Answers
Using implicit differentiation, the rates are given as follows:
2. dV/dt = 144π cm³/sec.
3. dh/dt = 2/5π cm/s.
4. dr/dt = -0.2/π cm/day.
What is the rate of change of the volume of an sphere?The volume of an sphere of radius r is given by the following equation:
[tex]V = \frac{4}{3}\pi r^3[/tex]
Applying implicit differentiation, differentiating both variables relative to t, the rate of change is given as follows:
[tex]\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}[/tex]
For item 2, the parameters are given as follows:
[tex]\frac{dr}{dt} = 2, r = 4[/tex]
Hence the rate is given as follows:
dV/dt = 4π x 4² x 2 = 144π cm³/sec.
For item 4, the parameters are:
[tex]\frac{dV}{dt} = -0.2, r = 5[/tex]
Negative because the orange is shrinking.
Hence the rate of the radius can be found as follows:
[tex]\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}[/tex]
[tex]-0.2 = 4\pi (5)^2\frac{dr}{dt}[/tex]
dr/dt = -0.2/100π
dr/dt = -0.2/π cm/day.
What is the rate of change of the volume of an cylinder?The volume of a cylinder of radius r and height h is given as follows:
[tex]V = \pi r^2h[/tex]
The rate of change of the volume as a function of time is given by:
[tex]\frac{dV}{dt} = 2\pi rh\frac{dr}{dt} + \pi r^2\frac{dh}{dt}[/tex]
For item 3, the parameters are given as follows:
[tex]r = 5, \frac{dV}{dt} = 10, \frac{dr}{dt} = 0[/tex]
The radius is of 5 as r² = 25, due to the area of the base.
Hence the rate of change of the height is found as follows:
[tex]\frac{dV}{dt} = \pi r^2\frac{dh}{dt}[/tex]
10 = 25π dh/dt
dh/dt = 10/25π
dh/dt = 2/5π cm/s.
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Related Questions
The number of hours it takes for ice to melt varies inversely with the air temperature. A block of ice melts in 2.5 hours whenthe temperature is 54 degrees.How long would it take for the same block of ice to melt if the temperature was 45 degrees?
Answers
Let t be the time in hours it takes a block of ice to melt at T degrees.
Since both variables have an inverse relation, we can say that:
[tex]t=\alpha\cdot\frac{1}{T}[/tex]Where alpha is a constant value.
Since we know that a block of ice melts in 2.5 hours when the temperature is 54 degrees, we can plug in such data and find alpha as following:
[tex]\begin{gathered} 2.5=\alpha\cdot\frac{1}{54}\rightarrow2.5\cdot54=\alpha \\ \Rightarrow\alpha=135 \end{gathered}[/tex]Therefore, our expression would be:
[tex]t=\frac{135}{T}[/tex]For T = 45,
[tex]\begin{gathered} t=\frac{135}{45} \\ \\ \Rightarrow t=3 \end{gathered}[/tex]We can conclude that it would take 3 hours for the block of ice to melt.
5 tons of rocks cost $7,000.00. What is the price per pound ??
Answers
Answer: Per rock or pound?
which number is the closest approximation to the value of 3/94?
Answers
Answer:
[tex] \frac{1}{3} [/tex]
Step-by-step explanation:
[tex] \frac{3}{94} = 0.31 \\ \\ \frac{1}{3} = 0.33[/tex]
(Solve the problem down below & simplify the answer. Round to the nearest hundredth as needed.)
Answers
From the given information, we know that the decreasing function values is
[tex]V(t)=3600(3^{-0.15t})[/tex]and we need to find the time t when V(t) is equal to $1200. Then by substituting this values into the function, we have
[tex]1200=3600(3^{-0.15t})[/tex]By dividing both sides by 3600, we get
[tex]3^{-0.15t}=\frac{1200}{3600}=\frac{1}{3}[/tex]So we have the equations
[tex]3^{-0.15t}=\frac{1}{3}[/tex]From the exponents properties, we know that
[tex]3^{-0.15t}=\frac{1}{3^{0.15t}}[/tex]so we have
[tex]\frac{1}{3^{0.15t}}=\frac{1}{3}[/tex]or equivalently,
[tex]3^{0.15t}=3[/tex]This means that
[tex]0.15t=1[/tex]Then, by dividing both sides by 0.15, we obtain
[tex]t=\frac{1}{0.15}=6.6666[/tex]So, by rounding to the nearest hundreadth, the answer is 6.67 years
Suppose 72% of students chose to study French their freshman year, and that meant that there were 378 such students. How many students chose not to take French their freshman year?
Answers
As per the given percentage, there are 147 students are not take French their freshman year.
Students:
The students are the person formally engaged in learning, especially one enrolled in a school or college.
Given,
Suppose 72% of students chose to study French their freshman year, and that meant that there were 378 such students.
Here we need to find the number of students who are not take French as their freshman year.
Let x be the total number of students in the freshman year.
So according to the given question, we know that,
72% of x = 378
Therefore, we have to find the value of x, so we have to move the others,
Then we get,
x = 378/72%
x = 378 x 100/72
x = 525
So, the total number of students are 525.
Then the students who are not taken French is,
=> 525 - 378 = 147
Therefore, the students who are not taken French is 147.
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Can you please help me solve this questions. I want a clear explanation thank you.
Answers
Given:
[tex]\begin{gathered} \text{Mean}=\mu=21.4 \\ \text{Standard deviation=}\sigma=5.0 \\ \text{Smple size=n=66} \end{gathered}[/tex]The z-score for 90 percent is 1.64.
To find the 90 percent confidence interval,
[tex]\begin{gathered} \mu+(\frac{\sigma}{\sqrt[]{n}}\times1.64)=21.4+(\frac{5}{\sqrt[]{66}}\times1.64) \\ =21.4+1.0094 \\ =22.4094\ldots\ldots(\text{ upper limit)} \\ \text{And} \\ \mu-(\frac{\sigma}{\sqrt[]{n}}\times1.64)=21.4-(\frac{5}{\sqrt[]{66}}\times1.64) \\ =21.4-1.0094 \\ =20.3906\ldots\ldots\ldots(\text{lower limit)} \end{gathered}[/tex]Answer: The 90 % confidence interval is ( 20.3906 , 22.4094 )
Rajan climbed to a height of 15 ft from the bottom of a climbing wall. He then climbed up an additional 10 ft.
What must he do to return to the bottom of the climbing wall?
In the first blank, say whether Rajan must climb up or down. In the second blank, state the number of feet Rajan must travel up or down.
Rajan must climb_____ft to return to the bottom of the climbing wall.
Answers
Rajan must climb down 25 ft to return to the bottom of the climbing wall
Height climbed from the bottom of the wall = 15ft
Height: The measurement of an object's height from the base to the top is known as height. It is sometimes referred to as altitude in geometry. The vertical distance between the lowest and highest point is measured. In coordinate geometry, it is often the measurement along the y-axis of an object.
Additional height climbed = 10ft
Total height climbed = 25ft
He should climb down the distance, as climbing up will further move him away from the bottom of the wall
Distance to climb down to the bottom of the wall = 25ft
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How many times larger is 8.1 than 2.7
Answers
Helping my son with homework.. I have no clue what this is
Answers
Answer:
45 times
Step-by-step explanation:
it is 45 times because it can land on blue or green 5 out of 8 times so u do 72 divided by 8 and u get 9 so I do 9 times 5 and u get 45
If f (x) = x
2 − 2 x , g (x) = x − 2
1) prove that : f(2) = g(2)
2) If g (K) = 7 , find : the value of k
Answers
1. The value of f(2) = 0 and g(2) = 0
2. The value of k is 9
What is a function?A function can simply be described as a law, rule, theorem or expression that explains the relationship between two terms.
These terms are known as;
The dependent variableThe independent variableGiven the functions;
f(x) = x^2 - 2xg(x) = x - 21To prove that f(2) = g(2)
Substitute the value of x as 2 in the functions, we have;
f(2) = 2^2 - 2(2)
expand the bracket
f(2) = 4 - 4
f(2) = 0
Then for g(2), we have;
g(2) = 2 - 2
g(2) = 0
Then, f(2) = g(2) = 0
If g(k) = 7
Then, we have;
k - 2 = 7
collect like terms to determine the value of k
k = 7 + 2
k = 9
Hence, the value is 9
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I dont know how to answer this pls help
Answers
The hedgehog's total change in weight is -2.5 ounces.
What is the total change in weight?A relationship is proportional if the ratio of the variables is constant. The variables can either increase or decrease at a constant rate. A proportional relationship can be modelled with a linear equation.
Ratio = change in weight / days of hibernation
-0.18 / 9 = -0.02
-0.56 / 28 = -0.02
-1.44 / 76 = -0.02
-1.96 / 98 = -0.02
Weight when the day of hibernation is 125 = number of days x ratio
125 x -0.02 = -2.5
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can you help me factor the expression x^2+4x-21
Answers
To factor this expression we need to find two numbers which sum is 4 and the product is -21.
Now, 21 has only 2 factors (besides 1 and 21), 7 and 3, se we need to write them in a way such that they sum 4, so
[tex]x^2+4x-21=(x+7)(x-3)[/tex]we see that 7-3=4 and 7(-3)=-21, so the factorization is correct.
Four of the twelve members of the panel were late for
the meeting. What fraction of the members were on
time?
Answers
Answer:
2/3
Step-by-step explanation:
If 4 were late, then 8 were on time.
8/12 Divide the top and bottom by 4
2/3
on a map where each unit represents one kilometer two marinas are located at p(4,2) and q(8,12). if a boat travels in a straight line from one marina to the other how far does the boat travel. Answer choices: 14 kilometers 2sqrt296 kilometer 2sqrt5 kilometers
Answers
Solution:
Given the points below
[tex]p\left(4,2\right)and\text{ }q\left(8,12\right)[/tex]To find the distance between two points, the formula is
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Where point p represents coordinates 1 and point q represents coordinates 2
Substitute the coordinates into the formula above
[tex]\begin{gathered} d=\sqrt{(8-4)^2+(12-2)^2} \\ d=\sqrt{4^2+10^2} \\ d=\sqrt{16+100} \\ d=\sqrt{116} \\ d=\sqrt{4\times29} \\ d=2\sqrt{29}\text{ units} \end{gathered}[/tex]Since, 1 unit represents 1 kilometer on the map,
Hence, the answer is
[tex]2\sqrt{29}\text{ km}[/tex]Choose the equation that represents a line that passes through points (−3, 2) and (2, 1)
5x + y = −13
5x − y = 17
x − 5y = −13
x + 5y = 7
Answers
The equation that represents a line that passes through points (−3, 2) and (2, 1) is x + 5y = 7.
How to find the equation of a line?The equation of the line passes through the points (−3, 2) and (2, 1). The equation of the line can be represented as follows;
y = mx + b
where
m = slopeb = y-interceptTherefore, let's find the slope using (−3, 2) and (2, 1)
m = slope = 1 - 2 / 2 + 3
m = - 1 / 5
Therefore,
slope = - 1 / 5
Let's find the y-intercept of the line using (2, 1)
y = - 1 / 5 x + b
1 = - 1 / 5 (2) + b
b = 1 + 2 / 5
b = 5 + 2/ 5
b = 7 / 5
Therefore, the equation of the line can be represented as follows:
y = - 1 / 5 x + 7 / 5
5y = -x + 7
5y + x = 7
x + 5y = 7
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In the distribution shown, state the mean and the standard deviation. Hint: The vertical lines are 1 standard deviation apart.
Answers
We are given a distribution graph.
The mean of the distribution is the center that is 125.
[tex]\operatorname{mean}=\mu=125[/tex]The standard deviation is given by
[tex]std=\sigma=153-125=28[/tex]Therefore, the mean of the distribution is 125 and the standard deviation is 28.
I just need to know how to work it out for future problems. (Business Calc)
Find the price elasticity of the demand function as a function of p.
1. q=D(p)=600-3p
A)E(p)= p/p-200
B)E(p)= p/600-3p
C)E(p)= p/200-p
D)E(p)= p(200-p)
Answers
The price elasticity of the demand function as a function of p is C. E(p)= p/200-p
How to illustrate the information?It should be noted that the price elasticity of demand simply means the ratio of the percentage change in quantity demanded to the percentage change in price.
In this case, the function is given as:
q = D(p) = 600-3p
Differentiate with respect to p
D'(p) = 3
The elasticity of demand will be:
= (P / 600 - 3p) × (-3)
= P / (200 - P)
In conclusion, the correct option is C.
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What is the exact decimal equivalent of 7/12
Answers
Answer:
0.58333333333
convert these measurements to centimeters : (a)73.1 cm and (b) 0,0012km
Answers
Answer:
(a) 73.1 cm (b) 120 cm
Step-by-step explanation:
Step-by-step explanation:
Part (a)
73.1 cm [It remains the same as it is already given in cms]
Part (b)
To convert 0.0012km to cm we multiply x 100000
=> 0.0012 x 100000
=> 120 cm
Write the inequality shown by the shaded region in the graph with the boundary line y = -3x +5
Answers
Inequality is an inequality comparison of numbers or expressions.
Given inequality:
y < -3x+5, y < x+2, y < -1
Graph the above inequality using Geogebra's online graphing calculator.
Answer:
...................... Hoffe er u
(3ax - n) ÷ 5
Solve for X.
Answers
Answer:
x = 10/a; restrictions a ± 0
Answer:
[tex]x = \cfrac{n+5}{3a}[/tex]=============================
Given expression
[tex](3ax - n) : 5 = 1[/tex]Solve it for x in steps below:
[tex]3ax - n = 5[/tex][tex]3ax = n + 5[/tex][tex]x = \cfrac{n+5}{3a}[/tex]Find sinθ, tanθ, and secθ, where θ is the angle shown in the figure. Give exact values, not decimal approximations.sinθ=tanθ=secθ=
Answers
1) We have here a right triangle, so we can use the Pythagorean Theorem to find the adjacent leg. Note that the hypotenuse is leg opposite to the right angle. In this case, a=4.
a²=b²+c²
4²=b²+3²
16=b²+9
16-9=b²
b=√7
2) Let's find the trigonometric functions. The first one is the sine (θ), which relates the opposite side and the hypotenuse:
[tex]\sin (\theta)\text{ =}\frac{3}{4}[/tex]2.2)Moving on, we can deal with the tangent of theta, relating the opposite side over the adjacent, for this one we have to rationalize it:
[tex]\begin{gathered} \tan (\theta)\text{ =}\frac{3}{\sqrt[]{7}} \\ \tan (\theta)=\frac{3\sqrt[]{7}}{7} \end{gathered}[/tex]Finally, let's find the reciprocal function of the cosine function by setting this as the reciprocal of the cosine, and then calculating it:
[tex]\begin{gathered} \sec \text{ (}\theta)=\frac{1}{\cos(\theta)}\Rightarrow\sec (\theta)=\frac{1}{\frac{\sqrt[]{7}}{4}}\Rightarrow\sec (\theta)\text{ =}\frac{4}{\sqrt[]{7}} \\ \sec (\theta)\text{ =}\frac{4\sqrt[]{7}}{7} \end{gathered}[/tex]2. A membership to the theme park is $48 per year. Marcos cancelled his membership early and is only required to pay for 0.25 of a year. How much will he need to pay?
Answers
Marcos needs to pay $12 for 0.25 of a year due to cancelling the membership
Membership cost to the theme park per year = $48
Payment required to be made by Marcos = 0.25 of a year
When we divide a whole into smaller parts, we get decimals. Then, there are two parts to a decimal number: a whole number part and a fractional part. The whole component of a decimal number has the same decimal place value system as the complete number. However, as we proceed to the right following the decimal point, we obtain the fractional portion of the decimal number.
Converting the decimal into a fraction we get:
=0.25
= 25/100
= 1/4 of a year
So, Marcos needs to pay the membership fees for only 1/4 of a year
According to which we get: 48*1/4 = $12
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it says on top that the image is not drawn to scale.
Answers
D; Both statements are correct
Here, we want to check the correct statement
Basically from the diagram, it is confirmed that what we have is a 30-60-90 triangle
Now, for a 30-60-90 degrees triangle, the side lengths are in a particular ratio
From the diagram, MO faces the right angle and thus, it is the longest side and the hypotenuse
Now, let us take a look at the ratio of the sides in both submissions
For a 30-60-90 degrees triangle, the ratio of the sides is;
[tex]1\colon\sqrt[]{3}\text{ :2}[/tex]Hence, one side must be twice the length of the other
From Kellie's statement we can see MO is twice NO
From Eddie's statement, we can also see that MO is twice NO
Lastly, we can also see in both that dividing MN by NO yields the root of 3 in both statemets which confirms the existence of the theoretical ratio
So the conclusion here is that;
Both students are correct
Jolie is redesigning a water bottle to fit better in her cup holder. The diameter will have to be no larger 2.5 in. The water bottle is straight up and down. What would the height of the bottle be if she wants it to have a 99.34 in3 volume. (Round to nearest whole number)
Answers
20in
Explanations:The water bottle in question is known to be cylindrical in shape. The formula for calculating the volume of a cylinder is expressed as:
[tex]V=\pi r^2h[/tex]where;
r is the radius
h is the height
Given the following parameters
diameter = 2.5in
radius = 2.5/2 = 1.25in
volume = 99.34 in^3
Required
Height of the water bottle
Substitute the given parameters intothe formula
[tex]\begin{gathered} 99.34=3.14(1.25)^2\times h \\ 99.36=4.90625h \\ h=\frac{99.36}{4.90625} \\ h=20.25\approx20in \end{gathered}[/tex]The height of the water bottle will be 20in
A person tosses a coin 14 times. In how many ways can he get 10 heads?
Answers
14 times 14 =196 divided by 2 = 98
10 times 10 = 100 divided by 2 = 50
He can get 10 heads 50 different ways
Find X.
Cirucumscribed angles
Answers
Answer:
x = 64
Step-by-step explanation:
to find x add make an equation to solve for x. the equation is 90+90+44+2x+8=360. after solving for x, x will equal 64.
Question 10The table below shows the amount of water that is in a bathtub that is being filled.Minutes1234Gallons ofWater12172227Which of the following functions could model the amount of gallons of water, f(x), that the bathtub will contain after x minutes of being filled?Аf(x) = 5x + 7Bf(x) = 5x + 12сf(x) = 7x + 5Df%) = 12x + 5No
Answers
PLS HELP WILL MARK YOU BRAINLIEST!!
Point B is the midpoint of AC. AB=2x and AC=3x+20.
What is the value of x?
x=-20
x=20
x=4
x=6
Answers
Answer:
X=4
Step-by-step explanation:
i got the answer on a different app
Answer: The correct answer is 20 or -20
Step-by-step explanation:
its between 20 and -20 since i selected x=6 and x=4 and they were both wrong lol
To rent a certain meeting room, a college charges a reservation fee of $46 and an additional fee of $6.70 per hour. The math club wants to spend less than$106.30 on renting the meeting room.What are the possible amounts of time for which they could rent the meeting room?Use t for the number of hours the meeting room is rented, and solve your inequality for t.
Answers
As given by the question
There are given that the college charges a reservation fee of $46.
Now,
According to the question, the inequality is:
[tex]46+6.70t<106.30[/tex]Then,
To solve the above inequality, collect the constant terms on the right side
So,
[tex]\begin{gathered} 46+6.70t<106.30 \\ 6.70t<106.30-46 \\ 6.70t<60.3 \\ t<\frac{60.3}{6.70} \\ t<9 \end{gathered}[/tex]Hence, the room could be rented for up to 9 hours less than the stated amount.