When the radius is 10 inches and the height is 20 inches
(a) rate change of the volume of the cylinder is 5340.7 inch²/min.
(b) rate of change of the surface area of the cylinder is 1068.14 inch²/min.
Volume and surface area of the closed cylinder both increase with time.
Let r be the Radius and h be the height of the cylinder in inches at time t.
[tex]\frac{dr}{dt} =5[/tex] inch/min and [tex]\frac{dh}{dt} =-3[/tex] inch/min
Let V be the volume of the cylinder then
Differentiating with respect to t,
[tex]\frac{dV}{dt} =2\pi r(\frac{dr}{dt} )h+\pi r^{2} \frac{dh}{dt}[/tex]
At r=10 inch and h=20 inch,
[tex]\frac{dV}{dt} =1700\pi =[/tex] 5340.7 inch²/min ≈ 5000 inch²/min
Thus, it is increasing with time as it has a positive sign.
Let S be the surface area of a closed cylinder
[tex]\frac{dS}{dt} =2\pi r(\frac{dr}{dt} )h+2\pi r(\frac{dr}{dt} )+4\pi (\frac{dr}{dt} )r[/tex]
At r=20 inch and h=20 inch
[tex]\frac{dS}{dt} =340\pi =[/tex] 1068.14 inch²/min ≈ 1000 inch²/min
It is also increasing with time as it is also having a positive sign.
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a 13-ft ladder rests against a vertical wall. if the bottom of the ladder slides away at 1 ft/s, at what rate is the top of the ladder sliding down the wall when the bottom of the ladder is 5 ft from the wall? g
The given situation can be solved using the differential equation and the top of the ladder slides downward at 0.417 ft/s.
Let:
x be the distance from the bottom of the ladder to the base of the wall
y be the distance from the top of the ladder to the bottom of the wall
When the ladder is still, apply the Pythagorean Theorem,
x² + y² = 13²
5² + y² = 13²
y² = 169 - 25 = 144
y = 12 ft
When the ladder slides, apply the differential equation:
x² + y² = 13²
2x . dx/dt + 2y . dy/dt = 0
Substitute x = 5, dx/dt = 1, y = 12
2 . 5 . 1 + 2 . 12 . dy/dt = 0
24 dy/dt = -10
dy/dt = -5/12 = - 0.417 ft/s
The minus sign indicates that the ladder is sliding downward.
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Whats the midpoint of line segment (-1,6), (3,0)
Answer:
(1,3)
Step-by-step explanation:
You can use the distance formula to help you solve the answer
Answer:
(1; 3)
Step-by-step explanation:
x = (-1 + 3)/2 = 1
y = (6 + 0)/2 = 3
Given diagram is a solid prism of a triangular base. if the base of the prism is 480 cm cube. find its height .
Answer:
20 cm
Step-by-step explanation:
Given a prism with a right triangle base and a volume of 480 cm³, you want the height of the prism. The base has one side 8 cm, and hypotenuse 10 cm.
Base edgeThe missing edge of the right triangle base can be found using the Pythagorean theorem. It tells us the square of the hypotenuse is the sum of the squares of the other two sides:
b² = c² +a²
10² = 8² +a² . . . . . . . use given lengths
a² = 100 -64 = 36 . . . . subtract 8², simplify
a = 6 . . . . . . . . . . . . . length of side BC
Base areaThe area of the right triangle base is ...
A = 1/2bh . . . . . . . . b is the triangle base; h is its height
A = 1/2(6 cm)(8 cm) = 24 cm²
VolumeThe volume of the prism is ...
V = Bh . . . . . . . . . . . . . . . where B is the base area, and h is the height
480 cm³ = (24 cm²)h . . . use known values
h = 20 cm . . . . . . . divide by the coefficient of h
The height of the prism is 20 cm.
What is the value of in if the remainder of /is 2?
O A. 1
OB. i
O C. -1
O D. -1
The value of i where remainder of n/ 4 is 2 is :
i² = -1 .
What is i?I has the value √-1.The complex numbers are expressed using the imaginary unit number, where I is defined as imaginary or unit imaginary. An imaginary number is a complex number component that can be written as a real number multiplied by the imaginary unit I where i² = -1. When the imaginary number is multiplied by itself, the result is negative. Consider the imaginary number 3i, which, when multiplied by itself or divided by its square, yields 9i², or -9.z = a + ib .i² = -1
i⁰ = 1
i¹ = i
i² = -1
i³ = i² x i
= -1 x i
= -i
i⁴ = i² x i²
= -1 x -1
= 1
The pattern repeats.
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What is one solution of this system?
Answer:
C (0,3)
Step-by-step explanation:
Plot the two equations and the points. See the attached graph. The blue area reflects the inequality of y+3x<=8: many points satisfy this equation and they are all blue.
The red line is equation y-3=x. There is only one point on this line that is also in the blue area: (0.3). The other three points (A, B, and D) do not satisfy either equation. The metric term for them is losers.
a lab technician is tested for her consistency by taking multiple measurements of cholesterol levels from the same blood sample. the target accuracy has an average of 2.78 or less with a standard deviation of 1.17. if the lab technician takes 16 measurements and the variance of the measurements in the sample is 2.45, does this provide enough evidence to reject the claim that the lab technician's accuracy is within the target accuracy?
The proper test statistic has a value of 13.21.
As a result, repeated measurements of cholesterol levels from the same blood sample are taken to assess a lab technician's consistency.
The desired accuracy is a measurement variance of 2.78 or less. In the event that the lab technician conducts 16 measurements, and the sample's measurement variance is 2.45.
1) n=16 samples were used.
The desired accuracy is a measurement variance of 2.78 or less.
The sample's measurements' variance is 2.45 or less.
We list both the null and alternative hypotheses in response to the query.
Null hypothesis [tex]H_{0}[/tex] = [tex]var^{2} \geq[/tex] 2.78
Alternative hypothesis [tex]H_{a}[/tex] = [tex]var^{2} <[/tex] 2.78
We claim the alternative hypothesis.
2)Calculate the appropriate test statistic's value.
Using Chi-square,
X= ((n-1)(2.45)/(2.78)
X= 15(2.45)/(2.78)
X=36.75/2.78
X= 13.21
Therefore, The value of the appropriate test statistic is 13.21
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Which of the following best describes the slope of the line below?
Answer:
D. Negative
Step-by-step explanation:
Always read the graph from left to right
The graph is going down thus it is negative
Answer: Negative.
Step-by-step explanation: The graph of the line falls from left to right which makes it negative.
Find p(-1) and p(2) for p(x)=4-3x
Answer:
p(-1) = 7
p(2) = -2
Step-by-step explanation:
We have p(x) = 4 - 3x
p(-1) = 4 - 3(-1) = 4 + 3 = 7
p(2) = 4 - 3(2) = 4 - 6 = -2
The cone of the volcano has a height of 414 meters and a diameter of 416 meters. Find the volume of the cone. Round your answer to the nearest hundred thousand. Use 3.14 for 1. The volume of the cone is about m3.
Volume of a cone is given by;
[tex]V=\frac{1}{3}\pi^{}r^3h[/tex]From the queston,
h = 414
diameter = 416, this implies r = d/2 = 416/2 = 208
π = 3.14
substitute the values into the formula
[tex]V=\frac{1}{3}\times3.14\text{ }\times208^2\times414[/tex][tex]=18747156.48[/tex][tex]\approx18700000m^3[/tex]Simplify: I3 - 6| - (12 ÷ 6 + 1)3
Solution
The question would like us to evaluate the following expression
[tex]|3-6\left|-(12\div6+1\right)^2[/tex]- We should deal with the modulus and bracket separately.
- After that, we can then perform the subtraction operation on the results of the modulus and bracket.
- This is done below:
[tex]\begin{gathered} |3-6|=|-3\left|\right? \\ |-3\left|\right?=3\text{ \lparen Because the modulus always returns a positive number\rparen} \end{gathered}[/tex]Also,
[tex]\begin{gathered} \lparen12\div6+1)^2 \\ By\text{ the rules of PEDMAS,} \\ Division\text{ comes before Addition, thus, we should perform the division operation first} \\ 12\div6=2 \\ \\ \lparen2+1)^2=3^2=9 \end{gathered}[/tex]Thus, combining both results, we have:
[tex]\begin{gathered} 3-9 \\ =-6 \end{gathered}[/tex]Final Answer
The answer is -6
Answer:
-6 hope it helps
thank you
A set of data is summarized by the stem and leaf plot below.
A Stem and Leaf Plot is a special table where each data value is split into a "stem" (the first digit or digits) and a "leaf" (usually the last digit).
With this in mind we conclude that:
There are 8 values in the data set which are greater than or equal to 20 and less than or equal to 29. This comes from the fact that we have to count how many leafs are in the stem 2.
There are 6 values in the data set which are greater than or equal to 10 and less than or equal to 19. This comes from the fact that we have to count how many leafs are in the stem 1.
There are 14 values in the data set which are greater than or equal to 40 and less than or equal to 49. This comes from the fact that we have to count how many leafs are in the stem 4.
70 points plss help composite functions
Answer:
[tex](f \circ g)(2)=2[/tex]
Step-by-step explanation:
Given functions:
[tex]\begin{cases}f(x)=-2x+8\\g(x)=\sqrt{x+7}\end{cases}[/tex]
The composite function (f o g)(x) means to substitute the function g(x) in place of the x in function f(x).
Therefore (f o g)(2) means to substitute the result of g(2) in place of the x in the function f(x).
[tex]\begin{aligned}\implies (f \circ g)(2) &=f[g(2)]\\&=f(\sqrt{2+7})\\&=f(\sqrt{9}) \\& = f(3)\\& = -2(3)+8\\&=-6+8\\&=2\end{aligned}[/tex]
A red ballon is 40 feet above the ground and rising at 2 ft/s. At the same time, a blue balloon is at 60 feet above the ground and descending at 3 ft/s. What will the height of the balloons be when they are the same height above the ground
Answer: 48 ft
Step-by-step explanation:
The height gap between the balloons is 60 -40 = 20 feet. That gap is being closed at the rate of 2 + 3 = 5 ft/s, so will be gone in ...
(20 ft)/(5 ft/s) = 4 s
At that time, the red balloon will have risen (2 ft/s)(4 s) = 8 ft to a height of ...
40 ft +8 ft = 48 ft
The blue balloon will have descended (3 ft/s)(4 s) = 12 ft to a height of ...
60 ft -12 ft = 48 ft
The balloons at at 48 ft when they are both the same height.
_____
Time and speed and distance are related by the formula you see on every speed limit sign:
speed = distance/time . . . . . . . (on the sign, it's "miles per hour")
or
time = distance/speed
or
distance = speed × time
_____
If you want equations, you can write them as ...
h = 40 +2t
h = 60 -3t
where h is the altitude the balloons have when they are at the same height, and t is the number of seconds it takes to get there.
We're only interested in h, so we can cancel t by multiplying the first equation by 3 and adding that to the second equation multiplied by 2:
3(h) + 2(h) = 3(40 +2t) +2(60 -3t)
5h = 120 +6t +120 -6t
h = 240/5 = 48 . . . . the height in feet at which the balloons are the same height
Factor the monomial 16x²y
The factors of the given monomial are given below
What is a monomial?
A monomial is, broadly speaking, a polynomial with just one term in mathematics. There are two definitions of a monomial: A monomial, often known as a power product, is a product of powers of variables with nonnegative integer exponents, or a product of variables with repeats. The constant 1 is a monomial, which means that it is equivalent to the empty product and to for any variable x. If just one variable, x, is examined, a monomial is either 1 or a power xn of x, where n is a positive integer. If many variables, say x,y,z, are examined, each can be assigned an exponent, such that each monomial has the form xaybzc with a,b,c non-negative integers.
The factors of the monomial 16x²y are 2.2.2.2.x.x.y
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The factors of the monomial are given below
What is a monomial?
A monomial is, broadly speaking, a polynomial with just one term in mathematics. there are two definitions of a monomial: A monomial, often known as a power product, is a product of powers of variables with nonnegative integer exponents, or a product of variables with repeats. The constant 1 is a monomial, which means that it is equivalent to the empty product and to for any variables x. If just one variable, x, is examined, a monomial is either 1 or a power xn of x, where n is a positive integer, if many variables, say x, y, z, are examined, each can be assigned an exponent, such that each monomial has the form xaybzc with a, b, c non-negative integers.
The factors of the monomial 16x²y are 2.2.2.2.x.x.y
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A $350,000 house is assessed at % of its value. If the yearly tax rate is $3.25 per hundred of assessed value, what is the yearly tax on this property?
F. $2,275
G. $1,600
H. $5,687
J. $6,405
K. $8,000
Please answer quick! i'm stuck on th is and it is due today! (equation is on the bottom on the picture)
4 - 1/3x = x + 2x
10 Km to 20 m convert into ratio
Answer:
10 : 0.02 (20m)
Based on the given diagram, complete the sentence below.
Point D is the centroid of ΔABC, since we know that A.F, BC, and CE are all medians.
What is the Median of a Triangle?A median of a triangle is the line segment that connects the vertex of a triangle to the midpoint of the opposite side. There are usually three medians of a triangle.
What is the Centroid of a Triangle?The point where the three medians of a triangle intersect each other or meet at is referred to as the centroid of the triangle.
The diagram shows a triangle with three medians that meets at point D in the triangle. Therefore, point D is the centroid of the tringle because lines A.F, BC, and CE are all medians of the given triangle.
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Help please. Was absent due to medical issues and trying to catch myself up and learn it now.
We paint with:
• red the regions where the function is decreasing,
,• green the regions where the function is increasing.
From the graph, we see that the function is:
• decreasing in the intervals (-∞, -1.5) and (2, ∞),
,• increasing in the interval (-1.5, 2).
Answer
• Increasing on the interval(s): ,(-∞, -1.5), (2, ∞)
,• Decreasing on the interval(s): ,(-1.5, 2)
help!!
I'm not sure if i'm right...
are these the steps to solve for x?
-5 *4 +3 /2
Yes, you are right!
First of all, we will subtract [tex]5[/tex] from each side.
[tex]\frac{2x-3}{4}=5[/tex]Then we multiply both sides by [tex]4[/tex].
[tex]2x-3=20[/tex]Then add [tex]3[/tex] to each side.
[tex]2x=23[/tex]Finally, we divide each side by [tex]2[/tex].
[tex]x=\frac{23}{2}=11.5[/tex][tex]\boldsymbol{\sf{\cfrac{2x-3}{4}+5=10 }}[/tex]
Multiply the two sides of the equation by 4.
2x - 3+20 = 40
Add −3 and 20 to get 17.
2x+17 = 40
It remains 17 on both sides.
2x = 40−17
Subtract 17 of 40 to get 23.
2x = 23
Divide both sides by 2.
[tex]\boldsymbol{\sf{x=\dfrac{23}{2} \ \ \longmapsto \ Answer }}[/tex]
Solve for x in the diagram below
If two angles be 10x + 5 and 15x - 30 then the value of x = 7.
How to find the value of x?Let the two angles be 10x + 5 and 15x - 30.
simplifying the above two equations, we get
10x + 5 = 15x - 30
Subtract 5 from both sides
10x + 5 - 5 = 15x - 30 - 5
Simplifying the above equations,
10x = 15x - 35
Subtract 15x from both sides of the equation
10x - 15x = 15x - 35 - 15x
Simplifying the above equation, we get
-5x = -35
Divide both sides by -5
[tex]$\frac{-5 x}{-5}=\frac{-35}{-5}[/tex]
x = 7
Therefore, the value of x = 7.
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What is the area of a triangle whose vertices are D(1, 1), E(3, -1), and F(4, 4)?
Enter your answer in the box.
Answer:
6
Step-by-step explanation:
Trust
Mid point of -2,11 and 18,-1
The midpoint of (-2, 11) and (18, -1) is:
[tex]\begin{gathered} \bar{x}=\frac{-2+18}{2}\rightarrow\bar{x}=8 \\ \\ \bar{y}=\frac{11-1}{2}\rightarrow\bar{y}=5 \end{gathered}[/tex]Point (8,5)
Distribute:
-8(9 - 4x)
Answer:
32x – 72
OR
–72 + 32x
Step-by-step explanation:
To use the Distributive Property, just simply follow my steps:
Write the equation:
–8(9 – 4x)
To distribute, multiply the numbers inside the parenthesis by the number outside of the parenthesis SEPARATELY. It should look like this:
(–8 • 9)(–8 • –4x)
(–72)(32x)
Since the 32 is positive, the equation could be written either way:
–72 + 32x
OR
32x – 72
If you want the equation to be fully solved...
Set the equation equal to zero:
32x – 72 = 0
Isolate the variable. Subtract 72 from both sides:
32x – 72 + 72 = 0 + 72
32x = 0 + 72
32x = 72
Divide:
32x/32 = 72/32
x = 9/4 (Improper Fraction Form)
OR
x = 2 (and) 1/4 (Mixed Number Form)
Convert 9/4 to a decimal:
x = 9/4
x = 2.25
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Sailman A gets paid $300 a month plus $8 for every sale he makes. Salesman B gets paid $1500 a mont. Write and equation and solve to see how many salesman a must make in order to make the same amount of salesman b
the number of sales by b must be 150 to make the same amount of salesman B.
What is Algebraic expression ?
Algebraic expressions are the idea of expressing numbers using letters or alphabets without specifying their actual values. The basics of algebra taught us how to express an unknown value using letters such as x, y, z, etc. These letters are called here as variables. An algebraic expression can be a combination of both variables and constants. Any value that is placed before and multiplied by a variable is a coefficient.
given that :
Salesman A gets paid $300 a month plus $8 for every sale he makes :
let the no. of sales A made be : x
then, in order to make the same amount of salesman b we equate btoth the amount :
300 + 8x = 1500
8x = 1500-300
8x = 1200
x = 1200/8
x = 150
Therefore, the number of sales by A must be 150 to make the same amount of salesman b.
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What is the value of logx?y3lodwhen given the following:Zlog(x) = 3log(y)=2log(z)= -1
Given the question
[tex]\log (\frac{x^2y^3}{z})[/tex]To resolve this, we can follow the steps below
Step1: Apply the logarithm rule
[tex]\log (\frac{x^2y^3}{z})=\log x^2+\log y^3-\log z[/tex]This will give=>
[tex]\log x^2+\log y^3-\log z=2\log x+3\log y-\log z[/tex]Since we have been given that
[tex]\begin{gathered} \log x=3 \\ \log y=2 \\ \log z=-1 \end{gathered}[/tex]Step2: Substitute the given values into the equation
[tex]2\log x+3\log y-\log z=2(3)+3(2)-(-1)[/tex]=>
[tex]6+6+1=13[/tex]Answer = 13
The perimeter of a square can be found using p=n+n+n+n
Answer: Well, yes. It's more like p = s + s + s + s. "S" = side. "P" = perimeter.
So, this is theoretically true if your variable for a side just happened to be "n".
You also put, "The perimeter of a square", A simpler way to put this is
P = s4 or in your case P = n4.
Because this shape is a square all side lengths and angles will be congruent, and therefore will all be the same (which is why we are able to multiply one side by four.)
Hope this helps. :)
On the spinner show
We want to know what's the probability of spinning a number greater than two
There are 2 spinnings greater than two, of a total of 8. therefore, the probability I will be
[tex]\frac{\text{favorable cases}}{\text{all cases}}=\frac{2}{8}=\frac{1}{4}[/tex]The probability is 1/4
x + 6Identify the vertical asymptotes of f(x)(1 point)- 9x + 18O x= -6 and x = -3O x= -6 and x = 3oO x = 6 and x = -3O x = 6 and x = 3
So we want to find the vertical asymptotes of the function:
[tex]f(x)=\frac{x+6}{x^2-9x+18}[/tex]Remember that there's a vertical asymptote at points of x where the denominator of the rational function is 0.
So, equal the denominator to zero:
[tex]x^2-9x+18=0[/tex]And solve this equation for x as follows:
We could factor
[tex]\begin{gathered} x^2-9x+18=0 \\ (x-6)(x-3)=0 \end{gathered}[/tex]Now, notice that any of both factors could be zero, so:
[tex]\begin{cases}x-6=0\to x=6 \\ x-3=0\to x=3\end{cases}[/tex]Therefore, the function has vertical asymptotes at x=6 and x=3.
Question is in the image. I need help with #19
EXPLANATION
Replacing the given functions into the following:
[tex]\frac{f(x+h)-f(x)}{h}[/tex][tex]\frac{x^2-1+h-(x^2-1)}{h}[/tex]Removing the parentheses and simplifying:
[tex]\frac{x^2-1+h-x^2+1}{h}[/tex]Simplifying like terms:
[tex]\frac{h}{h}=1[/tex]Hence,
[tex]\frac{f(x+h)-f(x)}{h}=1[/tex]