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)

The expression of 2B - 3C in standard form is -3n^2 + 24n - 2

Given 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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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

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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Answer:

Answer is 32

Step-by-step explanation:

150-22=128

128÷4=32

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:

The area of the shaded region shown in the figure is 80π unit²

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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Answer:

1/2

Step-by-step explanation:

The dimensions are cut in half ....scale factor 1/2

The expression that represents the width is w = (P-2l)/2

Given 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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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]

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

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.

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:

= [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]

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]

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.

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.

Answer:

Yes, it is a linear equation.

Step-by-step explanation:

   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.

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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An isosceles triangle is one with two equal-length sides. The correct option is C.

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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Answer:

[tex]x=10^{4}[/tex]

Step-by-step explanation:

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]

Think 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.

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 description below proves that the perpendicular drawn from the vertex angle to the base bisect the vertex angle and base.

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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Answer:

Step-by-step explanation:

Answer:10.2km

He took the route eastwards then southwards

Answer:

Step-by-step explanation:

(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.

Answer: (99/13, 150/13)

Step-by-step explanation:

I'm too lazy to explain so here's a screen shot from desmos.

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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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:

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.

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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