, 6 feet
45×(−10)×45 STEP explanation
Answer:
-20,250
Step-by-step explanation:
PEMDAS
Multiply left to right
45 * -10 * 45
45* -10
-450 * 45 =
-20,250
Please help me out, I don’t usually ask for answers but this will mean the world too me
56) A tire retailer plans to sell
$100,000 worth of tires in
January. Planned reductions are
$15,000; planned inventory for
January Ist is $300,000 and for
January 31st is $350,000. As of
January 8th, the retailer has
ordered $15,000 for January
delivery. What is the open-to-
buy for the rest of the month?
Answer:
/
Step-by-step explanation:
You start at (2, 5). You move left 5 units and right 8 units. Where do you end?
Answer:
(5,5)
Step-by-step explanation:
left 5 is -5 and right 8 is +8.
-5+8=3
2+3=5
When talking about left and right, you change the x-coordinate, when up and down, y-coordinate.
Overall, (5,5)
Answer:
5,5
Step-by-step explanation:
since we are not changing the height of the point and just moving side to side, the y intercept does not change. Once you move 5 units to the left, your new point is -3,5 but then when you move 8 to the right it switches to positive again and the new point is 5,5
The spinner in a game based on chance has 25 equal sectors. The prize is worth $50, and the cost to play is $2.50. A player wins if the arrow lands on the sector he or she chose.
Which table shows the correct outcomes for a player who plays the game one time?
Answer:
________________ WIN _____ LOSE
Outcome(x) ______ $47.50 ____ - $2.50
PROBABILITY P(x) _____ __________
Step-by-step explanation:
Given that :
The cost of play = $2.50
The winning price = $50
Hence, if a player emerges victorious or wins; the net amount won is :
Winning price - cost of play
$50 - $2.50 = $47.50
If the player loses :
Then he wins $0
Net amount lost equals :
Amount won - cost of play
$0 - $2.50 = - $2.50
________________ WIN _____ LOSE
Outcome(x) ______ $47.50 ____ - $2.50
PROBABILITY P(x)
Answer:
Win Lose
47.50 -2.50
Step-by-step explanation:
Got it right on the test.
Define a variable, write an inequality, and solve the problem : Four less than nine times a number is greater than eleven more than two times the same number.
Answer:
x > 15/7
Step-by-step explanation:
four less than nine times a number:
9x - 4
Eleven more than two times the same number:
2x + 11
Thus:
9x - 4 > 2x + 11
7x > 15
x > 15/7
I hope this helps you :)
Perform the function compositions
Answer:
1. 4x^2-24x^2+37
2.16x^4+8x^2-2
Step-by-step explanation:
I need help figuring out what: f(r-2)=6x+7 is
Answer:
f(r-2) = 6r -5
Step-by-step explanation:
f(r-2) is another notation for x = r-2
so we’ll plug in r-2 for x and solve:
f(r-2) = 6(r-2)+7
f(r-2) = 6r -12 +7
f(r-2) = 6r -5
Write the number 7,070,000,000 in scientific notation.
Answer:
7.07 x 10^9
Hope this Helps!! Have a fabulous day!
~PumpkinSpice1
Answer:
7.07 x 10^9
Step-by-step explanation:
hope it helps!
the process of finding factors
8)Four added to twice a number yields. Find the number.
Answer:
X=-2
Step-by-step explanation:
Let no =X
So 2x+4=0
2x=-4
X=-4/2
X=-2
So the answer is -2
ILL GIVE BRAINLIEST FOR BEST AWNSER!!!
A football player kicks a ball with a force of 50 N
How does the impulse above change if he doubles the amount of time that his foot is in contact with the ball?
Answer: AP= 30 kgm/s c) Calculate the final velocity of the object, if it was initially at rest. A 0.5kg baseball experiences a 10N force for a duration of 0.1s.
Step-by-step explanation:
Select ALL the true statements
Answer:
2 and 5 because dilation always have the same angles and they can increase and decrease in size
Step-by-step explanation:
But I am not sure so if I am wrong I am sorry I did a test like this
ksdfjsbjgdfg f sndm f
Answer:
Nice
Step-by-step explanation:
Answer:
hahhhh funny one nice :-)
Needdd helpp noww please
Giving extra points
Answer:
b
20
10
Step-by-step explanation:
I NEED HELP PLZ
Write an equation in point slope and slope-intercept that includes the points (0, 0) and (9, -3).
Ten percent of the engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested, what is the probability the fourth defective engine will be found on the second trial?
Answer:
0
Step-by-step explanation:
From the given information;
Let x be the number of trails on which the [tex]r^{th}[/tex] defective occurs.
Let the probability that an occurrence of a defective be p = 0.10
Suppose x is a random variable that follows a negative binomial distribution with parameters r and p.
Then the probability mass function of X can be expressed as:
[tex]P(X=x) = \bigg (^{x-1}_{r-1} \bigg)p^r (1 - p) ^{x-r} \ \ \ \ ; x=r, \ r+1 , \ r+2 , ...[/tex]
[tex]P(X=x) = \bigg (^{x-1}_{r-1} \bigg)(0.10)^r (1 - 0.10) ^{x-r}[/tex]
[tex]P(X=x) = \bigg (^{x-1}_{r-1} \bigg)(0.10)^r ( 0.90) ^{x-r} \ \ ... \ \ (1)[/tex]
We are to find the probability of the fourth defective engine will be found on the second trial.
i.e. r = 4 and x = 2
[tex]P(X=2) = \bigg (^{2-1}_{4-1} \bigg)(0.10)^4 ( 0.90) ^{2-4}[/tex]
[tex]P(X=2) = \bigg (^1}_{3} \bigg)(0.10)^4 ( 0.90) ^{-2}[/tex]
[tex]P(X=2) = \bigg (\dfrac{1!}{3!(1-3)!} \bigg)(0.10)^4 ( 0.90) ^{-2}[/tex]
[tex]\mathbf{P(X=2) = 0}[/tex]
Please help 30 points plus brainlyist who does first
Determine which expressions can be simplified further, and which cannot. Sort the expressions into the correct
category
2x + 3y
Can Be Simplified
Cannot Be Simplified
x + x
4r +
7y + 1
4y + 4x
y + 2y
Answer:
Step-by-step explanation:
expresion can be simplified is they have like terms such as
x+x=2x
y+2y=3y
expresions can NOT be simplified if they have difrerent variables or just one number suchh as
2x+3y
7y+1
4y+4x
I do not know what is 4r+
Answer:
can - y +2y
9x+6x
4x+x
can't 4y+4x
7y+1
2x+3y
Consider the function. f(x) = 2x(x - 1)
What are the values of
f(0)=
f(3)=
f(-4)=
Answer:
3
Step-by-step explanation:
Directions : Find the indicated term of each arithmetic sequence.
Question :14th term: a = 4; d = -5
Answer:
The 14th term is -61.
Step-by-step explanation:
T(n) = 9-5n
T(14) = 9 - 5 * 14 = 9 - 70 = -61
Write the fractional equivalent (in reduced form) to each number
10
0.125 =
0.16
simplify the given involving the indicated division.
2(3x+10)=6x+20 does this equation have one solution or many ? Explain why
Answer:
2(3x+10) = 6x+20
6x+20 = 6x+20
6x = 6x
x = x
This has an infinite amount of solutions
Step-by-step explanation:
cube m holds a maximum of 64 unit cubes. each unit cube has an edge length of 1/2 centimeter. what is the length of cube m?
Answer:
136 or 34
Step-by-step explanation:
not sure if its right because my math is bad
twice a number k plus the quantity s minus 2
Answer:
2k+s-2
im assuming
Please help me!!!!!!!!!!!!!!!!!!!!
Answer:
70
Step-by-step explanation:
4060/58=70
