Water runs into a conical tank at the rate of [tex]9ft^{3}/min.[/tex] The tank stands point down and has a height of 10 ft and a base of 5 ft. How fast is the water level rising when the water is 6 ft deep?
Answers
Answer:
[tex]\frac{1}{\pi} \frac{\text{ft}}{\text{min}}[/tex] is the rate at which the water is rising when the water is 6 ft deep.
Step-by-step explanation:
See the attached diagram that I drew to represent this problem.
To solve related rates problems, let's use the recommended steps:
Draw a diagram.Label all quantities and their rates of change.Relate all quantities in the same equation.Differentiate (implicitly) with respect to time.Use the resulting equation to answer the question in context.Step 1:I already drew the diagram; see attached image.
Step 2:I labeled the quantities we are given in the problem. h = 10 ft, and r = 5 ft. We are also told that the change in volume is 9 ft³/min; dV/dt = 9 ft³/min.
We want to find dh/dt when h = 6.
[tex]\frac{dh}{dt}\ \vert \ _h_=_6 =\ ?[/tex] Step 3:We know that we are dealing with a cone in this problem, and we are given the volume of the cone. Therefore, we can use the formula for the volume of a cone in order to relate all of the quantities in the same equation.
[tex]V=\frac{1}{3} \pi r^2 h[/tex]Since we only want the two variables, V and h, we can solve for r in terms of h and substitute this value for r in the formula.
This is because when we perform implicit differentiation, we do not have the change in r (dr/dt) but we do have dh/dt, which is what we are trying to solve for.
We know that r = 5 ft, and h = 10 ft. Therefore, we can say that [tex]\frac{r}{h}=\frac{5}{10} \rightarrow \frac{r}{h} = \frac{1}{2}[/tex]. Multiply h to both sides to solve for the variable r: [tex]r=\frac{h}{2}[/tex].
Substitute this into the volume of a cone equation:
[tex]V=\frac{1}{3} \pi(\frac{h}{2})^2 h[/tex]Simplify this equation.
[tex]V=\frac{\pi}{3}\cdot \frac{h^2}{4} \cdot h[/tex] [tex]V=\frac{\pi}{12}h^3[/tex] Step 4:Perform implicit differentiation on the volume equation.
[tex]\frac{dV}{dt} =\frac{\pi}{12}3h^2 \cdot \frac{dh}{dt}[/tex] Step 5:Substitute known values and solve for dh/dt to find the change in height, or the rise in water level, when the water is 6 ft deep (h = 6).
We know that:
[tex]\frac{dV}{dt} =9[/tex] [tex]h=6[/tex]Plug these values into the implicitly differentiated volume equation.
[tex]9=\frac{\pi}{12}3(6)^2\cdot \frac{dh}{dt}[/tex] [tex]9=\frac{3\pi}{12}\cdot 36 \cdot \frac{dh}{dt}[/tex] [tex]9=9\pi\frac{dh}{dt}[/tex] [tex]\frac{dh}{dt} =\frac{9}{9\pi}= \frac{1}{\pi}[/tex]Answering the question in context:
Since dh/dt = 1/π when h = 6 ft, we can say that the water is rising at a rate of [tex]\frac{1}{\pi} \frac{\text{ft}}{\text{min}}[/tex] when the water is 6 ft high.
![Water Runs Into A Conical Tank At The Rate Of [tex]9ft^{3}/min.[/tex] The Tank Stands Point Down And](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/q5bk33DP5eAhpKysA0iDwDDhihHjU15n.jpeg)
Hey there!
We can create some variables to denote some of the things we are working with.
v = volume
h = height
r = radius
We know that [ dv/dt = (π/3)r²h ]. We also know that [ r/h = 5/10 = 1/2 ] which resembles the parts of a triangle.
Solution:
v = π/3*h/2²
h = πh³/(4)(3)
~Differentiate both sides
dv/dt = (3πh³/4*3)(dh/dt)
~Simplify
dh/dt = (4dv/dt)/πh²
~Use chain rule
(4)(9)/π6²
1/π
Thus, the speed of the water level rising is [ 1/π ft/min ] when the water is 6ft deep.
Best of Luck!
Related Questions
determine the axis of
symmetry for y=x²+2
Answers
Answer:
x=0
Step-by-step explanation:
The axis of symmetry always passes through the vertex of the parabola . The x -coordinate of the vertex is the equation of the axis of symmetry of the parabola. For a quadratic function in standard form, y=ax2+bx+c , the axis of symmetry is a vertical line x=−b/2a
in our case
y=x^2+2
a=1 b=0 c=2
so x=-b/2a=-0/2*1=0
x=0 is axis of symmetry
70 POINTS
Select the correct answer.
Here are four decimal numbers:
0.39, 0.85, 0.731, 0.773
Choose the list that shows the numbers in order from smallest to largest.
A.
0.773, 0.39, 0.85, 0.731
B.
0.773, 0.85, 0.39, 0.731
C.
0.39, 0.731, 0.773, 0.85
D.
0.85, 0.773, 0.731, 0.39
Answers
Answer: A
Step-by-step explanation:
Answer:
d
Step-by-step explanation:
Tito opened a savings account with $100.
He saves $25 per month. Which of the
inequalities below can be used to find the
least number of months it will take Tito to
have $700 saved?
PLEASE HELP!! INEED A EQUATION
Answers
Answer:
100+25m=700
Step-by-step explanation:
i just took a good guess so don't trust it :)
Answer: 700/25=28
28x25=700 28 months
Step-by-step explanation:
since he saves $25 per month you have to divide 700 and 25 and you get 28
if You aren’t sure Its right then multiply 25 and 28 and you get 700
Get this right and I'll give Brainlist!!!!!!
Answers
Step-by-step explanation:
So, we have f / -(2/3) = -(1/3), right?
Well, this means that f = -(1/3) * -(2/3) = positive 2/9 !
Your answer is f = 2/9 :)
Answer:
F = 2/9
Step-by-step explanation:
5. A life insurance policy costs $16.00 for every $1,000.00 of insurance. At
this rate, what is the total policy cost for $50,000.00 of insurance.
Answers
Answer:
800.00
Step-by-step explanation:
so you give me a hint that was for every 16.00 it was 1000.00
i did 16x50=800
i got 50 from 50 in 50000 and the 16 from 16.00 so then i times 16 from 50 which is 800
HELPO aSaPpp!!! thank you!!!
Answers
You could say "7328 dollars and 42 cents", but the shorter format is preferred in my opinion.
====================================
Work Shown:
A = P*e^(r*t)
A = 6000*e^(0.04*5)
A = 7328.41654896153
A = 7328.42
The symbol 'e' is a special number in math similar to how pi = 3.14 is another important symbol. If your calculator doesn't have a button labeled 'e', then replace 'e' with 2.718 to use its approximation.
Five kittens share 6 cups of milk equally. How much milk does each kitten get?
Answers
Answer:
They get .83 cup per.
Step-by-step explanation:
Divide 6 by 5. So do 6 / 5, which gets you 0.833333333
Triangle ABC has 3 medians drawn creating centroid D. If a centroid creates a 2:1 ratio and one median is 24 inches, what is the larger piece?
PLS HELP ASAP! WILL MARK BRAINLEST!! THX :)
Answers
Answer: The larger piece is 16 inches
Step-by-step explanation: Median of a triangle is a line segment uniting a vertex and the midpoint of the opposite side. A triangle has 3 medians and the point where these 3 intersect is called centroid.
The ratio of one median is 2:1, which means the line segment is divides by the centroid into two parts, one the double of the other. So:
2x + x = 24
3x = 24
x = 8
Then, the larger piece is
2x = 2(8) = 16
The larger part of the median measuring 24 inches is 16 inches
The number of kids in a school decreased by 20% between last year and this year. Last year there were 80 kids in the school. What is the population this year? If you get stuck, consider making a table.
Answers
Answer:
64
Step-by-step explanation:
Multipy 80 by 80% or 0.80 to find the new population of 20% less
Two girls and two boys went to the festival. The total cost of their tickets was $20. Each child paid the same amount for a ticket. What was the cost of each ticket? Solve this problem any way you choose
Answers
Answer: $5
Step-by-step explanation: 4 children divided by 20 equals 5
2(3x-(2x-3(3x+1)+5x)
Answers
Answer: Is the answer is 10x+6? You didn’t even show what can I do for your math expression, I thought it was simplified expression.
Step-by-step explanation:
[tex]13b+7b+7a+13a = ?[/tex]
Answers
Answer: 20a + 20b
Step-by-step explanation:
Combine the like terms.
13b + 7b = 20b
7a + 13a = 20a
Our answer to the question is 20a + 20b.
A tennis shoes regularly sells for $20.00. The store advertises a 15% discount. What is the sale price of the necklace in dollars?
Answers
Answer:
Shoes with discount = $17
Necklace = Price * 0.85
Step-by-step explanation:
-If the tennis shoes store sells other things than shoes-
Shoes = $20.00
Discount = 15%
*Price expressed as a percentage*
Price without discount = 100%
Price after discount = 100% - 15%
Price after discount = 85%
*Normal price*
Price without discount = 20
Price after discount = 20 * 0.85
Price after discount = 17
can someone help me please and thank you!!
Answers
Answer:
Perimeter: perimeter = (12 x 2) + (x-2 times 2)
Area: 12 x (x-2)
Step-by-step explanation:
I hope im right im only in grade 6 so wouldnt trust me with that :/
Write 6.384 correct to 1 decimal place ?
Answers
Answer:
Write 6.384 correct to 1 decimal place ?=6
Answer:
6.4
Step-by-step explanation:
Rounding 6.384 to one decimal place would be the same as rounding it to the nearest tenth. The tenth in this number is 3. To round a number up, look at the number to the right of it - if the number is 5 or greater, then you increase the value of the number you're rounding by 1. If the number to the right of it is 4 or less, then the value of the number you're rounding stays the same.
In this case, the number we're rounding is 3 and the number to the right of it is 8. Since 8 is greater than 5, we have to increase the value of 3 by 1, which means that the number in the tenths position is now 4.
The number rounded to one decimal place is 6.4.
A runner is running a 10k race. The runner completes 30% of the race in 20 minutes. If the runner continues at this pace, how long will it to finish?
Answers
Answer:
Step-by-step explanation:
20 Min=30% complete
40 min=60% complete
60 min (1 hr)= 90% complete
67 min=100% complete
Help ASAP Find the measure of Z.
45°
75°
105
Answers
Answer:
75°
Step-by-step explanation:
m∠W = m∠Z = 180° - 105° = 75°
Answer:
75°
Step-by-step explanation:
A golfer aims for a hole that is 120 feet away and, when he hits the golf ball off the ground, its highest point reaches 80 feet up in the air. Noemi’s team is working together to write the quadratic equation to model the path of the golf ball in graphing form. She calculated the value of a = - 1/90 . Is she correct?
Answers
Answer:
No she isn't. The value of [tex]a[/tex] is [tex]a=-\frac{1}{45}[/tex]
Step-by-step explanation:
We know that all quadratic equation can be written in the following way :
[tex]y=ax^{2}+bx+c[/tex] (I)
Where [tex]a,b[/tex] and [tex]c[/tex] are real numbers.
I will attach a drawing with the quadratic graph to understand the situation.
We know by looking at the drawing and analyzing it that the parabola passes through the points : [tex](0,0) ; (60,80)[/tex] and [tex](120,0)[/tex]
[tex](0,0)[/tex] because we put our coordinates origin there.
[tex](120,0)[/tex] because that's where the golf hole is.
And [tex](60,80)[/tex] because we know that its highest point reaches 80 feet up in the air at the middle of the distance between its roots (property of a negative parabola).
Finally, we work with the three points and the equation (I) in order to find the values of [tex]a,b[/tex] and [tex]c[/tex] ⇒
The parabola passes through [tex](0,0)[/tex] ⇒
[tex]y=ax^{2}+bx+c[/tex] ⇒
[tex]0=a(0)^{2}+b(0)+c[/tex] ⇒ [tex]c=0[/tex]
The parabola passes through [tex](60,80)[/tex] ⇒
[tex]80=a(60)^{2}+b(60)[/tex] ⇒
[tex]80=3600a+60b[/tex] (II)
The parabola passes through [tex](120,0)[/tex] ⇒
[tex]0=a(120)^{2}+b(120)[/tex] ⇒
[tex]0=14400a+120b[/tex]
[tex]120b=-14400a[/tex]
[tex]b=-120a[/tex] (III)
Now if we use (III) in (II) ⇒
[tex]80=3600a+60(-120a)[/tex]
[tex]80=3600a-7200a[/tex]
[tex]3600a=-80[/tex]
[tex]a=-\frac{1}{45}[/tex] ⇒ [tex]b=\frac{8}{3}[/tex]
Finally the equation of the parabola is
[tex]y=-\frac{x^{2}}{45}+\frac{8}{3}x[/tex]
Where the value of [tex]a[/tex] is [tex]a=-\frac{1}{45}[/tex]
father bought 2kg 200g of potatoes and 1kg 500g of tomatoes.find the total weight of vegetables he bought
Answers
Answer:
3kg 700g
hope it helps:-) :-)
use the number line to model the equation -5 + -3 = -3+ (-5)
Answers
Answer: I don’t have a number line, but...
Step-by-step explanation:
If you add -1 to any number, you lose one, right?
So, do that five times, and use the properties to show that there is no difference between them.
7. Lin is reading a 146-page book. She read the first 20 pages in 35 minutes. 1 point How many minutes does it take her to finish reading the book?
Answers
Answer:
255.5 minutes or 255 minutes and 30 seconds
Step-by-step explanation:
146 pages/ x minutes
20 pages/ 35 minutes
All you have to do is cross-multiply to find the answer.
146x35= 5110
5110/20=x
5110/20= 255.5
x= 255.5
the audience thinks 30% of Brandon's jokes are funny. what fraction of brandons jokes does the audience think are funny??
Answers
Answer: 3/10
Step-by-step explanation:first conversation percent to fraction 30/100 then reduce by taking out ten and you get 3/10
1. The ratio of women to men at a construction site is 2 to 4. There are a
total of 60 workers. How many of the workers are women?
Answers
Logan ate g out of 16 gumdrops. Write an expression that shows how many gumdrops Logan has left.
Answers
Answer:
16-g=x
Step-by-step explanation:
You know you need to subtract because it says ate out of. Next you don't know how many gumdrops Logan ate so they gave a the variable g. Since you don't know how many gumdrops he ate you don't know how many are left. You need another variable to represent how many gumdrops are left.
In the figure, what is the value of X?
Answers
Answer: I think it's 12
PLZZZZZZ HELP!!!!!!!!!!!
Answers
Answer:
greater number is 8 lesser number is 3
what is the coefficiant of (x^4)( y^3) iterm in the expansion of (x-2y)^7
Answers
Answer:
The coefficient of [tex]x^{4}[/tex][tex]y^{3}[/tex] is -280
Step-by-step explanation:
Solving the Binomial expansion using the Pascal's triangle, it would be observed that an expansion to the power of 7 can be solved using;
1 7 21 35 35 21 7 1
Thus;
[tex](x - 2y)^{7}[/tex] = 1. [tex]x^{7}[/tex] + 7.[tex]x^{6}[/tex](-2y) + 21.[tex]x^{5}[/tex][tex](-2y)^{2}[/tex] + 35.[tex]x^{4}[/tex][tex](-2y)^{3}[/tex] + 35.[tex]x^{3}[/tex][tex](-2y)^{4}[/tex] + 21.[tex]x^{2}[/tex][tex](-2y)^{5}[/tex] + 7.x[tex](-2y)^{6}[/tex] + [tex](-2y)^{7}[/tex]
= [tex]x^{7}[/tex] -14[tex]x^{6}[/tex]y +84[tex]x^{5}[/tex][tex]y^{2}[/tex] -280[tex]x^{4}[/tex][tex]y^{3}[/tex] + 560[tex]x^{3}[/tex][tex]y^{4}[/tex] - 672[tex]x^{2}[/tex][tex]y^{5}[/tex] + + 448x[tex]y^{6}[/tex] - 128[tex]y^{7}[/tex]
Therefore, the coefficient of [tex]x^{4}[/tex][tex]y^{3}[/tex] is -280
• What is the total number of pencils that Mr. Moretti gives to his students
if he puts 3 mechanical pencils in each bag?
Answers
Answer:162
Step-by-step explanation:
162
How many distinct permutations can be formed using the letters of the word "PAPER"?
Answers
Answer:
2 i think
Step-by-step explanation:
if the function g has the factors (x-7) and (x+6), what are the zeros of function g?