Given that
y=1/27x^3
express each of the following in the form kx^n where k and n are constants.
(a) y^1/3
(b) 3y^-1
(c) √(27y)
Answers
Express the following in the form kx^n where k and n are constants.
(a) y=x^3
(b)y=3x^-1
(c) y=x^3/27
To express each of the following in the form kx^n where k and n are constants.
What is algebra?A part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.
Given that:
y=1/27x^3
express each of the following in the form kx^n where k and n are constants.
(a) y^1/3
Let y^1/3=x
by cubing both sides we get,
y=x^3
The equation in the form of kx^n
where,
k=1
n=3
(b) 3y^-1
Let 3y^-1=x
3/y=x
y=3/x
By simplify
y=3x^-1
The equation in the form of kx^n
where,
k=3
n=-1
(c) √(27y)
Let (27y)^1/3=x
by cubing both sides we get,
27y=x^3
y=x^3/27
The equation in the form of kx^n
where,
k=1/27
n=3
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Related Questions
Y is inversely proportional to the square root of x if y=3 when x=25 find y when x is 9
Answers
Answer:
[tex]y=5[/tex]
Step-by-step explanation:
[tex]\textsf{If }y \textsf{ is \underline{inversely proportional} to the square root of }x, \textsf{ then}:[/tex]
[tex]y \propto\dfrac{1}{\sqrt{x}} \implies y=\dfrac{k}{\sqrt{x}}\quad \textsf{(where k is some constant)}[/tex]
[tex]\textsf{When }x=25, y = 3:[/tex]
[tex]\implies 3=\dfrac{k}{\sqrt{25}}[/tex]
[tex]\implies 3=\dfrac{k}{5}[/tex]
[tex]\implies k=15[/tex]
Inputting the found value of k into the equation:
[tex]\implies y=\dfrac{15}{\sqrt{x}}[/tex]
To find the value of y when x is 9, substitute x = 9 into the found equation:
[tex]\implies y=\dfrac{15}{\sqrt{9}}[/tex]
[tex]\implies y=\dfrac{15}{3}[/tex]
[tex]\implies y=5[/tex]
how to solve part ii and iii
Answers
(i) Given that
[tex]\tan^{-1}(x) + \tan^{-1}(y) + \tan^{-1}(xy) = \dfrac{7\pi}{12}[/tex]
when [tex]x=1[/tex] this reduces to
[tex]\tan^{-1}(1) + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}[/tex]
[tex]\dfrac\pi4 + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}[/tex]
[tex]2 \tan^{-1}(y) = \dfrac\pi3[/tex]
[tex]\tan^{-1}(y) = \dfrac\pi6[/tex]
[tex]\tan\left(\tan^{-1}(y)\right) = \tan\left(\dfrac\pi6\right)[/tex]
[tex]\implies \boxed{y = \dfrac1{\sqrt3}}[/tex]
(ii) Differentiate [tex]\tan^{-1}(xy)[/tex] implicitly with respect to [tex]x[/tex]. By the chain and product rules,
[tex]\dfrac d{dx} \tan^{-1}(xy) = \dfrac1{1+(xy)^2} \times \dfrac d{dx}xy = \boxed{\dfrac{y + x\frac{dy}{dx}}{1 + x^2y^2}}[/tex]
(iii) Differentiating both sides of the given equation leads to
[tex]\dfrac1{1+x^2} + \dfrac1{1+y^2} \dfrac{dy}{dx} + \dfrac{y + x\frac{dy}{dx}}{1+x^2y^2} = 0[/tex]
where we use the result from (ii) for the derivative of [tex]\tan^{-1}(xy)[/tex].
Solve for [tex]\frac{dy}{dx}[/tex] :
[tex]\dfrac1{1+x^2} + \left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} + \dfrac y{1+x^2y^2} = 0[/tex]
[tex]\left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} = -\left(\dfrac1{1+x^2} + \dfrac y{1+x^2y^2}\right)[/tex]
[tex]\dfrac{1+x^2y^2 + x(1+y^2)}{(1+y^2)(1+x^2y^2)} \dfrac{dy}{dx} = - \dfrac{1+x^2y^2 + y(1+x^2)}{(1+x^2)(1+x^2y^2)}[/tex]
[tex]\implies \dfrac{dy}{dx} = - \dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2) (1 + x^2y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2) (1+x^2y^2)}[/tex]
[tex]\implies \dfrac{dy}{dx} = -\dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2)}[/tex]
From part (i), we have [tex]x=1[/tex] and [tex]y=\frac1{\sqrt3}[/tex], and substituting these leads to
[tex]\dfrac{dy}{dx} = -\dfrac{\left(1 + \frac13 + \frac1{\sqrt3} + \frac1{\sqrt3}\right) \left(1 + \frac13\right)}{\left(1 + \frac13 + 1 + \frac13\right) \left(1 + 1\right)}[/tex]
[tex]\dfrac{dy}{dx} = -\dfrac{\left(\frac43 + \frac2{\sqrt3}\right) \times \frac43}{\frac83 \times 2}[/tex]
[tex]\dfrac{dy}{dx} = -\dfrac13 - \dfrac1{2\sqrt3}[/tex]
as required.
Suzanne walks four miles every third day. What is the fewest number of miles she can walk in February
Answers
36 miles is the fewest number of miles she can walk in February
There are 28 days in February.
Every third day is 28/3 = 9.333 = 9 days of walking.
since she walks four miles,
9*4 = 36
Hence, 36 miles is the fewest number of miles she can walk in February
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URGENT PLEASE ANSWER THESE
Answers
Answer:
1A) 2 gallons of 20% solution and 3 gallons 15% solution needed
1B) 4 gallons of 20% solution and 1 gallons 15% solution needed
Step-by-step explanation:
1A) adding 20% salt and 15% water making 5 gallons of 17%
20% salt + 15% salt = 5 gallons of 17% salt
0.20x + 0.15(5 - x) = 0.17(5)
0.20x + 0.75 - 0.15x = 0.85
0.20x + 0.75 - 0.75 - 0.15x = 0.85 - 0.75
0.20x - 0.15x = 0.10
0.05x = 0.10
0.05x/ 0.05 = 0.10/0.05
x = 2 gallons (amount of 20% solution needed)
5 - x = 5 - 2 = 3 gallons (amount of 15% solution needed)
1B)
0.20x + 0.15(5 - x) = 0.19(5)
0.20x + 0.75 - 0.15x = 0.95
0.20x + 0.75 - 0.75 - 0.15x = 0.95 - 0.75
0.20x - 0.15x = 0.20
0.05x = 0.20
0.05x / 0.05 = 0.20/0.05
x = 4 gallons (amount of 20% solution needed)
5 - x = 5 - 4 = 1 gallons (amount of 15% solution needed)
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Independent Practice
An image on a slide is similar to its projected image. A slide is 35 mm wide and 21 mm high. Its projected image is 85 cm wide. To the nearest centimeter, how high
is the image?
A. 51 cm
B. 5.1 cm
C.
D.
142 cm
9 cm
Answers
Answer:
Projected height = (85 *21) / 35 = 51 cm
Step-by-step explanation:
Since the images are similar, we can set up the proportional equation like this:
Slide width / Slide height = Projected width / Projected height
35 mm / 21 mm = 85 cm / Projected height (in cm)
Projected height = (85 *21) / 35 = 51 cm
You design a logo for your soccer team. The logo is 3 inches by 5 inches. You decide to dilate the logo to 1.5 inches by 2.5 inches. What is the scale factor of this dilation?
Answers
Answer:
1/2
Step-by-step explanation:
The dimensions are cut in half ....scale factor 1/2
A tennis tournament has 2n contestants. We want to pair them up for the first round of singles matches. Show that the number of different ways in which first round matches can be conducted is
Answers
There are
[tex]\dbinom{2n}2 = \dfrac{(2n)!}{2! (2n-2)!}[/tex]
ways of pairing up any 2 members from the pool of [tex]2n[/tex] contestants. Note that
[tex](2n)! = 1\times2\times3\times4\times\cdots\times(2n-2)\times(2n-1)\times(2n) = (2n-2)! \times(2n-1) \times(2n)[/tex]
so that
[tex]\dbinom{2n}2 = \dfrac{(2n)\times(2n-1)\times(2n-2)!}{2! (2n-2)!} = \boxed{n(2n-1)}[/tex]
The number of different ways in which first-round matches can be conducted is n (2n - 1).
We have,
Number of contestant = 2n
Number of contestants in each match = 2
Now,
The number of different ways in which first-round matches can be conducted.
A combination formula is used.
= [tex]^{2n}C_2[/tex]
= 2n! / 2! (2n - 2)!
= 2n (2n - 1)(2n - 2)! / [2 x (2n - 2)!]
= 2n (2n - 1) / 2
= n (2n - 1)
Thus,
The number of different ways in which first-round matches can be conducted is n (2n - 1).
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True or False? If ab<0, then a<0, b>0 or a>0, b<0
Answers
Answer:
True
Step-by-step explanation:
If ab < 0, then ab = negative #.
In order for ab to be a negative #, one of them has to be negative while the other one needs to be positive.
Example:
a = -2, b = 1
ab < 0
(-2)(1) < 0
-2 < 0, TRUE
a < 0
-2 < 0, TRUE
b > 0
1 > 0, TRUE
If I switch a = -2 to 1 and b = 1 to -2, a > 0 and b < 0 is true too.
change the unit of length 6ft 2in= ___ft?
Answers
Answer:
Step-by-step explanation:
Comment
You don't have to do anything to the 6 feet. It already is in feet. It's the 2 inches you have to worry about.
2 inches = 2/12 of a foot.
2/12 = 1/6 = 0.167 feet
Answer
6 feet 2 inches = 6.167 feet
Step-by-step explanation:
from two points one on each leg of an isosceles triangle perpendicular are drawn to the base prove that the triangles formed are similar
Answers
The description below proves that the perpendicular drawn from the vertex angle to the base bisect the vertex angle and base.
How to prove an Isosceles Triangle?Let ABC be an isosceles triangle such that AB = AC.
Let AD be the bisector of ∠A.
We want to prove that BD=DC
In △ABD & △ACD
AB = AC(Thus, △ABC is an isosceles triangle)
∠BAD =∠CAD(Because AD is the bisector of ∠A)
AD = AD(Common sides)
By SAS Congruency, we have;
△ABD ≅ △ACD
By corresponding parts of congruent triangles, we can say that; BD=DC
Thus, this proves that the perpendicular drawn from the vertex angle to the base bisect the vertex angle and base.
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helllppppppppppp please
Answers
Answer:
No, it will not pass specifications.
Step-by-step explanation:
It says that the max slope of the wheelchair ramp should be 1/12.
To calculate slope, you do rise over run or y values over x values. In this case, the rise over run is 5/50. 5/50 simplifies to 1/10.
1/10 > 1/12, meaning that it is more than the maximum slope.
what is the decimal of 1/2
Answers
Answer:
0.5
one divide by two is zero point five
[tex]\frak{Hi!}[/tex]
[tex]\orange\hspace{300pt}\above2[/tex]
We need to find the decimal form of [tex]\boldsymbol{\sf{\displaystyle\frac{1}{2}}}[/tex].
But first we need to convert it into a fraction whose
denominator is 10, or 100, or 1,000, etc.
The closest number that's divisible by 2 is 10.
So that's the new denominator of the fraction.
Now we should also multiply the top times the number
that we multiplied the bottom by. See how this works?
[tex]\boldsymbol{\sf{\displaystyle\frac{1\times5}{2\times5}}}[/tex]. This yields
[tex]\boldsymbol{\sf{\displaystyle\frac{5}{10}}}[/tex]. And 5/10 in decimal-form yields
[tex]\boldsymbol{\sf{0.5}}[/tex].
[tex]\orange\hspace{300pt}\above3[/tex]
[tex]\LARGE\boldsymbol{\sf{calligraphy}}[/tex]
Point Z is equidistant from the vertices of ΔTUV.
Point Z is equidistant from the vertices of triangle T U V. Lines are drawn from point Z to the sides of the triangle to form right angles and line segments Z A, Z B, and Z C.
Which must be true?
Line segment T A is-congruent-to line segment T B
Line segment A Z is-congruent-to line segment B Z
AngleBTZ Is-congruent-to AngleBUZ
AngleTZA Is-congruent-to AngleTZB
Answers
An isosceles triangle is one with two equal-length sides. The correct option is C.
What is an isosceles triangle?An isosceles triangle is one with two equal-length sides. It is sometimes stated as having exactly two equal-length sides, and sometimes as having at least two equal-length sides, with the latter form containing the equilateral triangle as a particular case.
The diagram as per the given conditions is drawn below.
In ΔUTZ, since the sides UZ and TZ are equal because the point z is equidistance from T and U, therefore, the triangle is an isosceles triangle. Thus, ∠BTZ≅TZB.
Hence, the correct option is C.
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A courier travels due North at an average speed of 40km/h for 6 minutes to collect a parcel, before travelling 10km due East to deliver it. He then travels due South at an average speed of 30km/h for 12 minutes to collect another parcel. Find the shortest distance between the courier and his starting point.
Answers
Answer:
Step-by-step explanation:
Answer:10.2km
He took the route eastwards then southwards
Kirsten is knitting and hates to donate to a local charity. Each hat takes 3 hours to make. If Kirsten knitted 3 hats and a third of another hat. how many hours did she spend knitting?
Answers
so she’d spend 10 hours
The formula for the perimeter of a rectangle is P=2(l+w). Part A. Rewrite the formula for the perimeter of a rectangle in terms of the width, w. In your final answer, include all of your work. Part B. In two or more complete sentences, describe the process you followed while isolating the variable w in the equation P=2(l+w).
Answers
The expression that represents the width is w = (P-2l)/2
Subject of formulaGiven the formula for calculating the perimeter of a triangle expressed as:
P = 2(l + w)
where
l is the length
w is the width
Make w the subject of the formula
Given
P = 2(l + w)
Expand
P = 2l + 2w
Subtract 2l from both sides
P - 2l = 2w
Divide both sides by2
2w/2 = (P-2l)/2
w = (P-2l)/2
Hence the expression that represents the width is w = (P-2l)/2
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Area=
Help me please thanks
Answers
The area of the shaded region shown in the figure is 80π unit²
What is an equation?An equation is an expression that shows the relationship between two or more variables and numbers.
Area of the shaded region = area of semicircle IL - area of semicircle JK + area of semicircle IJ + area of semicircle KL
Area of the shaded region = area of semicircle IL + area of semicircle IJ = (0.5 * π * 12²) + (0.5 * π * (12/3)²) = 80π
The area of the shaded region shown in the figure is 80π unit²
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Explain weather this equation is a linear equation
4. y = 1 - x
Answers
Answer:
Yes, it is a linear equation.
Step-by-step explanation:
Linear equation:An equation of the form ax + by +c = 0, where x and y are two variables and a, b,c are the non-zero real numbers is called a linear equation and the degree of the equation is 1.
y = 1 - x
x + y - 1 = 0
Where a = 1 ; b=1 and c = -1
So, y = 1 -x is a linear equation.
Solve the questions by factoring! Please help asap!
Answers
Answer:
Step-by-step explanation:
If B=3n-10B=3n−10 and C=n^{2}-6n-6,C=n
2
−6n−6, find an expression that equals 2B-3C2B−3C in standard form.
Answers
The expression of 2B - 3C in standard form is -3n^2 + 24n - 2
Functions and valuesGiven the following expressions
B = 3n - 10
C = n^2-6n-6
Required
2B - 3C
Substitute
2B - 3C = 2(3n -10) - 3(n^2-6n-6)
2B - 3C = 6n - 20 -3n^2+18n+18
2B - 3C = -3n^2 + 24n - 2
Hence expression of 2B - 3C in standard form is -3n^2 + 24n - 2
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Which exponential equation is equivalent to this logarithmic equation? log x = 4
Answers
Answer:
[tex]x=10^{4}[/tex]
Step-by-step explanation:
Switching from Logarithmic to Exponential form:An exponential base is the inverse of a logarithm with the same base. Given the equation log x = 4, note that the base of the logarithm isn't written in (it has no subscript). By default, the base of the logarithm function is base-10. So, to rewrite the equation using an exponential, we need to undo the logarithm with an exponential of the same base.
[tex]\log(x)=4[/tex]
[tex]10^{\log{(x)}}=10^{4}[/tex]
[tex]x=10^{4}[/tex]
Analogies with Addition/SubtractionThink about the equation x + 7 = 10.
I could tell you this is the equation in "addition form". What if I asked you to write an equation in "subtraction form"? While neither "addition form" nor "subtraction form" have been defined explicitly, one could undo the addition, and get an equation with subtraction in it, which one could argue is a "subtraction form" of the equation.
[tex]x+7=10[/tex]
Subtracting 7 from both sides to undo the addition...
[tex](x+7)-7=(10)-7[/tex]
Simplifying the left side since the "+7" and "-7" cancel...
[tex]x=10-7[/tex]
This is arguably in a "subtraction form" (there's a subtraction in it), whereas the original equation had addition in it.
While we could simplify this particular equation's right-hand side, we might not always be able to (what if the 10 had been a "y"... then the "y" and the "7" aren't like terms, and they would have to remain separate)
Similarly, in the logarithm problem, one could simplify 10^4 (it's 10,000), but one doesn't have to, and one won't always be able to.
For the logarithm problem, x=10^4 is the exponential form, as requested.
2. A map is drawn so that every 1 inch on the map represents 50 actual miles. The table below can be used to record the actual distances between towns on the map. Actual Distance (mi.) Map Distance (in.) Actual Distances and Map Distances 50 1 Complete the table. What is the actual distance between two towns that are 4 inches apart on the map? 2. A map is drawn so that every 1 inch on the map represents 50 actual miles . The table below can be used to record the actual distances between towns on the map . Actual Distance ( mi . ) Map Distance ( in . ) Actual Distances and Map Distances 50 1 Complete the table . What is the actual distance between two towns that are 4 inches apart on the map ?
Answers
Answer:
200miles
Step-by-step explanation
4 x 50 = 200
As for the table, just multiply each inch by 50.
For example; 7in = 350 miles because 7x50 = 350
The formula s = startroot startfraction s a over 6 endfraction endroot gives the length of the side, s, of a cube with a surface area, sa. how much longer is the side of a cube with a surface area of 180 square meters than a cube with the surface area of 120 square meters? startroot 30 endroot minus 4 startroot 5 endroot meters m startroot 30 endroot minus 2 startroot 5 endroot meters m startroot 10 endroot meters m 2 startroot 15 endroot meters m
Answers
Step-by-step explanation:
so, let me get this straight :
s = sqrt(sa/6)
this is because the surface area of a cube consists of 6 squares (as everybody knows that dice have 6 equal sides).
and the area of a single square is side².
for a cube with sa = 180 m² we have
s = sqrt(180/6) = sqrt(30)
for a cube with sa = 120 m² we have
s = sqrt(120/6) = sqrt(20) = sqrt(4×5) = 2×sqrt(5)
so, the side length of the larger cube is
sqrt(30) - 2×sqrt(5) meters
longer than the side length of the smaller cube.
in other words, the difference between both side lengths.
Answer:
answer above is correct, its B
Step-by-step explanation:
The equation y = ax describes the graph of a line. If the value of a is negative,
the line:
Answers
Answer:
The line has a downward slope
Step-by-step explanation:
If y = ax, it means that a represents the slope of the line. So, if A is negative, the line has a downward slope. In other words, for every increase in the value of x, there is a decrease in the value of y (increase in the negative direction). You can imagine the line y = -x, where the value of a is -1. When x is 1, y is -1; when x is 2, y is -2; when x is 5, y is -5. You can imagine that the graph looks like a diagonal line, much like y = x, except the function values are positive in the second quadrant and negative in the fourth (it slants downward to the right side). I hope that answers your question.
A group of summer campers went on a trip.
22 campers rode in cars with their parents
while the rest filled four buses. How many
campers were on each bus if a total of 150
campers went on the trip?
Answers
Answer:
Answer is 32
Step-by-step explanation:
150-22=128
128÷4=32
Simplify the expression to a polynomial in
standard form:
(3x-1)(3x^2+3x+8)
Answers
Answer:
9x^3+6x^2+21x-8
Step-by-step explanation:
(3x-1) (3x ^ 2 + 3x + 8) =3x(3x ^ 2 + 3x + 8) +(-1)(3x ^ 2 + 3x + 8) =9x^3+9x^2+24x-3x^2-3x-8 = 9x^3+6x^2+21x-8
Answer:
[tex]\huge\boxed{\sf 9x^3+6x^2+21x-8}[/tex]
Step-by-step explanation:
Given expression:= [tex](3x-1)(3x^2+3x+8)[/tex]
[tex]Expand[/tex]
[tex]= 3x(3x^2+3x+8)-1(3x^2+3x+8)\\\\Multiply\\\\=9x^3+9x^2+24x-3x^2-3x-8\\\\Combine \ like \ terms\\\\= 9x^3+9x^2-3x^2+24x-3x-8\\\\= 9x^3+6x^2+21x-8\\\\\rule[225]{225}{2}[/tex]
Line I has a slope of 13/7. The line through which of the following pair
of points is perpendicular to l?
Answers
Answer:
Step-by-step explanation:
You need to find the set of points that will yield a slope that is the negative reciprocal of the slope of Line L because perpendicular lines have negative reciprocal slopes. The negative reciprocal of 13/7 is -7/13. Which set of points will produce this result? The formula for finding the slope is:
m = (y2 - y1)/(x2 - x1)
Consider the second set of coordinates.
(2 - (-5))/(-7 - 6) = (2 + 5)/(-13) = -7/13
The second set of coordinates satisfy the condition.
Researchers at a food company are interested in how a new ketchup made from green tomatoes (and green in color) will compare to their traditional red ketchup. They randomly assign participants to either the green or red ketchup condition. Participants indicate the tastiness of the sauce on a 20-point scale. Tastiness scores tended to be skewed (not normal). What statistical test should be used to analyze that data
Answers
The statistical test which can be used to analyze the data is chi square test.
Given two categories of tomatoes , one is red and other one is green.
When we want to compare more than two categorical variables, we can use the chi-square test. The main objective of chi square test is to compare the distribution of responses or the proportions of participants in each response category. This test does not require mean, standard deviation or anything. In the given problem we have not given sample mean, population mean, population standard deviation, etc. So we cannot use z test, t test, f test because they require sample mean, population mean , standard deviation in their calculation. The formula to be use in chi square test is as under:
[tex]X^{2}=[/tex]∑[tex](f_{0} -f_{e}) ^{2} /f_{e}[/tex]
where [tex]f_{0}[/tex]= observed frequency of each of response category.
[tex]f_{1}[/tex]=expected frequency in each of the response categories.
We can find the critical value in a table of probabilities for the chi square distribution with degree of freedom df=K-1.
Hence we can use chi square test to analyze the data.
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help asap 20 points
Select the correct answer from each drop-down menu.
Answers
Answer: (99/13, 150/13)
Step-by-step explanation:
I'm too lazy to explain so here's a screen shot from desmos.
The polynomial 2x3 − 5x2 + 4x − 10 is split into two groups, 2x3 + 4x and −5x2 − 10. The GCFs of each group is then factored out.
Answers
Answer:
(x² + 2)(2x - 5)
Step-by-step explanation:
assuming the expression has to be factorised
2x³ - 5x² + 4x - 10 ← rearranging terms as indicated
= 2x³ + 4x - 5x² - 10 ( factor the first/second and third/fourth terms )
= 2x(x² + 2) - 5(x² + 2) ← factor out (x² + 2) from each term
= (x² + 2)(2x - 5) ← in factored form