Answer:
[tex]\frac{dh}{dt} = \frac{216}{1875\pi}[/tex]
Step-by-step explanation:
Given
Represent radius with r and height with h
[tex]h = 12ft[/tex]
[tex]r = 10ft[/tex]
[tex]Rate = \frac{8ft^3}{min}[/tex] when [tex]h = 10ft[/tex]
Express radius in terms of height
[tex]\frac{r}{h} = \frac{10}{12}[/tex]
[tex]\frac{r}{h} = \frac{5}{6}[/tex]
[tex]r = \frac{5}{6}h[/tex]
First, we need to determine the volume of the cone in terms of height.
[tex]Volume = \frac{1}{3}\pi r^2h[/tex]
Substitute [tex]r = \frac{5}{6}h[/tex]
[tex]V = \frac{1}{3} * \pi * (\frac{5}{6}h)^2 * h[/tex]
[tex]V = \frac{1}{3} * \pi * \frac{25}{36}h^2 * h[/tex]
[tex]V = \frac{1}{3} * \pi * \frac{25}{36}h^3[/tex]
[tex]V = \frac{25}{108} \pi h^3[/tex]
Differentiate both sides w.r.t time (t)
[tex]\frac{dV}{dt} = 3 * \frac{25}{108} \pi h^{3-1} * \frac{dh}{dt}[/tex]
[tex]\frac{dV}{dt} = \frac{75}{108} \pi h^{2} \frac{dh}{dt}[/tex]
Solve for [tex]\frac{dh}{dt}[/tex]
[tex]\frac{dh}{dt} = \frac{dv}{dt} * \frac{108}{75\pi h^2}[/tex]
At this point, [tex]h = 10[/tex] and [tex]\frac{dv}{dt} = \frac{8ft^3}{min}[/tex]
So, we have:
[tex]\frac{dh}{dt} = 8 * \frac{108}{75\pi 10^2}[/tex]
[tex]\frac{dh}{dt} = \frac{8 * 108}{75\pi 10^2}[/tex]
[tex]\frac{dh}{dt} = \frac{864}{7500\pi}[/tex]
[tex]\frac{dh}{dt} = \frac{216}{1875\pi}[/tex]
Hence:
The rate is [tex]\frac{216}{1875\pi} ft/min[/tex]
Using implicit differentiation, it is found that the depth of water changes at a rate of -0.0764 feet per minute.
The volume of a cone of radius r and height h is given by:
[tex]V = \frac{\pi r^2h}{3}[/tex]
Applying implicit differentiation, we can find it's rate of change, thus:
[tex]\frac{dV}{dt} = \frac{2\pi rh}{3}\frac{dr}{dt} + \frac{\pi r^2}{3}\frac{dh}{dt}[/tex]
In this problem:
Height of 12 ft and radius of 10 ft, thus [tex]h = 12, r = 10[/tex].Radius does not change, thus [tex]\frac{dr}{dt} = 0[/tex].Water leaks at a rate of 8 cubic feet per minute, thus [tex]\frac{dV}{dt} = -8[/tex]Then:
[tex]\frac{dV}{dt} = \frac{2\pi rh}{3}\frac{dr}{dt} + \frac{\pi r^2}{3}\frac{dh}{dt}[/tex]
[tex]-8 = \frac{\pi (10)^2}{3}\frac{dh}{dt}[/tex]
[tex]\frac{dh}{dt} = -\frac{24}{100\pi}[/tex]
[tex]\frac{dh}{dt} = -0.0764[/tex]
The depth of water changes at a rate of -0.0764 feet per minute.
A similar problem is given at https://brainly.com/question/9543179
PLEASE HELP MEEEEEEE I NEED THIS QUICK
Which of the following best defines the two quantities that are measured to find the call rate of a phone company? (5 points)
Group of answer choices
The quantity of price measured in dollars depends on the quantity of time measured in minutes.
The quantity of time measured in minutes depends on the quantity of price measured in dollars.
The quantity of time measured in minutes depends on the quantity of speed measured in calls per minute.
The quantity of speed measured in calls per minute depends on the quantity of time measured in minutes.
Answer:
A: The quantity of the price would be measured in dollars depends on the quantity of time measured in minutes.
Step-by-step explanation:
A seemed like the best answer to me. I hope I helped
mike is 46 yrs old. He is 4 yrs older than 3 times his son Jared's age. Find Jared's age
Answer:
Jared is 14 years old
Step-by-step explanation:
Will give brain if correct segment addition
Answer:
13) 28
14) 22
15) 24
16)13
Step-by-step explanation:
13) Because they all lie on the same line, AC would be AB+BC, or 16+12, or 28.
14) Same thing as 13, 13+9=22
15) Same thing as others but make an equation
-1+2x+11=3x+3
Simplifying the equation, we get x=7. Now, we plug in x=7 to 3x+3 to get AC.
3(7)+3=21+3=24
16) build the equation:
x+14+x+10=22
2x+24=22
2x=-2
x=-1
Now plug in x=-1 to x+14.
-1+14=13
the volume of a cube is 125 in cúbicas. How long is each side?
3m - 3(m + 8) > 3m
A) m 8
C) m > -8
D) m < -8
lp assap
A system of equations is shown.
- x + 3y = 9
x = 1 - 2y
What is the solution of the system?
A. (-19, 10)
B. (-6, 7/10)
C. (-9/2, 9/2)
D. (-3,2)
Answer:
c
......................
Step-by-step explanation:
.....................
Answer:
c
Step-by-step explanation:
Michelle has a piece of yarn measuring 7 yards. How many 1-inch strips can be cut from the yarn?
Answer:
252
Step-by-step explanation:
7 yards is equivalent to 252 inches.
Answer:
252 inches
Step-by-step explanation:
There are 3 feet in 1 yard. Multiply 7 yards x the 3 feet in each 1 yard.
You should now have 21 feet.
Multiply 21 feet x the 12 inches in each foot.
You should come up with 252 inches as your final product.
g The average lifespan for a certain type of vehicle is 8 years and follows an exponential distribution. A lot contains 200 of these vehicles, brand new. What is the probability that 50 or more of them fail in the first 2 years
Answer:
The probability is [tex]P(A \ge 50 ) = 0.14321[/tex]
Step-by-step explanation:
The average life span is [tex]\pi = 8[/tex]
The sample size is n = 200
Generally we are told that average lifespan for a certain type of vehicle follows an exponential distribution
i.e
[tex]X \~ \ \ \ exp(8)[/tex]
Gnerally the probability distribution function for exponential distribution is
[tex]f(x) = \left \{ {{\lambda * e^{-\lambda * x }\ \ \ x\ge 0 } \atop {0 \ \ \ \ \ \ \ \ \ \ x < 0}} \right.[/tex]
Now [tex]\lambda[/tex] is the constant which is evaluated as
[tex]\lambda = \frac{1}{\pi}[/tex]
=> [tex]\lambda = \frac{1}{8}[/tex]
=> [tex]\lambda = 0.125[/tex]
Generally the that the vehicles will fail in less than 2 years is mathematically represented as
[tex]P(X \le 2 )= p= \int\limits^2_0 {\lambda e^{- \lambda * x} } \, dx[/tex]
[tex]P(X \le 2 )= p= [ {0.125 * - \frac{1}{0.125 } e^{- 0.125 * x} } ]|\left 2 } \atop {0}} \right.[/tex]
[tex]P(X \le 2 )= p= [ {0.125 * - \frac{1}{0.125 } e^{- 0.125 * 2} } ] - [ {0.125 * - \frac{1}{0.125 } e^{- 0.125 * 0} } ][/tex]
[tex]P(X \le 2 )= p= -0.77880 + 1[/tex]
[tex]P(X \le 2 )= p= 0.2212[/tex]
Generally the probability that the vehicle will fail in the first 2 years follows a binomial distribution with the mean evaluated as
[tex]\mu = n * p[/tex]
=> [tex]\mu = 200 * 0.2212[/tex]
=> [tex]\mu = 44.24[/tex]
The standard deviation is
[tex]\sigma = \sqrt{np(1 - p)}[/tex]
[tex]\sigma = \sqrt{200 * 0.2212 (1 - 0.2212)}[/tex]
[tex]\sigma = 5.86[/tex]
Using normal approximation of binomial distribution the probability that 50 or more of them fail in the first 2 years is mathematically represented as
[tex]P(A \ge 50 ) = 1 - P(A < 50 )[/tex]
Applying continuity correction
[tex]P(A \ge 50.5 ) = 1 - P(A < 50.5 )[/tex]
Here
[tex]P(A < 50.5 ) = P(\frac{A - \mu }{\sigma} < \frac{50 - 44.24}{5.87} )[/tex]
Generally
[tex]\frac{A - \mu }{\sigma} = Z (The \ standardized \ value \ of A )[/tex]
=> [tex]P(A < 50.5 ) = P( Z < 1.0664 )[/tex]
From the z-table the probability value of ( Z < 1.0664 ) is
[tex]P( Z < 1.0664 ) = 0.85679[/tex]
So
[tex]P(A \ge 50 ) = 1 - 0.85679[/tex]
=> [tex]P(A \ge 50 ) = 0.14321[/tex]
If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 194 feet
Complete Question
Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 220 feet and a standard deviation of 40 feet. Let X = distance in feet for a fly ball.
If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 194 feet
Answer:
The value is [tex]P(X < 194 ) = 0.25785[/tex]
Step-by-step explanation:
From the question we are told that
The mean is [tex]\mu = 220 \ feet[/tex]
The standard deviation is [tex]\sigma = 40 \ feet[/tex]
Generally the probability that this ball traveled fewer than 194 feet is mathematically represented as
[tex]P(X < 194 ) = P(\frac{X -\mu}{\sigma } < \frac{194 - 220}{40} )[/tex]
Generally [tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]
[tex]P(X < 194 ) = P(Z < -0.65 )[/tex]
From the z-table the probability of [tex](Z < -0.65 )[/tex]
[tex]P(Z < -0.65 ) = 0.25785[/tex]
So
[tex]P(X < 194 ) = 0.25785[/tex]
Given f(x) = -x + 7, evaluate f(-2).
Answer:
f(x)=9
Step-by-step explanation:
f(x)=-(-2)+7
f(x)=9
A=4 b= -5 and c=-8
HELLPPPPPP
HELP ILL GIVE BRAINLIEST HELPPPP
Can someone please help me!!! Also does anyone know if there is a website or calculator that may give me the answers? Please help me! Thank you!
Answer:
[tex] x = 17 [/tex]
Step-by-step explanation:
[tex] 2x + 7 = 3x - 10 [/tex] (alternate exterior angles are congruent when a transversal cuts across two straight lines)
[tex] 2x - 3x = -7 - 10 [/tex] (collecting like terms)
[tex] - x = -17 [/tex]
[tex] \frac{-x}{-1} = \frac{-17}{-1} [/tex] (dividing both sides by -1)
[tex] x = 17 [/tex]
prime factorization of 60 error
Answer:
2*2*3*5
Step-by-step explanation:
One fruit drink advertised that it contained "10% real fruit juice." In one bottle, this was found to be 4.8 oz of real juice. How many ounces is something other than fruit juice?
Answer:
43.2 oz
Step-by-step explanation:
4.8/x=10/100
simplify:
4.8/x=1/10
cross multiply:
x=48
there is 48 oz juice in total
to find the rest:
48-4.8=43.2
43.2 oz is something other than fruit juice.
Round 47.58051 to the nearest thousandths
Answer:
47.581
Step-by-step explanation:
can't explain I just know how to do it
In 1902, the yearly attendance at a major league baseball park was 3.4 × 105 people.
One hundred years later, the yearly attendance was 1.7 × 106 fans.
How many times greater was the attendance in 2002 than in 1902? Enter your answer in the box. helpppppp
Answer: 1902 had 357 people and 2002 had 180.2 people so, the answer would be
176.8
Step-by-step explanation: i hope this helps
plz tell me if its wrong so i can fix it
Which is the last operation when evaluating (8-2x)2+4 for x=3?
Answer:
addition.
Step-by-step explanation:
(8-2x)2+4
(8-2(3))2+4
(8-6)2+4
(2)2+4
4+4 (last operation)
8
Hector is visiting a cousin who lives 360 miles away. He has driven 70 miles. How many more miles does he need to drive to reach his cousin's home? Let d be how many miles left to reach his cousin's home.
Answer:
260 miles.
Step-by-step explanation:
350-90=260
If you see the word difference in an equation, it means to subtract. :) hope that helps
If M is the midpoint of AB and AM = 3x -1 and AB =5 x + 2, find x, AM, MB, and AB.
AB = 2x + 3
Just subtract the 3x - 1 from 5x + 2 = 2x + 3 is AB
x = 2
AB = 12
MB = 7
AM = 5
Hope this helps.
The perimeter, P, of a square whose side length is n.
Answer: p = 4n
Step-by-step explanation:
To find the perimeter of a shape, you need to add the length of the 4 sides. A square's 4 sides are equal. Therefore, you add n+n+n+n which is the same as 4n.
Answer:
P = n4
Step-by-step explanation:
in order to find the perimeter of an object you must add up all the sides. since we are dealing with a square all the sides are n. so in conclusion, 4 times n would you get you the perimeter, P.
Annabel wants to buy a $50.00 jacket that is marked 30% off. What is the final cost of the jacket (after the discount)?
Question 1 options:
$15.00
$35.00
No Solution
$65.00
Answer:
$ 35.00
Step-by-step explanation:
50.00 ÷ 10 = 5.00 (10%)
5.00 x 3 = 15.00 ( 30%)
50 - 15 = 35.
Sorry I couldn't explain more, But I hope this helps!
Help ASAP). Which relation is a function. Where the hand is at thats graph A across is graph B. The graph under graph a is ( graph C). Under graph B is (graph D). Need answer now please and thank you. [ I chose B but not 100% sure]
What are the x and y-intercepts from the table shown?
Can some please tell me what this is?
Step-by-step explanation:
This is mean domain is all real numbers
Someone, please help me.
Help Please I Really Need Help
Answer:
X=5
Step-by-step explanation:
3x+4=19
3x+4-4=19-4
3x=15
15/3x=
X=5
Hope this helps!
Can anybody graph y= —2x + 5.
Answer:
The y-intercept is 5 so that is where it will start then just keep going down by 2 since the slope is (-2)
hope this helps =3
Find the GCF of 28p4 and 44p2q.
Find the value(s) of "k" that will cause the equation 4x^2 +kx + 4 to have one real solution.
Answer: k = {8, -8}
Step-by-step explanation:
In order for a quadratic equation to have exactly one solution, the discriminant must equal zero. → b² - 4ac = 0
4x² + kx + 4 = 0
↓ ↓ ↓
a=4 b=k c=4
b² - 4ac = 0
k² - 4(4)(4) = 0
k² = 64
k = √64
k = ± 8