The amount of room he has for more games is 35. 3 times 40 = 120 - 85 = 35.
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]
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A school wants to buy some new desks and chairs. If it buys 40 desks and 100 chairs, the
total cost is $1100; if it buys 50 desks and 80 chairs, the total cost is $1150. What is the
price of a desk and a chair
Answer: The price of desk is 15 and the price of the chairs 5
Step-by-step explanation:
I need help finding the circled ones
#12 is 17, #3 is 59, #8 is 28, #22 is 78, #32 is 90, #10 is 59, #21 is 26
For the equation
y - 3 = 2(x + 4),
the line has a slope of 2 and contains what point?
O (3,-4)
O (-3,-4)
0 (-4,-3)
O (-4.3)
We can use the equation of tangent as a structure.
[tex]y - f(a) = f'(a)(x - a)[/tex]
Our f(a) is y-point and Our a is x-point..
Looking back at the equation, our f(a) would be 3 and a would be - 4
Therefore the answer is (-4,3)
How tall is Becky? Can you please help
Answer:
Becky is 5 1/4 ft tall
Simplify 6258 x 52 + 25
Answer:
325441
Step-by-step explanation:
6258*52 = 32416
325416 + 25 = 325441
Answer:
325441
Step-by-step explanation:
6258(52)=325416
325416+25=325441
What’s the answer, if you don’t know please do not answer.
Answer:
Yes, the bathroom has enough water and shampoo for all of them.
Step-by-step explanation:
70L+ 60S < 5600 70(8) + 60(7) = 560 + 420 = 980 There is enough water.0.02L + 0.01S 0.02(8) + 0.01(7) = 0.16 + 0.07 = 0.23 There is enough shampoo.Answer:It takes 250 g of pasta and 240 ml of sauce to make one batch of spaghetti and it takes 600g of pasta and 700ml of sauce to make one batch of lasagna.
Case 1: Pasta
The chef wants to make spaghetti and lasagna using more than 4000 g of pasta.
S denote the number of batches of spaghetti he makes and L the number of batches of lasagna he makes.
So the equation becomes :
Case 2: Sauce
chef wants to make spaghetti and lasagna using more than 5000 ml of sauce.
S denote the number of batches of spaghetti he makes and L the number of batches of lasagna he makes.
So the equation becomes:
Answer :
The two equations are :
For pasta:
For sauce:
Step-by-step explanation:
what is the answer to 3(4x-5)
Answer:
12x - 15
Step-by-step explanation:
3 times 4x is 12x
3 times -5 is -15
therefore, 12x - 15
What is the answer to -9 = 2/7x + 5
Answer: x = -49
Step-by-step explanation:
hope this helped
BRAINLIEST TO RIGHT ANSWER! ASAP!
Answer:
y=-1/2x+1
Step-by-step explanation:
The length of a dog on a scale drawing is 6 cm. If the scale is 1 cm : 7 cm, what is the actual length of the dog? (A) 15 cm (B) 30 cm (C) 42 cm (D) 108 cm
Answer: (C) 42 cm
Work Shown:
(length on paper)/(real length) = (length on paper)/(real length)
(1 cm)/(7 cm) = (6 cm)/(x cm)
1/7 = 6/x
1*x = 7*6
x = 42
The actual length of the dog is 42 cm.
PART OF A STUDY SET!!!! NEED THE ANSWER ASAP BEING TIMED GIVING BRAINLY AND THANKS FOR HELP!!!!!!!!
1.)
What value of x makes the following equation true?
2(6 - 2x) - 2x = 2(5 + x) - 4
A) 3/4
B) 1/4
C) -1/4
D) -3/4
2.)
What is the solution to the following equation?
3(7 + 2x) + 3 = 3(5 + x) - 9(x + 1)
A) 3/2
B) 2/3
C) -2/3
D) -3/2
Answer:
1.) A
2.) D
Step-by-step explanation:
Answer:
1. 3/4
2.-3/2
Step-by-step explanation:
For the first problem, distribute the 2(6-2x) and 2(5+x) to simplify the equation. This means, multiply what's inside the parentheses by 2. The simplified version of the equation is: 12-4x-2x=10+2x-4.
Once you have distributed, then combine like terms and solve for x.
You do the same thing for the second problem.
The temperature at 10 A.M. was -4 °F. By noon the temperature had risen 8 °F. At 8 P.M., the temperature had dropped 10° F. What was the temperature at 8 P.M.? *
Answer:
-6
Step-by-step explanation:
-4+8=4
4-10= -6
If f(x) = 9-2 |x+2|, find f(-7).
-15
-1
1
19
Step-by-step explanation:
f(x) = 9 - 2|x + 2|
When x = 7, f(-7) = 9 - 2|(-7) + 2|
= 9 - 2|(-5)|
= 9 - 2(5)
= 9 - 10
= -1.
9514 1404 393
Answer:
-1
Step-by-step explanation:
Put -7 where x is and do the arithmetic.
f(-7) = 9 -2|-7+2|
= 9 -2|-5| = 9 -2·5 = 9 -10
f(-7) = -1
The recycling bin for the school is in the shape of a rectangular prism. The bin has a length of 16 inches, a width of 14 inches, and a height of 15 inches. Whatis the volume of the recycling bin?
Answer:
3150 inches is the volume
Step-by-step explanation:
i hoped this helped
This is the last question from my questions before
Answer:
Just right the numbers they put
Answer:
see explanation
Step-by-step explanation:
Under a reflection in the line y = - x
a point (x, y ) → (- y, - x ) , thus
M(- 7, 1 ) → M'(- 1, 7 )
N(- 4, 1 ) → N'(- 1, 4 )
O(- 4, - 4 ) → O'(4, 4 )
P(- 7, - 4 ) → P'(4, 7 )
4. write the following in standard form
four hundred, thirty - two thousand, six hundred eight
6 points
11) Use synthetic division to divide the two polynomials.
(2x3 – 12x — 2) = (x – 5)
A. (x - 5)(2x2 - 10x + 38) - 190
B. (x - 5)(2x2 + 10x + 38) + 188
C. (x + 5)(x2 + 50x + 190) + 188
D. (x + 5)(2x2 - 5x - 90) - 190
Answer:
the is a
Step-by-step explanation:
The 6% state income tax on a $40,000 salary
Answer:
2400
Step-by-step explanation:
because 6 percent of 40000 is 2400 please mark brianliest
Atrain travels at a speed of 58.6 km'hi Express this speed
a.ms
b. m/min
Answer:
16.27
Step-by-step explanation:
if 1000m rep 1km
therefore :
58.6km/h in ms = 58.6km×1000/3600
=16.277 ms
how many solutions does the equation 2(9x-6)=18x+2 have??? Please help
Answer:
x=5.889
Step-by-step explanation:
2(9x6)=18x+2
108=18x+2
106=18x
x=106÷18
x=5.888888889
An item on the sale rack at a local supermarket was marked $3.42. The original price of the item was $4.75.
What was the percent of the discount?
Answer:
$475
Step-by-step explanation:
The percentage of discount of the item is 72 %
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the percentage of discount in the price be = A
The marked price of an item at the supermarket = $ 3.42
The original price of an item at the supermarket = $ 4.75
Now , the equation will be
The percentage decrease = ( marked price of an item at the supermarket / original price of an item at the supermarket ) x 100
Substituting the values in the equation , we get
The percentage decrease A = ( 3.42 / 4.75 ) x 100
The percentage decrease A = 0.72 x 100
The percentage decrease A = 72 %
Therefore , the value of A is 72 % and the percentage of discount is 72 %
Hence , The percentage of discount of the item is 72 %
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Using known properties, determine if the statements are true or not. Select True or False for each statement. If one pair of consecutive sides of a parallelogram is congruent, then the parallelogram is a rectangle. The diagonals of a parallelogram bisect each other. If a quadrilateral has four right angles, then it is a square.
Answer:
The diagonals of a parallelogram bisect each other.Step-by-step explanation:
If one pair of consecutive sides of a parallelogram is congruent, then the parallelogram is a rectangle.
False. This is rhombus.The diagonals of a parallelogram bisect each other.
TrueIf a quadrilateral has four right angles, then it is a square.
False. It is rectangle.Determine whether the lines are parallel, perpendicular, or neither. 5x + 2y = 14 and y = −5x + 9
Answer: Neither
Step-by-step explanation:
5x + 2y = 14
2y = -5x + 14
y = -5/2 x + 7; slope m = -5/2
y = –5x + 9 ; slope m = -5.
Parallel lines slopes are equal and perpendicular slopes are opposite and reciprocal.
So both lines are neither parallel nor perpendicular
Answer
C. Neither.
Answer: Neither
Step-by-step explanation:
5x + 2y = 14
2y = -5x + 14
y = -5/2 x + 7; slope m = -5/2
y = –5x + 9 ; slope m = -5.
Parallel lines slopes are equal and perpendicular slopes are opposite and reciprocal.
So both lines are neither parallel nor perpendicular
Answer
C. Neither.
what’s the value of x in the solution to this system of equitations:
3x + 4y =-3
x= -2y- 1
Answer:
x = -1
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightEquality Properties
Algebra I
Solving systems of equations using substitution/eliminationStep-by-step explanation:
Step 1: Define Systems
3x + 4y = -3
x = -2y - 1
Step 2: Solve for y
Substitution
Substitute in x: 3(-2y - 1) + 4y = -3Distribute 3: -6y - 3 + 4y = -3Combine like terms: -2y - 3 = -3Isolate y term: -2y = 0Isolate y: y = 0Step 3: Solve for x
Define equation: x = -2y - 1Substitute in y: x = -2(0) - 1Multiply: x = 0 - 1Subtract: x = -1Pam used the following process to find the y intercept of the line described by the equation 2y-y=11.
According to Pam's method, which expression gives the y-intercept of the line described by the equation ax+by=c?
a. -a/b
b. b/a
c. -b/c
d. c/d
Answer:
c/bStep-by-step explanation:
Given the expression ax+by = c, the y intercept of the line occurs at when x = 0
Substitute x = 0 into the expression and get the value of y
ax+by = c
a(0)+by = c
0+by = c
by = c
divide both sides by b
by/b = c/b
y = c/b
Hence the y intercept of the line occurs at y c/b
Find the minimum coordinate due in 5 min
9514 1404 393
Answer:
(4, -6)
Step-by-step explanation:
The minimum is the vertex of the parabolic section in the 4th quadrant. The coordinates appear to be (4, -6).
You bought a video game for $35. You got it on sale
for 25% off. What was the price before the sale?
Which formula do we use to solve this?
A.Final Price Formula
B.Original Price Formula
Answer:
Sorry I feel bad I don't kt the answer sorry hope you find the answer tho
Step-by-step explanation:
sorry
A substance has a density of 0.70 g/mL, if it has a volume of 3.0 mL, what is its mass?
Answer:
2.1 gStep-by-step explanation:
The mass of a substance when given the density and volume can be found by using the formula
mass = Density × volume
From the question we have
mass = 0.70 × 3
We have the final answer as
2.1 gHope this helps you
could someone please help me with my math? i really need help and i am struggling
https://brainly.com/question/20134003
Answer:
yes
Step-by-step explanation: