The depth of the water is increasing at a rate of approximately is 0.14 ft/min
To determine how fast the depth of the water is increasing when the water is 13 feet deep, we need to relate the volume of the water in the conical tank to its depth. We can use related rates and the geometry of the cone.
Given:
- The radius of the tank at the top R = 10 feet
- The height of the tank H = 27 feet
- The rate of water flow into the tank [tex](\(\frac{dV}{dt}\))[/tex] = 10 ft[tex]\(^3\)/min[/tex]
- The depth of the water h = 13 feet
First, let's find the relationship between the radius r of the water's surface at depth h.
Since the water forms a smaller cone similar to the tank, we can use the concept of similar triangles:
[tex]\[\frac{r}{h} = \frac{R}{H} \implies \frac{r}{h} = \frac{10}{27} \implies r = \frac{10}{27}h\][/tex]
Next, we use the volume formula for a cone:
[tex]\[V = \frac{1}{3} \pi r^2 h\][/tex]
Substitute [tex]\(r = \frac{10}{27}h\):[/tex]
[tex]\[V = \frac{1}{3} \pi \left( \frac{10}{27}h \right)^2 h = \frac{1}{3} \pi \frac{100}{729} h^3 = \frac{100\pi}{2187} h^3\][/tex]
Differentiate both sides with respect to t:
[tex]\[\frac{dV}{dt} = \frac{100\pi}{2187} \cdot 3h^2 \frac{dh}{dt}\][/tex]
Simplify:
[tex]\[\frac{dV}{dt} = \frac{300\pi}{2187} h^2 \frac{dh}{dt}\][/tex]
Given [tex]\(\frac{dV}{dt} = 10 \) ft\(^3\)/min[/tex], and [tex]\(h = 13\) ft:[/tex]
[tex]\[10 = \frac{300\pi}{2187} \cdot (13)^2 \cdot \frac{dh}{dt}\][/tex]
Solve for [tex]\(\frac{dh}{dt}\):[/tex]
[tex]\[10 = \frac{300\pi}{2187} \cdot 169 \cdot \frac{dh}{dt}\][/tex]
[tex]\[10 = \frac{50700\pi}{2187} \cdot \frac{dh}{dt}\][/tex]
[tex]\[\frac{dh}{dt} = \frac{10 \cdot 2187}{50700\pi}\][/tex]
[tex]\[\frac{dh}{dt} = \frac{21870}{50700\pi}\][/tex]
[tex]\[\frac{dh}{dt} \approx \frac{21870}{159252} \approx \frac{1}{7.29} \approx 0.137 \text{ ft/min}\][/tex]
Therefore, the depth of the water is increasing at a rate of approximately: [tex]\[\boxed{0.14 \text{ ft/min}}\][/tex]
The sum of two numbers is 35. Three times the larger number is the same as 4 times the smaller number. Find the smaller number
(HINT. Let x= larger number; 35- x= smaller number
I need help solving this!!!
Numbers 3 and 4, solve algebraically. I don't get #5 question.
Find the exact value of the following. Do NOT use labels in your answer and do NOT round your answer. -3(1.2)^2 =
A cube has a surface area of 900 cm what is the number of cubic cm in the volume of the cube
X+y+z=1
2x+4y+6z=2
-x+3y-5z=11
The question revolves around solving a system of three linear equations with three variables, using algebraic techniques such as substitution or elimination.
Explanation:The system of equations provided can be solved using methods such as substitution, elimination, or matrix operations. The equations given are:
x + y + z = 1 2x + 4y + 6z = 2 -x + 3y - 5z = 11
To solve this system, we can perform operations such as eliminating one variable at a time or using matrices to find the values of x, y, and z that satisfy all three equations simultaneously.
Kiko is coasting on her bicycle down a hill. After 3 s, her speed is 10 m/s. After 8 s, her speed is 25 m/s. What is her acceleration during this time?
Kiko is coasting on her bicycle down a hill. After 3 s, her speed is 10 m/s. After 8 s, her speed is 25 m/s. Her acceleration during this time will be 5 m/s².
What is acceleration?Acceleration is the rate of change in both speed and direction of velocity over time. When anything moves faster or slower in a straight line, it is said to have been accelerated. Because the direction is always shifting, motion on a circle accelerates even while the speed is constant.
It is given that, Kiko is coasting on her bicycle down a hill. After 3 s, her speed is 10 m/s. After 8 s, her speed is 25 m/s.
Acceleration is found as the ratio of the change in speed to the change in a time interval.
Acceleration = change in speed/ change in a time interval
Acceleration = (25-10) / (8-5)
Acceleration = (15)/(3)
Acceleration = 5 m/s²
Thus, Kiko is coasting on her bicycle down a hill. After 3 s, her speed is 10 m/s. After 8 s, her speed is 25 m/s. Her acceleration during this time will be 5 m/s².
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What is an angle that is complementary to angle dge
Why does a horizontal line have a slope of 0, but a vertical line has anundened slope?
Find the range of the logarithmic function f(x) = log4 x.
Consider the function f(x) = (x − 3)2(x + 2)2(x − 1).
The zero ______? has a multiplicity of 1.
The zero −2 has a multiplicity of________? .
we have
[tex]f(x)=(x-3)^{2}(x+2)^{2}(x-1)[/tex]
we know that
The zero of the function is the value of x when the value of the function is equal to zero
[tex]f(x)=(x-3)^{2}(x+2)^{2}(x-1)[/tex]
rewrite the function
[tex]f(x)=(x-3)(x-3)(x+2)(x+2)(x-1)[/tex]
The zero's of the function are
[tex]x=3\ x=3\ x=-2\ x=-2\ x=1[/tex]
so
[tex]x=3[/tex] ------> has a multiplicity of [tex]2[/tex]
[tex]x=-2[/tex] ------> has a multiplicity of [tex]2[/tex]
[tex]x=1[/tex] ------> has a multiplicity of [tex]1[/tex]
therefore
the answer is
The zero [tex]1[/tex] has a multiplicity of [tex]1[/tex]
The zero [tex]-2[/tex] has a multiplicity of [tex]2[/tex]
From the given function, f(x) = (x − 3)2(x + 2)2(x − 1);
The zero 1 has a multiplicity of 1.The zero −2 has a multiplicity of 2Zeros of a functionThe function f(x) = (x − 3)²(x + 2)²(x − 1) given can be expanded so that we have;
f(x) = (x − 3)(x-3)(x + 2)(x+2)(x − 1)Hence, the zeros of the function f(x) are;
x = 3, 3, -2, -2, 1We can conclude that the zero 1 has a multiplicity of 1 and the zero -2 has a multiplicity of 2.
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Why is it important to collaborate?
Discuss a project [either at home or at school] in which you have collaborated. Explain the benefits and drawbacks of working with someone else on collaboration projects.
One number is 3 less than twice another. If their sum is 39, find the numbers.
Which of the following systems of equations represents the word problem?
A) y=2x-3 and y=x+39
B) y=2x-3 and x+y=39
C) y=2(x-3) and x+y=39
Find mCE
A. 40 DEGREES
B. 50 DEGREES
C. 75 DEGREES
D. 80 DEGREES
Your cruise ship is leaving for a 610 league adventure. how many nautical miles is this
Final answer:
A 610-league adventure is equivalent to 1830 nautical miles, calculated by multiplying the number of leagues by the conversion factor, where one league is equal to approximately 3 nautical miles.
Explanation:
To convert the distance from leagues to nautical miles, we need to know the conversion factor. One league is approximately 3 nautical miles. Thus, by multiplying the given number of leagues by this factor, we can determine the distance in nautical miles.
If the cruise ship is embarking on a 610-league adventure, we calculate the equivalent distance in nautical miles as follows:
Multiply the number of leagues (610) by the conversion factor (3).
The calculation would be 610 leagues * 3 nautical miles/league = 1830 nautical miles.
Therefore, a 610-league voyage is equivalent to 1830 nautical miles.
Help me with this please
Find the remainder when f(x) is divided by (x - k).
f(x) = 4x3 - 5x2 + 2x + 11; k= 3
Answer:
The remainder is 80
Step-by-step explanation:
Given the polynomial
[tex]f(x)=4x^3-5x^2+2x+11[/tex]
we have to find the remainder when the above polynomial f(x) is divided by x-3
By remainder theorem when we put x=3 in f(x) we get the remainder i.e f(3)
[tex]f(3)=4(3)^3-5(3)^2+2(3)+11[/tex]
[tex]=108-45+6+11[/tex]
[tex]=80[/tex]
which is required remainder.
Hence, the remainder is 80
The measure of an angle is 9°. what is the measure of a complementary angle?
If the standard deviation of a population increases, what else must necessarily increase? (1 point)
the population's mean
the population's median
the population's mode
the population's variance
Identify the property that justifies the following statement: if 16m=160,then m=10
Division Property of Equality
16m = 160
divide both sides by 16 to find m
m = 160/16 = 10
Answer:
Property of equality for division.Step-by-step explanation:
The given expression is
[tex]16m=160[/tex]
So, we use one of the properties of equalities for division, which states that if we divide one side of the equation, we also must divide the other side by the same number, to not mess up the balance of the equation. Applying this property, we have
[tex]\frac{16m}{16}=\frac{160}{16}\\ m=10[/tex]
Which is true about the functional relationship shown below?
PLEASE HELP!!!! WILL GIVE 99 PTS! :)Sam is observing the velocity of a car at different times. After two hours, the velocity of the car is 54 km/h. After four hours, the velocity of the car is 58 km/h.
Part A: Write an equation in two variables in the standard form that can be used to describe the velocity of the car at different times. Show your work and define the variables used. (5 points)
Part B: How can you graph the equation obtained in Part A for the first six hours? (5 points)
t = time
v = velocity
you have (2,54) & (4,58)
v-54 = (58-54) / (4-2) *(t-2)
v=4/2(t-2)+54
v =2(t-2)+54
v = 2t-4+54
b - Graph by plotting the given points (2,54) & 4,58) and drawing a straight line through them
Write and equation of the line that passes through the point (2,4) and is perpendicular tothe line hows equation is 3y-6x+3
Describe which of the two points lies on the graph of the line
-4x+y=-11
A.3,1
B.1,3
estimate 63 times 39
estimate 62.93 times 73
estimate 34.44 times 57
Estimate of 63 times 39: Approximately 2400.
Estimate of 62.93 times 73: Roughly 4410.
Estimate of 34.44 times 57: About 2040.
To estimate the products, you can round the numbers and then perform the multiplication.
Here are the estimates for the given multiplication problems:
Estimate of 63 times 39:
Round 63 to 60 and 39 to 40.
Calculate 60 times 40, which is 2400.
Estimate of 62.93 times 73:
Round 62.93 to 63 and 73 to 70.
Calculate 63 times 70, which is 4410.
Estimate of 34.44 times 57:
Round 34.44 to 34 and 57 to 60.
Calculate 34 times 60, which is 2040.
These are rough estimates and may not be entirely accurate, but they provide a quick way to get an idea of the approximate result of the multiplication.
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If 1 card is drawn at random from a standard 52-card deck, what is the probability that the card drawn is a face card?
Four more than the quotient of a number and 3 is at least 9
Explanation of an inequality involving addition and division
The given inequality is: 4 + (x/3) ≥ 9
To express the statement algebraically:
Add four to the quotient of a number and 3.
Set the expression greater than or equal to 9.
Solve for x to find the number that satisfies the inequality.
Find the slope of the line passing through the two points.
(–7, 8), (–4, 3)
(12, 1), (36, 42)
At Taft high school 59% of all students participate in after school sports and 23% participate in both after school sports and volunteer activities. What is the probability that a student participates in volunteer activities given that the student participates in after school sports. A. 82% B. 36% C. 59% D. 39%
BRIANLIEST ~
Part A: Amir rented a scooter at $43 for 3 hours. If he rents the same scooter for 8 hours, he has to pay a total rent of $113.
Write an equation in the standard form to represent the total rent (y) that Amir has to pay for renting the scooter for x hours. (4 points)
Part B: Write the equation obtained in Part A using function notation. (2 points)
Part C: Describe the steps to graph the equation obtained above on the coordinate axes. Mention the labels on the axes and the intervals. (4 points)
how to solve bh+hr=25 when solving for h