The differentiable function f with the given properties is [tex]f(x)=4x^3-3x^2+10[/tex].
What is a differentiable function:
A function with a differentiable value of one real variable is one whose domain contains a derivative. In other words, each interior point in the domain of a differentiable function's graph has a non-vertical tangent line.
The given properties for the differentiable function f are:
[tex](i) f'(x)=ax^2+bx\\ (ii) f'(1)=6, f''(1)=18\\ (iii) \int\limits^2_1 {f(x)} \, dx =18[/tex]
From (i) we can get [tex]f''(x)=2ax+b[/tex]
By substituting (ii) in (i) we will get:
a+b=6 and 2a+b=18. By solving these two equations we will get the following:
a=12, b=-6.
We will integrate (i) on both sides, we will get:
[tex]f(x)=\frac{ax^3}{3} +\frac{bx^2}{2} +c[/tex] where c is the integration constant.
In this equation, we will substitute a,b.
[tex]f(x)=\frac{12x^3}{3} +\frac{-6x^2}{2} +c\\ \\ f(x)= 4x^3-3x^2+c[/tex]...................(iv)
Now we will substitute (iv) in (iii), and we will get:
[tex]\int\limits^2_1 {( 4x^3-3x^2+c)} \, dx =18\\ \\ (2^4-2^3+2c)-(1^4-1^3+c)=18\\ \\c=10[/tex]
Therefore [tex]f(x)=4x^3-3x^2+10[/tex].
Complete question:
Let f be a differentiable function, defined for all real numbers x, with the following properties. Find f(x). Show your work.
[tex](i) f'(x)=ax^2+bx\\ (ii) f'(1)=6, f''(1)=18\\ (iii) \int\limits^2_1 {f(x)} \, dx =18[/tex]
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"A number n equals 7 more than half the number"
The number 'n' is equal to 14
In the question, we have been given that the number n is equal to 7 more than half of the number 'n'.
So in these linear equation-solving types of problems, first of all, we convert the written sentence into an equation.
So the equation for the following problem statement is -
n = 7 + n/2
{since we have been given the number equals 7 more than half of the number }
Solving the equation we have,
n = 14.
We can easily see that 14 is the number which is 7 more than half of itself.
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3/4 of a peice of metal has a mass of 15kg. What is the mass of 2/5 of the peice of metal?
Answer:
Step-by-step explanation:
Let total mass be x
(3/4)*x=15
on solving
x=20
now,
mass of 2/5 of piece of metal is-
=> (2/5)*20
=> 8
Ans- 8kg
I need help, I’m trying to find the translation and scale factor.
Answer:
Translation: 4 to the right and 8 up.
Scale factor: 3
Step-by-step explanation:
If you move CDE 4 spaces to the right and 8 spaces up C and C' will match up. If you dilate (make bigger) CDE 3 times it will perfectly be C'D'E'.
Find the linearization of f(x) = √ x at a = 4
Answer:
Step-by-step explanation:
To find the linearization of the function f(x) = sqrt(x) at x = a, we can use the linearization formula:
f(x) ≈ f(a) + f'(a) (x - a)
To use this formula, we first need to find the value of f(a) and f'(a). At x = a = 4, the value of f(x) is f(4) = sqrt(4) = 2. The derivative of f(x) is f'(x) = 1/(2 * sqrt(x)), so the value of f'(a) is f'(4) = 1/(2 * sqrt(4)) = 1/4.
Substituting these values into the linearization formula, we get:
f(x) ≈ 2 + 1/4 (x - 4)
This is the linearization of f(x) = sqrt(x) at x = 4. It is an approximation of the function f(x) that is valid for values of x that are close to 4.
At a particular restaurant, each slider has
225 calories and each chicken wing has 70
calories. A combination meal with sliders
and chicken wings has a total of 10 sliders
and chicken wings altogether and
contains 1165 calories. Write a system of
equations that could be used to determine
the number of sliders in the combination
meal and the number of chicken wings in
the combination meal. Define the
variables that you use to write the system.
Let x be the number of sliders in the combination meal and y be the number of chicken wings in the combination meal. We can write the first equation in the system by equating the total number of sliders and chicken wings in the combination meal with the sum of the number of sliders and the number of chicken wings. Since the combination meal has 10 sliders and chicken wings altogether, we can write the equation x + y = 10.
The second equation in the system can be written by equating the total number of calories in the combination meal with the sum of the number of calories in the sliders and the number of calories in the chicken wings. Since each slider has 225 calories and each chicken wing has 70 calories, the total number of calories in the sliders is 225x and the total number of calories in the chicken wings is 70y. And since the combination meal contains 1165 calories, we can write the equation 225x + 70y = 1165.
Therefore, the system of equations that could be used to determine the number of sliders and chicken wings in the combination meal is x + y = 10 and 225x + 70y = 1165.
please help me!!!! put from least to greatest
Answer: -10,-8.7, -9.1, 8, 8.7, 8.06, 9
Step-by-step explanation:
77 squared rooted equals 8.7
-83 square rooted equals -9.1
-76 square rooted equals -8.7 ( remember negative numbers are greater than positive numbers)
65 square rooted equals 8.06
What is an equation of the line that passes through the points (-3, 8) and (6, 2)?
Answer:
y = (-2/3)x + 6
Step-by-step explanation:
To find the equation of the line that passes through the points (-3, 8) and (6, 2), we can use the slope-intercept form of the equation of a line, which is:
y = mx + b
where m is the slope of the line and b is the y-intercept, which is the point where the line crosses the y-axis.
To find the slope of the line, we can use the following formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line. Plugging in the coordinates of the two points given in the problem, we get:
m = (2 - 8) / (6 - (-3)) = -6/9 = -2/3
Next, we need to find the y-intercept of the line. This is the point where the line crosses the y-axis, which means that the x-coordinate of this point is 0. Since we know the slope of the line and one of the points that it passes through, we can use the point-slope form of the equation of a line to find the y-intercept:
y - y1 = m(x - x1)
Substituting the values we have calculated above and setting x = 0, we get:
y - 8 = (-2/3)(0 - (-3))
Solving for y, we find that the y-intercept is 8 - (2/3) * 3 = 8 - 2 = 6.
Therefore, the equation of the line that passes through the points (-3, 8) and (6, 2) is:
y = (-2/3)x + 6
4.
a) Find four consecutive even integers such that twice the sum of the second and third exceeds 3 times the first by 32.
b) The difference between two numbers is 24. Find the numbers if their sum is 88.
5.
a) Separate 60 into two parts so that 3 times the smaller added to 6 more than 6 times the smaller = 60.
b) A train traveled 300 miles. How long did the trip take if the train was traveling at a rate of: Note * Use the d=rt formula (distance = rate * time). NOTE: You may not be able to solve for the variable. If you do not have enough information to solve for the variable then write the equation.
1) 50 mph
2) 70 mph
3) x mph
4) (x+10)mph
5) (x-5)mph
The first, second, third, and fourth consecutive even integers must be greater than 10, 12, 14, and 16 respectively.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
Let the four consecutive even integers be n, (n + 2), (n + 4) and (n + 6).
Such that twice the sum of the second and third exceeds 3 times the first by 32.
2{(n + 2) + (n + 4)} > 3(n) + 32.
2n + 4 + 2n + 8 > 3n + 32.
4n - 3n > 32 - 12.
n > 10.
So, the consecutive even numbers must all be greater than 10.
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The average mass of Stephen, Shamsul and Saravanan is 64 kg. Stephen's mass is 72 kg Shamsul's mass is 1.5 times Saravanan's mass Calculate Shamsul's mass in kg.
Answer:
72kg
Step-by-step explanation:
Stephen = 72
Saravanan = x
Shamsul = (x • 1.5) since Shamsul's mass is 1.5 times Saravanan's mass
1. make an equation:
( 72 + x + (x • 1.5) ) ÷ 3 = 64
2. solve it:
( 72 + x + (x • 1.5) ) ÷ 3 = 64
( 72 + x + (x • 1.5) ) = 64 • 3
( 72 + x + (x • 1.5) ) = 192
x + 1.5x = 192 - 72
2.5x = 120
x = 120 ÷ 2.5
x = 48
Thus shamsul's mass is:
48 • 1.5 = 72kg
Ray AT bisects ∠MAR. If m∠MAT = (8x − 3)° and m∠RAT = (2x + 9)°, what is m∠MAR?
26°
13°
8°
2°
The measure of the angle ∠MAR will be equal to 26°. The correct option is A.
What is an angle bisector?In geometry, an angle bisector is a line that divides an angle into two equal angles. A bisector is something that divides a shape or object into two equal parts. An angle bisector is a ray that divides an angle into two equal parts of the same measurement.
Given that Ray AT bisects ∠MAR. If m∠MAT = (8x − 3)° and m∠RAT = (2x + 9)°,
The two angles ∠MAT and ∠RAT will be equal. Then calculate the value of x.
8x - 3 = 2x + 9
8x - 2x = 9 + 3
6x = 12
x = 2
The angle ∠MAR is calculated as,
∠MAR = 2 x ∠RAT
∠MAR = 2 x ( 2x + 9)
∠MAR = 2 x ( 2 x 2 + 9 )
∠MAR = 2 x ( 13 )
∠MAR = 26°
Option A is correct for the angle ∠MAR.
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Answer:
A all the way
Step-by-step explanation:
i need big help on this question please
make g the subject of the formula w=7-/sqrt g
Answer:
Below
Step-by-step explanation:
w = 7 - sqrt (g) ?? ( syntax is unclear in your post)
sqrt(g) = 7-w
g = (7-w)^2
An environmental engineer graphed the locations of a well and all of the drainage ditches in the vicinity. She positioned the well at (−6,7) and the farthest drainage ditch at (33,7). If each unit on the graph represents 1 foot, then how far away from the well is the farthest drainage ditch?
The farthest drainage ditch is 39 feet away from the well.
What is distance between two points?The length of the line segment bridging two points on a plane is known as the distance between the points.
The formula to find the distance between the two points is usually given by d=√{(x₂-x₁)² + (y₂-y₁)²}
Given, an environmental engineer graphed the locations of a well and all the drainage ditches in the vicinity.
She positioned the well at (−6,7) and the farthest drainage ditch at (33,7).
To find the distance:
Let d be the distance.
d = √{(x₂-x₁)² + (y₂-y₁)²}
Here, x₂ = 33, x₁ = -6, y₂ = 7 and y₁ = 7
Substituting the value to the distance formula,
d = √{(33+6)² + (7-7)²}
d = √{(39)² + (0)²}
d = √{(39)²
d = 39
Therefore, the value of d is 39.
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f(x) = 6x + 2 g(x) = -5x -9 Find the product of f and g.
Answer:30x^2 - 54x - 18
Step-by-step explanation: To find the product of f and g, we can simply multiply the expressions for f and g together. The product of f and g is:
(f * g)(x) = (6x + 2)(-5x - 9)
= -30x^2 - 54x - 18
So, the product of f and g is -30x^2 - 54x - 18
explain the meaning of 3^1/4 * 3^1/4 * 3^1/4 * 3^1/4 = 3 in terms of fractional exponents or radicals
The answer, based on the information provided, fractional exponents will be 3.
What are some exponent fundamentals?A product in which the same integer is used repeatedly as a factor is represented by a number elevated to a power. The exponent provides the power, and the integer is referred to as the base. The exponent indicates the number of factors, while the base is the repeating factor (the multiplied number).
What is the initial exponents rule?That number will be the outcome! One of the simplest exponent laws is this one. You'll see that this rule doesn't merely apply to numerical data. We might, for example, elevate a statistic to the initial power.
Briefing:[tex]=3^{\frac{1}{4} }\times3^{\frac{1}{4} }\times3^{\frac{1}{4} }\times3^{\frac{1}{4} }\\=3^{\frac{1}{4}+{\frac{1}{4}+{\frac{1}{4}+{\frac{1}{4}\\\\=3^{1} \\=3[/tex]
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Write a numerical expression to represent the rate of temperature change in degrees Fahrenheit per foot.
The numerical expression to represent the rate of temperature change in degrees Fahrenheit per foot is 0.14 degrees Celsius per foot.
What do you mean by numerical expression?
A group of numbers that have been written together using the arithmetic operations addition, subtraction, multiplication, and division is known as a numerical expression in mathematics. The number can be stated in a variety of ways, including verbally and numerically.
According to the given question,
To calculate the rate of temperature change in degrees Fahrenheit per foot,
We need to first convert degrees Fahrenheit to degrees Celsius.
To do this, we divide 100 by 5.9.
This yields a result of 0.14 degrees Celsius per foot.
Therefore, the numerical expression to represent the rate of temperature change in degrees Fahrenheit per foot is 0.14 degrees Celsius per foot.
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ANSWER NOW NEED HELP
A red die, a blue die, and a green die
are rolled. Each die is labeled 1 to 6.
Find the probability of each event:
a) a 2 on the red die, a 3 on the blue
die, and a 4 on the green die
b) a 4 on the red die, an even number
on the blue die, and a number less
then 3 on the green die
Answer:
a)1/216
b)1/36
Step-by-step explanation:
p(2r)and p(3b) and p(4g)
1/6×1/6×1/6=1/216
p(4r) and p(even b) and p(no less than 3)
1/6×3/6×2/6=1/36
HELP ASAP!! READ CAREFULLY
Answer:
Step-by-step explanation:
There is a scale factor of 2 so since the trapezoid is 9 we can divide by 2 so AD is 4.5 and CB is 4.5
YZ is the same as ZW so 8
Using the scale factor we can divide 8 by 2 to get 4
Perimeters can be found from the sums so
ABCD would be 19 and WXYZ would be 38
Solve for h. If you answer, you get 14 points
Answer: 24
Step-by-step explanation:
Answer:
h = 24
Step-by-step explanation:
Jorge needs to choose 3 people for his group project. There are 16 people in the class to choose from. How many different combinations of groups could he choose for his group project?
A. 4096
B. 560
C. 3360
D. 1120
The different combinations of groups for the group project is (b) 560
How to determine the different combinations of groups for the group project?From the question, we have the following parameters that can be used in our computation:
Total number of people, n = 16
Numbers to selection, r = 3 people
The number of ways of selection could be drawn is calculated using the following combination formula
Total = ⁿCᵣ
Where
n = 16 and r = 3
Substitute the known values in the above equation
Total = ¹⁶C₃
Apply the combination formula
ⁿCᵣ = n!/(n - r)!r!
So, we have
Total = 16!/(13! * 3!)
Evaluate
Total = 560
Hence, the number of ways is 560
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(x + 10)(3x² + 5x - 2)
To expand the expression (x + 10)(3x² + 5x - 2), we can use the distributive property, which states that for any numbers a, b, and c, the product (a + b)(c) can be written as a * c + b * c. Applying this property to the given expression, we get (x + 10)(3x² + 5x - 2) = (x * 3x² + 10 * 3x²) + (x * 5x + 10 * 5x) + (x * -2 + 10 * -2) = 3x³ + 30x² + 5x² + 50x - 2x - 20. Combining like terms, we get 3x³ + 35x² + 51x - 20. Thus, the expanded form of the expression (x + 10)(3x² + 5x - 2) is 3x³ + 35x² + 51x - 20.
The question asked is in the image attached below.
Answers with nice explanation and step wise order will be marked brainliest.
Thanks for your kind help.
Answer:
[tex]y' = \dfrac{1 + x \sin x - x \cos x + \sin x + \cos x}{(\cos x - x)^2}[/tex]
Step-by-step explanation:
[tex] y = \dfrac{x + \sin x}{\cos x - x} [/tex]
Derivative of a quotient:
[tex] \dfrac{d}{dx} \dfrac{f(x)}{g(x)} = \dfrac{g'h - h'g}{h^2} [/tex]
Recall:
[tex] \dfrac{d}{dx} \sin x = \cos x [/tex]
[tex] \dfrac{d}{dx} \cos x = - \sin x [/tex]
[tex] y = \dfrac{x + \sin x}{\cos x - x} [/tex]
[tex] y' = \dfrac{dy}{dx} = \dfrac{(1 + \cos x)(\cos x - x) - [(- \sin x - 1)(x + \sin x)]}{(\cos x - x)^2} [/tex]
[tex]y' = \dfrac{\cos x - x + \cos^2 x - x \cos x - (-x \sin x - \sin^2 x - x - \sin x}{(\cos x - x)^2}[/tex]
[tex]y' = \dfrac{\cos x - x + \cos^2 x - x \cos x + x \sin x + \sin^2 x + x + \sin x}{(\cos x - x)^2}[/tex]
[tex]y' = \dfrac{\sin^2 x + \cos^2 x + x - x - x \cos x + x \sin x + \sin x + \cos x}{(\cos x - x)^2}[/tex]
[tex]y' = \dfrac{1 + x \sin x - x \cos x + \sin x + \cos x}{(\cos x - x)^2}[/tex]
Stephen's lunch bill is currently at $ 8.33. Stephen orders a fruit salad for take-out, and wants to leave $ 2.25$ as a tip for his server. He has a $\$ 10$ bill and a $5 bill. How much change should he receive after paying for his lunch, the fruit salad, and the tip?
The amount of change that Stephen receives is $______
Using mathematical operations on the word problem, the amount he will receive as his change is $1.43
Word ProblemA word problem is a few words are presented in a form of problem and needs to be solved by way of a mathematical calculation.
This problem can be solved using mathematical operation such as addition and subtraction.
The total amount he has with him = $15
The current bill = $8.33
The fruit salad = $2.99
Tip = $2.25
Total amount spent = 8.33 + 2.99 + 2.25
The total amount spent = 13.57
The amount of change = total amount he has with him - total amount spent
The amount of change = 15 - 13.57
The amount of change = 1.43
The change he is going to receive is $1.43
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Elizabeth has a 1,200-word report to be finished in 3 weeks.
During the first week, she wrote every day (Sunday to Saturday) and wrote a total of 525 words. For the second week, she plans to write every day according to the equation y = 55z where y is
the total number of words written after x days.
Use the drop-down menus to correctly complete the statements below.
Last week, Elizabeth wrote at a pace of
This week, she plans to write at a pace of
She will still need to write To complete report
Answer:
Step-by-step explanation: for the first part, 525/7 = 75.
for the second part, 55y = x, which means 55 words per day for a total of (55*7) or 385
for the third part, 1200-525-385=290, not totally sure, but I think this is the answer, the text was a bit blurry and I couldn't understand everything, try not to following this blindly and just see if it makes sense. Hope I could help
3x + 2 = 5x - (2x + 1).
Answer:
No solution
Step-by-step explanation:
Help a girl out
Given: ray AC is parallel to ray DE, measure angle FED= measure angle GCA= 45°
Prove: ray FE is parallel to ray GC
We have proved that the ray FE is parallel to the ray GC.
What is meant by parallel lines?
In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance.
Angle DEF intersects at B
which results in angle DEF corresponding to the angle ACG.
As DEF = 45 so angle ACG = 45
So by the above discussion, we can say that the ray FE is parallel to the ray GC.
Hence, we have proved that the ray FE is parallel to the ray GC.
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I CAN DETERMINE IF A VALUE IS TRUE GIVEN AN EQUATION.
9. If x = -3, then which of the following equations are true? Put a checkmark on all that apply.
Χ
3-9= 15
-2x+18=24
6x +9=-13
4x-9=-21
-2(-3)+18=24
6+18 = 24
24 = 24
correct6(-3) +9=-13
-18+9 = -13
-9 ≠ - 13
Wrong4(-3)-9=-21
-12 -9 =-21
-21 = -21
Correct6 Bob is flying to Croatia for a holiday. When he gets to the airport, he discovers that they
hove a new pricing system for luggage. 2 items of hand luggage plus 1 item to go in the hold
costs £86. 1 item of hand luggage and a piece of luggage to go in the hold costs £66. How
much does it cost to put an item of luggage in the hold?
DP BOOT
Answer:
£46
Step-by-step explanation:
You want to know the cost of luggage in the hold when 2 in the hand and 1 in the hold is £86, and 1 in the hand and 1 in the hold is £66.
RelationsLet 'a' and 'b' represent the costs of a piece of hand luggage and one in the hold, respectively. Then the given relations are ...
2a +b = 86
a +b = 66
SolutionSubtracting the first equation from twice the second gives ...
2(a +b) -(2a +b) = 2(66) -(86)
b = 46 . . . . . . . simplify
It costs £46 to put an item of luggage in the hold.
Compute the following and write in the form x+iy :
[tex]\frac{1+2i}{3-4i} + \frac{2-i}{5i}[/tex]
[tex] \Large{\boxed{\sf \dfrac{1 + 2i}{3 - 4i} + \dfrac{2 - i}{5i } = - \dfrac{2}{5}}} [/tex]
[tex] \\ [/tex]
Explanation:Given sum:
[tex] \sf \: \dfrac{1+2i}{3-4i} + \dfrac{2-i}{5i}[/tex]
[tex] \\ [/tex]
We can simplify the sum only if the denominators of the two fractions are the same. Since they are different, we have to multiply the numerator and the denominator of each fraction by the denominator of the other one.
[tex] \sf \: \dfrac{1+2i}{3-4i} + \dfrac{2-i}{5i} = \dfrac{5i(1 + 2i)}{5i(3 - 4i)} + \dfrac{(3 - 4i)(2 - i)}{(3 - 4i)5i} \\ \\ \\ \sf \: = \dfrac{5i + 10 {i}^{2} }{15i - 20 {i}^{2} } + \dfrac{6 - 3i - 8i + 4 {i}^{2} }{15i - 20 {i}^{2} } [/tex]
[tex] \\ [/tex]
Replace i² with -1:
[tex] \sf \dfrac{5i + 10 {i}^{2} }{15i - 20 {i}^{2} } + \dfrac{6 - 3i - 8i + 4 {i}^{2} }{15i - 20 {i}^{2} } \: \\ \\ \\ \\ \sf \: = \dfrac{5i + 10( - 1)}{15i - 20( - 1)} + \dfrac{ 6 - 11i + 4( - 1)}{15i - 20( - 1)} \\ \\ \\ \\ \sf = \dfrac{5i - 10}{20 + 15i} + \dfrac{2 - 11i}{20 + 15i} [/tex]
[tex] \\ [/tex]
Simplify the expression:
[tex] \sf = \dfrac{5i - 10}{20 + 15i} + \dfrac{2 - 11i}{20 + 15i} \\ \\ \\ \\ \sf \: = \dfrac{5i - 10 + 2 - 11i}{ 20 + 15i} = \sf \dfrac{ - 8 - 6i}{20 + 15i}[/tex]
[tex] \\ [/tex]
To write our solution in the x + iy form, also known as the algebraic form, we have to understand what the conjugate of a complex number is.
[tex] \textsf{Let z be our complex number, and} \: \overline{\sf z} \: \textsf{its conjugate.} [/tex]
[tex] \\ [/tex]
The conjugate of z, [tex] \overline{ \sf z}, [/tex] is the complex number formed of the same real part as z but of the opposite imaginary part.
Since x is the real part of z, and y is its imaginary part, this can be expressed as:
[tex] \sf If \: z = x + iy \:, then \: \overline{ \sf z} = x - iy [/tex]
[tex] \\ [/tex]
Now, we have to multiple both the denominator and the numerator of our fraction by the conjugate of its denominator:
[tex]\sf \dfrac{ - 8 - 6i}{20 + 15i} = \dfrac{( - 8 - 6i)( \overbrace{20 - 15i}^{ \overline{z}}) }{ (20 + 15i)( \underbrace{20 - 15i}_{ \overline{z}}) } \\ \\ \\ \sf = \dfrac{ - 160 + 120i - 120i + 90 {i}^{2} }{400 - 300i + 300i - 225 {i}^{2} } \\ \\ \\ \sf = \dfrac{ - 160 + 90 {i}^{2} }{400 - 225 {i}^{2} }[/tex]
[tex] \\ [/tex]
One more time, substitute -1 for i²:
[tex] \sf \: \dfrac{ - 160+ 90 {i}^{2} }{400 - 225 {i}^{2} } \: = \dfrac{ - 160 + 90( - 1)}{400 - 225( - 1)} \\ \\ \\ \sf = \boxed{\sf - \dfrac{ 250}{625}} [/tex]
[tex] \\ [/tex]
Finally, let's simplify our result:
[tex] \sf - \dfrac{250}{625} = - \dfrac{2 \times 125}{5 \times 125} = \boxed{ \boxed{ \sf - \dfrac{2}{5}}}[/tex]
[tex] \\ \\ [/tex]
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please quick
giving brainliest
Answer:
see explanation
Step-by-step explanation:
(a)
given y is inversely proportional to x² then the equation relating them is
y = [tex]\frac{k}{x^2}[/tex]
to find k use any ordered pair from the table
using (1, 4 ) and substituting into equation
4 = [tex]\frac{k}{1^2}[/tex] = [tex]\frac{k}{1}[/tex] , thus
k = 4
y = [tex]\frac{4}{x^2}[/tex] ← equation of proportion
(b)
when y = 25 , then
25 = [tex]\frac{4}{x^2}[/tex] ( multiply both sides by x² )
25x² = 4 ( divide both sides by 25 )
x² = [tex]\frac{4}{25}[/tex] ( take square root of both sides )
x = ± [tex]\sqrt{\frac{4}{25} }[/tex] = ± [tex]\frac{2}{5}[/tex]
then
positive value of x when y = 25 is x = [tex]\frac{2}{5}[/tex]
the speed of water in a whirl pool varies inversely with the radius. if the water speed is 2.5 feet per second at a radius of 30 feet what is the speed of the water at a radius of 3 feet
Answer:
Step-by-step explanation:
Let, the Speed of the water be 'x'
and the Radius be 'r'
Now, the speed of Water Varies inversely with the Radius
So, x ∝ 1/r
x = k/r -----------(i)
Here, k is the proportionality constant
Now,
Given that, x = 2.5 ft per sec
then, 30 ft,
So, from eq. (i)
2.5 = k/30
So, k = 75
Again From eq. (i)
x = 75/r
Now, if r = 3ft
then, x = 75/3
x = 25
Hence, Speed of the water (x) = 25 ft per sec.