Answer:
$51.67
Step-by-step explanation:
Hope this helps! :3
plz mark as brainliest!
A constant volume of pizza dough is formed into a cylinder with a relatively small height and large radius. The dough is spun and tossed into the air in such a way that the height of the dough decreases as the radius increases, but it retains its cylindrical shape. At time t=k, the height of the dough is 13 inch, the radius of the dough is 12 inches, and the radius of the dough is increasing at a rate of 2 inches per minute.
(a) At time t=k, at what rate is the area of the circular surface of the dough increasing with respect to time? Show the computations that lead to your answer. Indicate units of measure.
(b) At time t=k, at what rate is the height of the dough decreasing with respect to time? Show the computations that lead to your answer. Indicate units of measure. (The volume V of a cylinder with radius r and height h is given by V=πr2h.)
(c) Write an expression for the rate of change of the height of the dough with respect to the radius of the dough in terms of height h and radius r.
Answer:
a) [tex]\frac{dA}{dt} = 48 \pi\frac{in^{2}}{min}[/tex]
b) [tex] \frac{dh}{dt} = - \frac{13}{3} \frac{in}{min}[/tex]
c) [tex]\frac{dh}{dt} = - 2\frac{h}{r} \frac {dr}{dt}[/tex]
Step-by-step explanation:
In order to solve this problem, we must first picture a cylinder of height h and radius r (see attached picture).
a) So, in order to find the rate at which the area of the circular surface of the dough is increasing with respect to time, we need to start by using the are formula for a circle:
[tex]A=\pi r^{2}[/tex]
So, to find the rate of change of the area, we can now take the derivative of this formula with respect to the radius r:
[tex]dA = \pi(2) r dr[/tex]
and divide both sides into dt so we get:
[tex]\frac{dA}{dr} = 2\pi r \frac{dr}{dt}[/tex]
and now we can substitute:
[tex]\frac{dA}{dr} = 2\pi(12in)(2\frac{in}{min})[/tex]
[tex]\frac{dA}{dt} = 48\pi\frac{in^{2}}{min}[/tex]
b) In order to solve part b, we can start with the formula for the volume:
[tex]V=\pi r^{2} h[/tex]
and solve the equation for h, so we get:
[tex]h=\frac{V}{\pi r^{2}}[/tex]
So now we can rewrite the equation so we get:
[tex]h=\frac{V}{\pi}r^{-2}[/tex]
and now we can take its derivative so we get:
[tex]dh=\frac{V}{\pi} (-2) r^{-3} dr[/tex]
we can rewrite the derivative so we get:
[tex]\frac{dh}{dt}=-2\frac{V}{\pi r^{3}}\frac{dr}{dt}[/tex]
we can take the original volume formula and substitute it into our current derivative, so we get:
[tex]\frac{dh}{dt}= -2\frac{\pi r^{2} h}{\pi r^{3}} \frac{dr}{dt}[/tex]
and simplify:
[tex]\frac{dh}{dt} =-2\frac{h}{r} \frac{dr}{dt}[/tex]
so now we can go ahead and substitute the values provided by the problem:
[tex]\frac{dh}{dt} =-2\frac{13in}{12in} (2\frac{in}{min})[/tex]
Which simplifies to:
[tex] \frac{dh}{dt} = - \frac{13}{3} \frac{in}{min}[/tex]
c)
Part c was explained as part of part b where we got the expression for the rate of change of the height of the dough with respect to the radius of the dough in terms of the height h and the radius r:
[tex]\frac{dh}{dt} =-2\frac{h}{r} \frac{dr}{dt}[/tex]
The rate of change of the height of the pizza with respect to (w.r.t.) time
can be found given that the volume of the pizza is constant.
(a) The rate of increase of the surface area with time is 4·π in.²/min(b) The rate at which the height of the dough is decreasing is [tex]\underline{4.\overline 3 \ in./min}[/tex](c) Rate of change the height of the dough with respect to the radius [tex]\dfrac{dh}{dr}[/tex], is [tex]\underline{-2 \cdot \dfrac{h}{r}}[/tex]Reasons:
The height of the dough when t = k is 13 inches
Radius of the dough = 12 inches
Rate at which the radius of the dough is increasing, [tex]\dfrac{dr}{dt}[/tex] = 2 in.²/min
(a) Required: The rate of increase of the surface area with time
Solution:
The circular surface area, A = π·r²
By chain rule of differentiation, we have;
[tex]\dfrac{dA}{dt} = \mathbf{\dfrac{dA}{dr} \times \dfrac{dr}{dt}}[/tex]
[tex]\dfrac{dA}{dt} = \dfrac{d ( \pi \cdot r^2)}{dr} \times \dfrac{dr}{dt} = 2 \cdot \pi \times 2 = 4 \cdot \pi[/tex]
The rate of increase of the surface area with time, [tex]\mathbf{\dfrac{dA}{dt}}[/tex] = 4·π in.²/min.
(b) Required: The rate of decrease of the height with respect to time
The volume of the pizza is constant, given by; V = π·r² ·h
Therefore;
[tex]h = \mathbf{ \dfrac{V}{\pi \cdot r^2}}[/tex]
[tex]\dfrac{dh}{dt} = \dfrac{d \left( \dfrac{V}{\pi \cdot r^2} \right)}{dr} \times \dfrac{dr}{dt} = \dfrac{-2 \cdot V}{\pi \cdot r^3} = \dfrac{-2 \cdot \pi \cdot r^2 \cdot h}{\pi \cdot r^3} \times \dfrac{dr}{dt} = \mathbf{-2 \cdot \dfrac{h}{r} \times \dfrac{dr}{dt}}[/tex]
[tex]\dfrac{dh}{dt} = -2 \cdot \dfrac{h}{r} \times \dfrac{dr}{dt} = -2 \times \dfrac{13}{12} \times 2 = \dfrac{13}{3} = 4. \overline 3[/tex]
The rate at which the height of the dough is decreasing, [tex]\mathbf{\dfrac{dh}{dt}}[/tex]= [tex]\underline{4.\overline 3 \ in./min}[/tex]
(c) Required:]The expression for the rate of change the height of the dough with respect to the radius of the cone.
Solution:
[tex]\dfrac{dh}{dr} = \dfrac{d \left( \dfrac{V}{\pi \cdot r^2} \right)}{dr} = \dfrac{-2 \cdot V}{\pi \cdot r^3} = \dfrac{-2 \cdot \pi \cdot r^2 \cdot h}{\pi \cdot r^3} = -2 \cdot \dfrac{h}{r}[/tex]
[tex]\dfrac{dh}{dr} = \mathbf{ -2 \cdot \dfrac{h}{r}}[/tex]
The rate of change the height of the dough w.r.t. the radius is [tex]\underline{\dfrac{dh}{dr} = -2 \cdot \dfrac{h}{r}}[/tex]
Learn more here:
https://brainly.com/question/20489729
I don’t understand so please HELPPPPPPPPPPPPPP
Answer:
C
Step-by-step explanation:
Okay so im going to gudie you thru So what it is asking you is to Count all the cubic squares looking things so what i can see is that You circled 169 aka C that would be correct because it cant be 11 cause it dosent look like 11 square units lol you get what im saying so the answer is C hope this HELPSS!!
By the way follow my insta. exotic. jaliyah if you need anymore help
using the elimination method to solve the system of equations.
3x+6y =36
3x-6y=0
a. 3,6
b. 6,-3
c. 6,3
d. 3,-6
Answer:
c. (6, 3)
Step-by-step explanation:
subtract
3x + 6y = 36
3x - 6y = 0
12y = 36, y = 3
3x + 6(3) = 36
3x + 18 = 36
3x = 18, x = 6
x = 6, y = 3
Answer:
the answer is b good luck
Please answer this as soon as possible!! ASAP Simplify to a single power of 2: 2^7/2^5
Answer:
i got 4
Step-by-step explanation:
Answer:
2²
Step-by-step explanation:
[tex]\frac{a^{m}}{a^{n}}=a^{m-n} , m>n\\\\\\\frac{2^{7}}{2^{5}}=2^{7-5}=2^{2}[/tex]
How many terms are in 12x - 3 = 18
Answer: 1.75
Step-by-step explanation:
12x-3=18
+3 +3
12x=21
12 12
x=1.75
Can someone help me ASAP !
Hope this helps! Have a nice day!
Determine the intercepts of the line
Answer:
x - intercept: -1.2
y - intercept: -0.7
An airplane pilot over the Pacific sights an atoll at an angle of depression of 5 . At this time, the horizontal distance from the airplane to the atoll is 4629 meters. What is the height of the plane to the nearest meter?
Answer:
404.98 m
Step-by-step explanation:
The angle of depression is 5°
The horizontal distance from the airplane to the atoll is 4629 meters.
We need to find the height of the plane.
If we consider a triangle. 4629 meters will be the base and angle is 5°. Let h is the height of the plane. Using trigonometry to find it :
[tex]\tan\theta=\dfrac{x}{4629 }\\\\\tan(5)=\dfrac{x}{4629 }\\\\x=\tan(5)\times 4629 \\\\x=404.98\ m[/tex]
So, the height of the plane is 404.98 m.
How do you do this for expanding and factoring expressions 8(4x-12)
Answer:
32x - 96
Step-by-step explanation:
Apply the distributed property:
8 × 4x = 32x8 × 12 = 96Re-write the expression: 32x - 96I hope this helps!
Helppppppppppppppppppppppppppppppppppppppp Pleaseeeeeeeeeeeeeeeee
Answer:
2. -1/3x > 12
3. 5+x > 7
4.10-x < 30
5. 2+5x ≤ 3
6.6-2x ≥ 17
8.$400+$49g=$750
at most the enthusiast can buy 7 games along with the player
Step-by-step explanation:
Jonathan is building a fence for his chickens. The length of a rectangle is
equal to triple the width. The perimeter(P=2L+2W) is 72 centimeters. What
is the width of the fence?
Answer:
Width = 9
Step-by-step explanation:
(P=2L+2W) = 72
L = 3W<- length is triple the width
(P = 2(3W) + 2W) = 72
P = 8W =72
W = 9
write the formula of mode for grouped data,and explain each term in it.
Answer:
Mode =Li+[f1-f0/2f1-f0-f2] *h
Li=lower limit.
F1= first frequency
F0= upper frequency of first frequency.
F2= second frequency.
H= difference between class intervals.
Hope it is helpful
Classify the triangle with side lengths of 16, 13, and 12 as acute, right, or obtuse
Answer:
16+13+12=41
Step-by-step explanation:
it is acute
-3,0 and 2 are the zeroes of the polynomial p(X)=x³+(a-1)x²+bx+c ,find a,c.
Answer:
a = 2
b = - 6
c = 0
Step-by-step explanation:
Since, -3,0 and 2 are the zeroes of the polynomial
[tex] Plug \:x = 0\: in \:p(X) \\
\implies p(0)= 0\\
\therefore 0= 0^3 +(a-1)(0)^2 +b(0)+c\\
\therefore 0= 0 +(a-1)\times 0 +b(0)+c\\
\therefore 0= 0 +0+0+c\\
\red{\bold{\therefore 0=c}}.....(1)\\\\
p(X)=x³+(a-1)x²+bx+c. \\
Plug \:x = -3\: in \:p(X) \\
[tex] \implies p(-3)= 0\\
\therefore 0= (-3)^3 +(a-1)(-3)^2 +b(-3)+c\\
\therefore 0= -27 +(a-1)\times 9 +b(-3)+c\\
\therefore 0= - 27 +9a-9 - 3b+c\\
\therefore 0= - 36+9a-3b+c\\
\therefore 36 =9a-3b+c... (2)\\
Plug \:c= 0\: in \:equation \: (2)\\
36= 9a - 3b + 0\\
36= 9a - 3b\\
36= 3(3a - b) \\
\purple {\bold{12 = 3a - b}} .... (3)\\\\
Plug \:x = 2\: in \:p(X) \\
\implies p(2)= 0\\
\therefore 0= 2^3 +(a-1)(2)^2 +b(2)+c\\
\therefore 0= 8 +(a-1)\times 4 +b(2)+c\\
\therefore 0= 8 +4a-4+2b+c\\
\therefore 0= 4 +4a+2b+c\\
\therefore - 4=4a+2b+c.....(4)\\
Plug \:c= 0\: in \:equation \: (4)\\
- 4 = 4a +2b+0\\
-4= 2(2a+b)\\
\orange{\bold{-2=2a + b}} .......(5)\\\\[/tex]
Adding equation (3) and (5), we find:
12 = 3a - b
-2 = 2a + b
_________
10 = 5a
10/5 = a
a = 2
Plug a = 2 in equation (5)
-2= 2(2) + b
-2 = 4 + b
-2 - 4 = b
-6 = b
b = - 6
What is the ratio of 4 red apples and 3 green apples
Answer:
4:3 8:6 12:9 so on
Step-by-step explanation:
Answer:
4 to 3 or 3 to 4
Step-by-step explanation:
depending on how the question is worded if its red the green then 4 to 3 but if it green to red then its 3 to 4.
PLS AWNSER BEST AWNSER GET BRAINLIST.
I need a good grade on this final for my pc lol
Answer:
no idea
Step-by-step explanation:
Jk the answers square
A fraction whose value is the same as 3/4 is
Answer:
[tex] \frac{6}{8} [/tex]
Step-by-step explanation:
[tex] \frac{3}{4} [/tex][tex] = \frac{3 \times 2}{4 \times 2} [/tex][tex] = \frac{6}{8} [/tex]
Graph the solution of the inequality on a number line.
2(x – 3) – 5x
What is the distance between point A and point B? Round to nearest tenth.
Answer:
[tex]d \approx 12.6[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra II
Distance Formula: [tex]d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Step-by-step explanation:
Step 1: Define
Find points from graph.
Point A (-4, 2)
Point B (8, 6)
Step 2: Find distance d
Simply plug in the 2 coordinates into the distance formula to find distance d.
Substitute [DF]: [tex]d = \sqrt{(8+4)^2+(6-2)^2}[/tex]Add/Subtract: [tex]d = \sqrt{(12)^2+(4)^2}[/tex]Exponents: [tex]d = \sqrt{144+16}[/tex]Add: [tex]d = \sqrt{160}[/tex]Simplify: [tex]d = 4\sqrt{10}[/tex]Evaluate: [tex]d = 12.6491[/tex]Round: [tex]d \approx 12.6[/tex]11. Write an equation in slope-intercept form that passes through (2,3) and 1 (1,5). *
Answer:13/6/7
Step-by-step explanation:
A line passes through point A(14,21). A second point on the line has an x-value that is 125% of the x-value of point A and a y-value that is 75% of the y-value of point A. Use point A to write an equation of the line in point-slope form.
Help i'll give you brainliest
Answer:Step-by-step explanation:
the first one and third one
Callum and Isobel have a total of 240 stamps.
The ratio of the number of Callum’s stamps to the number of Isabel’s stamps is 3:8.
Callum buys some stamps from Isabel.
The ratio of the number of Caullum’s stamps to the number of Isabel’s stamps is now 3:2
How many stamps does Callum buy from Isabel?
You must show all your working.
Answer:
callum bought 78.6 stamps
pls help
due at 11:59
Answer:
C. E.
Step-by-step explanation:
Parkside High School was having a talent show to raise money. It was x dollars for adults and $8 for students. Stephanie's family had 4 adults and 1 student. Her family paid $48 to see the show. Write an equation for the word problem. Use the variable x and leave no spaces.
Answer:
[tex]x = 10[/tex]
Step-by-step explanation:
Given
$x per adult
$8 per student
Population:
[tex]Students = 1[/tex]
[tex]Adults = 4[/tex]
[tex]Total = \$48[/tex]
Required
Determine the equation and solve for x
The system can be analyzed as thus:
The cost for 4 adults is: 4 * x
The cost for 1 students is: 8 * 1
Total Amount is: 48
From the analysis above, the equation is:
[tex]4 * x + 8 * 1 = 48[/tex]
[tex]4 x + 8 = 48[/tex]
Subtract 8 from both sides
[tex]4 x + 8 -8= 48 - 8[/tex]
[tex]4 x= 40[/tex]
Solve for x
[tex]x = 40/4[/tex]
[tex]x = 10[/tex]
Im asking for you're help Citizen I have a question for you Me
Answer:
No idea
Step-by-step explanation:
No idea for that either
Hope this helps, have a good day
Step-by-step explanation:
1. The bag contains fewer than 10 carrots
2.catherin3 biked farther than 10 miles
3.the chair is shorter than 20 in
4.carlos scored over 20 points
There are 123 goldfish in the tank. Goldfish like to swim 12 fish to a family. If 6 of the goldfish died before forming families, how many FULL families will be formed?
Answer:
Formed will be i think 9, but im not sure.
In only three days the temperature dropped by 24 degrees.How many degrees per day did the temperature drop?
Answer: 8° per day.
Step-by-step explanation:
Based on the information above, since we are informed that in three days, the temperature dropped by 24 degrees, the number of degrees per day that the temperature dropped will be go by dividing 24° by 3 days. This will be:
= 24° / 3 days
= 8° per day
Given:x-8>-3
Chose the solution set.
1.{x|x€R,x>-5}
2.{x|x€R,x>14}
3.{x|x€R,x>5}
4.{x|x€R,x>-9}
Step-by-step explanation:
x - 8 > -3
x > -3 + 8 (add 8 to both sides)
x > 5
Hence 3.{x|x€R,x>5} is correct.
Factor with the distributive property (no variables)
90+27=
Your answer is 9(10 + 3)
Tell me if you need an explanation
Suppose we collect a simple random sample of 200 registered voters from a large city. We find that 25% of the voters in the sample are Independents. Find a 95% confidence interval for the percentage of registered voters in the city that are Independents. Round your answer to at least two decimal places.
Answer:
0.05<x<0.06
Step-by-step explanation:
The formula for calculating the confidence interval is expressed as;
CI = p' ±z * √p(1-p)/n
z is the z score at 95% CI
p is the proportion of the independent sample
n is the sample size
p' = X/n
p' = 0.25/200
p' = 0.00125
Given
z = 1.96
p = 25% = 0.25
n= 200
Substitute into the formula;
CI = p' ±1.96 *√0.25(1-0.25)/200
CI = p' ±1.96 * √0.25(0.75)/200
CI = p' ±1.96 * 0.0306
CI = p' ±0.0605
CI = 0.00125 ±0.0605
CI = (0.048, 0.06175)
CI = (0.05, 0.06)
0.05<x<0.06