Related Rates Problem: The radius of a spherical watermelon is growing at a constant rate of 2 centimeters per week. The thickness of the rind is always one tenth of the radius. The volume of the rind is growing at the rate __?__ cubic centimeters per week at the end of the fifth week. Assume that the radius is initially zero. ...?
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so, if we grab the whole volume of the watermelon, and then subtract the volume of the inner sphere of 9/10, what we're left with is 1/10 of the volume, which is the volume of the rind, recall the rind has a radius of 1/10 of the radius, so we're simply taking off 1/10 from the radius and using the 9/10 and subtracting the volume with that radius.
now, after the 5th week, the radius of the watermelon has then grown to 2*5, or 10 cm, and the rind is 1/10 of that or just 1 cm.
[tex]\bf \cfrac{4\pi \left( \frac{9r}{10} \right)^3}{3}\implies \cfrac{4\pi \cdot \frac{729r^3}{1000}}{3}\implies \cfrac{\frac{2916\pi r^3}{1000}}{3}\implies \cfrac{2916\pi r^3}{3\cdot 1000}\implies \cfrac{243\pi r^3}{250}\\\\ -------------------------------\\\\ V_{rind}=\cfrac{4\pi r^3}{3}~~-~~\cfrac{243\pi r^3}{250}\implies V_{rind}=\pi r^3\left(\cfrac{4}{3}-\cfrac{243}{250} \right)[/tex]
[tex]\bf V_{rind}=\pi r^3\left( \cfrac{271}{250}\right)\implies \boxed{V_{rind}=\cfrac{271\pi }{250}r^3} \\\\\\ \cfrac{dV_{rind}}{dt}=\cfrac{271\pi }{250}\cdot \stackrel{chain~rule}{3r^2\cdot \cfrac{dr}{dt}}\implies \cfrac{dV_{rind}}{dt}=\cfrac{813\pi r^2\cdot \frac{dr}{dt}}{250} \\\\\\ \begin{cases} \frac{dr}{dt}=2\\\\ r=10 \end{cases}\implies \cfrac{dV_{rind}}{dt}=\cfrac{813\pi 10^2\cdot 2}{250}\implies \cfrac{dV_{rind}}{dt}\approx 3043.29[/tex]
Related Questions
which of the following three dimensional objects has Rotational symmetry about a line A.a car. B.a set of rectangular stairs. C. a rolling pin. D.a stapler
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Answer:
C. a rolling pin.Step-by-step explanation:
A rolling pin has a cylindric shape, which is a revolution surface. This means that the rolling pin is symmetric about a line or axis, therefore, it can be rotated around it.
A rolling pin has rotational symmetry about its longitudinal axis, making it the correct answer, while a car, a set of rectangular stairs, and a stapler do not demonstrate this symmetry.
Explanation:The question relates to the concept of rotational symmetry in three-dimensional objects, which is a part of the study of Physics, specifically dealing with rotational motion. The answer to which of the listed objects has rotational symmetry about a line is C. a rolling pin. A rolling pin has rotational symmetry because when it rotates about its longitudinal axis, its appearance is unchanged after a certain angle of rotation. In contrast, objects like a car, a set of rectangular stairs, and a stapler do not exhibit this symmetry because their appearance changes visibly when they are rotated about any line.
How many feet of fencing is needed to fence in a 130 ft by 225 ft area?
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Perimeter is:
2l + 2w
If l equals to length and w width.
So,
2 * 130 + 2 * 225
260 + 450
710
So the perimeter is 710 feet, so you need 710 feet to fence in a 130 ft by 225 ft area.
Gomez runs a small pottery firm. he hires one helper at $15,000 per year, pays annual rent of $6,500 for his shop, and spends $23,000 per year on materials. he has $40,000 of his own funds invested in equipment (pottery wheels, kilns, and so forth) that could earn him $6,000 per year if alternatively invested. he has been offered $20,500 per year to work as a potter for a competitor. he estimates his entrepreneurial talents are worth $5,000 per year (input cost for organizing resources). total annual revenue from pottery sales is $82,000. calculate the accounting profit and the economic profit for gomez's pottery firm.
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The accounting profit for Gomez's pottery firm is $37,500, calculated by subtracting explicit costs from total revenue. The economic profit is $6,000, figured out by subtracting both explicit costs and implicit costs from total revenue.
Explanation:To calculate the accounting profit for Gomez's pottery firm, we subtract all explicit costs (money spent) from the total revenue. In this case, we subtract the salary of the helper ($15,000), the rent of the shop ($6,500), and the amount spent on materials ($23,000) from the total annual revenue from pottery sales ($82,000). Therefore, the accounting profit is $82,000 - $15,000 - $6,500 - $23,000 = $37,500.
On the other hand, to obtain the economic profit, we need to also subtract implicit costs, which are the opportunity costs of using the resources in one way rather than the next best alternative way. Gomez could have made $6,000 by investing his own $40,000 elsewhere, and he could have earned $20,500 by working for a competitor, so these are both implicit costs. Additionally, he estimates his entrepreneurial talents to be worth $5,000. Therefore, "the economic profit is $37,500 (accounting profit) - $6,000 (returns from alternative investment) - $20,500 (salary from competitor) - $5,000 (value of his entrepreneurial talents) = $6,000".
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14.4% of what is 10.44
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Dividing both sides by 0.144, x = 10.44/0.144 = 72.5.
The "what?" is 72.5.
what is 62.19 x 32.5
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What algebraic property is demonstrated in the equation below? 5+3(2x-7)= 5+6x-21
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The distributive property is demonstrated in the equation.
Explanation:The algebraic property demonstrated in the equation is the distributive property. The distributive property states that when a number is multiplied by the sum or difference of two other numbers, you can distribute the multiplication to each term inside the parentheses.
In the given equation, 3(2x - 7) means multiplying 3 by both terms inside the parentheses. This results in 6x - 21. Therefore, the equation can be simplified to 5 + 6x - 21.
Anyone knows??? And like to help
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Let the price of a ticket be t.
t - 2 = 7
6b.
The number of gallons is g.
7 = 2g
6c.
Sorry, but I don't know what problem 5 was, so I can't answer.
A car can travel 30 mi on a gallon of gas and has20 gallon gas tanks let g be the number of gallons the car has in its tank the function d=30 g gives the distance d in miles that the car travels on g gallons
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[tex]30 \times 20 = 60[/tex]
What are the values of a, b, and c in the quadratic equation 0 =1/2 x2 – 3x – 2?
A. a =1/2 , b = 3, c = 2
B. a =1/2 , b = –3, c = –2
C. a =1/2 , b = 3, c = –2
D. a = 1/2, b = –3, c = 2
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Hope it helped :)
Answer:
B.
Step-by-step explanation:
The equation is [tex]\frac{1}{2} x^{2} -3x -2[/tex]. a is the coefficient of [tex]x^{2}[/tex], b is the coefficient of x and c is the independent term. So, for the equation we have a = 1/2, b= -3 and c = -2.
Stu hiked a trail at an average rate of 3 miles per hour. He ran back on the same trail at an average rate of 5 miles per hour. He traveled for a total of 3 hours. Which equation can be used to find the time it took Stu to hike the trail? 3t = 5 3t = 5(3 – t) 3 + t = 5 + (3 – t) 3t + 5(3 – t) = 3
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3t = 15 - 5t
3t + 5t = 15
8t = 15
t = 15/8
t = 1 7/8
7/8 * 60 mins = 52.50 minutes
t = 1 hour and 52.5 minutes
It took Stu 1 hour and 52.5 minutes to hike the trail one way.
Answer:
B
Step-by-step explanation:
Which line on the graph below has an undefined slope?
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Answer:
R
Step-by-step explanation:
A vertical line has slope that is "undefined". Line R is the vertical line on your graph.
_____
The slope of a vertical line is "undefined" because of the way slope is defined. Slope is the vertical change divided by the horizontal change. When the line is vertical, there is no horizontal change, so the computation of slope involves division by zero. The result of division by zero is "undefined."
Answer:
C. R
Good Luck!
Write an equation of the line passing through (-5,5) and having slope -6. Guve the answer in slope-intercept form.
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We want the answer in slope-intercept form.
** Remember! Slope-intercept form : y=mx+b where m=slope, b=y-intercept.
Simply plug everything in :)
y = mx + b
y = (-6)x + (5)
Simplify.
y = -6x + 5
~Hope I helped!~
Allison practices her violin for at least 12 hours per week. She practices for three fourths of an hour each session. If Allison has already practiced 3 hours this week, how many more sessions remain for her to meet or exceed her weekly practice goal?
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a rectangle has a side length 2x+9 and side width x. If the perimeter is 36 cm, what is the value of x?
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4x + 18 + 2x = 36
Combine like terms.
6x + 18 = 36
Subtract 18 from both sides.
6x = 18
Divide 6 on both sides.
x = 3
Hope this helps!
If f(x)=8x and g(x)=2x+1, find(f•g)(x).
A) 16x+1
B) 10x+1
C) 16x+8
D) 16x^2+8x
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8x(2x + 1)
16x^2 + 8x
(f•g)(x) = 16x^2 + 8x
A rectangle is 2 4/5 meters wide and 3 1/2 m long what is its area
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I like to convert my fractions to decimals if that's alright with you☺
2.8 * 3.5
^ ^
2 4/5 , 3 1/2
=9.8m
hope this helped!
The area of the rectangle whose dimensions are 2 ⁴/₅ meters wide and 3 ¹/₂ m long will be 49/5 square meters.
What is the area of the rectangle?Let W be the rectangle's width and L its length.
The area of the rectangle is the multiplication of the two different sides of the rectangle. Then the rectangle's area will be
Area of the rectangle = L×W square units
A rectangle is 2 ⁴/₅ meters wide and 3 ¹/₂ m long. Then the area of the rectangle will be
A = (2 ⁴/₅)(3 ¹/₂)
A = (14/5) (7/2)
A = 49/5 square meters
The area of the rectangle whose dimensions are 2 ⁴/₅ meters wide and 3 ¹/₂ m long will be 49/5 square meters.
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A boy walks for x hours at 33 mph and bicycles for (5−x) hours at 93 mph. If the total distance traveled is d miles, express d in terms of x.
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distance = speed * time
Walking:
d = speed * time
d1 = 33 mph * x hours = 33x miles
Bicycling:
d = speed * time
d2 = 93 mph * (5 - x) hours = 93(5 - x) miles = (465 - 93x ) miles
The total distance traveled is the distance walked plus the distance bicycled
d = d1 + d2
d = 33x + 465 - 93x
d = -60x + 465
Answer:
d = 465 - 60x
Step-by-step explanation:
1. Find the distances of walking and bicycling. We can express the walking distance as 33x, because x is the time and 33 is the amount of miles, and we can express the bicycling distance as 93(5-x), because 93 is the speed and 5-x is the amount of hours.
2. Construct an equation. We can say that the total distance is d = 33x + 93(5-x), the sum of both of the distances.
3. Simplify. If you simplify, you would get d = 465 - 60x
(This guy is going Super Saiyan)
Answer: d = 465 - 60x
a pile of newspapers in the recycling bin was 12 1/4 inches high. each week the height of the pile increases by 4 5/8 inches. what is the height of the pile after 4 weeks
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There were 36,000 people at a ballgame in los angeles. the day's receipts from $12 reserved seats and $6 general-admission seats were $258,000 how many of each type of seat were sold?
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A group of friends are going to a cottage for the weekend. The scale used is 1 : 250,000. The distance measures 8.5cm on the map. What is the actual distance
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So, the calculation for the actual distance would be:
8.5 cm x 250,000 = 2,125,000 cm
since 1 km = 100,000 cm
2,125,000 cm = 21,25 Km
Which equation represents a direct variation? y = 2x y = x + 4 y = x y =
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Which is the graph of the function f(x) = x2 + 2x – 6? Mark this and return
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Focus: ( -1, -27/4 )
Axis of Symmetry: x = -1
Directrix: y = -29/4
X | Y
———
-3| -3
-2| -6
-1 | -7
0 | -6
1 | -3
The graph of the function f(x) = x² + 2x − 6 is an upward-opening parabola with the vertex at (-1, -7) and a y-intercept at (0, -6). There are no asymptotes for this quadratic function.
Explanation:To find the graph of the function f(x) = x² + 2x − 6, we need to look at the characteristics of this quadratic equation. This is a parabola that opens upward because the coefficient of x² is positive. We can find the vertex of this parabola by using the formula -b/2a for the x-coordinate of the vertex.
Here, a = 1 and b = 2. Applying the formula, we get the x-coordinate of the vertex as -b/2a = -2/(2×1) = -1. Now, we plug this value back into the function to get the y-coordinate of the vertex: f(-1) = (-1)² + 2(-1) − 6 = 1 - 2 - 6 = -7. Therefore, the vertex of the parabola is at (-1, -7). The parabola will intersect the y-axis when x = 0, which gives us the y-intercept f(0) = -6. No asymptotes are present in this quadratic function.
The graph of f(x) is symmetrical about the line x = -1 and has a y-intercept at (0, -6). The vertex at (-1, -7) is the lowest point on the graph since the parabola opens upwards. Thus, we are looking for a graph with these characteristics to represent the function f(x) = x² + 2x − 6.
Which graph represents the solution set of the system of inequalities?
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So the first one (x + y < 1) will be changed to y < -x + 1.
The second one (2y ≥ x - 4) will be changed to y ≥ x/2 - 2.
Now we can analyze each graph.
In every single graph the first equation (y < -x + 1) is graphed correctly.
Now for the second equation, we can see that only the first and last graph correctly format to the equation.
Now for the shading:
The first equation shows us that y is less than -x +1, making the shading go under the dotted line. (to the left)
The second equation shows us that y is greater than or equal to x/2 - 2, making the shading go above the line. (also to the left)
Therefore, when we shade, the overlapping shading is correctly formatted in the first graph.
Hope this helped, comment any questions you have for me.
Find the absolute value of the jacobian, ∣∣∂(x,y)∂(s,t)∣∣, for the change of variables given by x=s+5t,y=3s+8t
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Which function represents the graph of f(x)=|2x| after it is translated 5 units to the left?
g(x)=|2(x−5)|
g(x)=|2x+5|
g(x)=|2x−5|
g(x)=|2(x+5)|
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Answer:
g(x) = |2(x+5)|
Step-by-step explanation:
To translate a function left 5 units, replace x with (x+5). In this function, that means
2x ⇒ 2(x+5) . . . . . matches the last choice
On Anne's bicycle, the ratio of pedal turns to rear- wheel turns in second gear is 4 to 7. If her rear wheel turns 875 times per mile, how many times does she turn the pedal in one mile? HINT: Set up and solve as a proportion to find x.
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cross multiply
(7)(x) = (4)(875)
7x = 3500
x = 3500/7
x = 500 <== she turns the pedal 500 times in 1 mile
Andrew has 10 more goldfish than todd.together they have 50 goldfish.how many goldfish each boy have?
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Todd has 20
Answer:
Andrew has 30 goldfish and Todd has 10 goldfish
Step-by-step explanation:
Lets call A the number of goldfish Andrew has and T the number of goldfish Todd has.
Andrew has 10 more goldfish than Todd:
A = T + 10
Together they have 50 goldfish:
A + T = 50
Lets replace A = (T + 10) in the las equation:
T + 10 + T = 50
Lets solve T in this equation:
2T + 10 -10 = 50 -10
2T /2 = 40 /2
T = 20
Todd has 20 goldfish
Now lets calculate A = T + 10 = 20 + 10 = 30
Andrew has 30 goldfish
Tiesha enjoys reading in her spare time. She reads 4 pages every 1/10 of an hour. The proportional relationship between the number of pages (p) and the number of hours (h) is represented by the equation ______. Write the equation in standard form with a constant of proportionality greater than 1.
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The equation in standard form with a constant of proportionality greater than 1 is p=40h.
Given that, Tiesha enjoys reading in her spare time. She reads 4 pages every 1/10 of an hour.
What is the proportional relationship?Proportional relationships are relationships between two variables where their ratios are equivalent. Another way to think about them is that, in a proportional relationship, one variable is always a constant value time the other. That constant is known as the "constant of proportionality".
Now, p∝h
⇒p=kh-----(i)
So, 4∝1/10
⇒4=k/10
⇒k=40-----(ii)
Substitute equation (ii) in equation (i).
That is, p=40h
Therefore, the equation in standard form with a constant of proportionality greater than 1 is p=40h.
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Mr. and mrs. harris have six kids aged 5, 6, 8, 10, 16, and 19. the median age of the harris children is _____.
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Answer:
The median age is 9 years old
Step-by-step explanation:
We can find the median by getting the average of the center number of the middle two numbers
8 and 10 are the two middle numbers in this scenario. 8 + 10 = 18
18/2 = 9
Find three consecutive even integers such that the sum of twice the smallest number and three times the largest is 42.
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Let n be the smallest of the three even integers.
Then the next larger even integer is n + 2.
The largest even integer is n + 4.
Now we follow what the problem tells us.
Twice the smallest is 2n
3 times the larges is 3(n + 4)
2n + 3(n + 4) = 42
2n + 3n + 12 = 42
5n + 12 = 42
5n = 30
n = 6
n + 2 = 8
n + 4 = 10
The integers are 6, 8, 10
To find the consecutive even integers, we created an equation and solved for x, which represents the smallest integer. Upon solving, we found that the smallest integer is 6, and hence, the three consecutive even integers are 6, 8, and 10.
To find three consecutive even integers such that the sum of twice the smallest number and three times the largest is 42, we can form an equation based on the description. Let the smallest even integer be x. Since we are looking for consecutive even integers, the next two will be x+2 and x+4.
According to the problem, twice the smallest plus three times the largest equals 42:
2x + 3(x+4) = 42
Solving the equation:
Distribute the 3 into the parentheses: 2x + 3x + 12 = 42
Combine like terms: 5x + 12 = 42
Subtract 12 from both sides: 5x = 30
Divide both sides by 5: x = 6
Therefore, the smallest even integer is 6, and the other two are 8 and 10, which are the consecutive even integers that satisfy the given condition.