convert fractions to a fraction out of 100.
15/60=?
30/60=?
25/75=?

Answer:

y = 80 -40x

Step-by-step explanation:

The given equation shows the friend arrives at the restaurant (y=0) when x=2. That is, half the initial distance to the restaurant is covered in each hour of travel time.

If I take 2 hours to travel the 80 miles from my house, my distance will decrease at the rate of 40 miles per hour. My distance from the restaurant can be represented by ...

... y = -40x +80

Answer:

D. 6 hours

Step-by-step explanation:

There are 3 ft in 1 yd, so 400 yd = 1200 ft.

At 200 ft/h, Jim's travel time for 1200 ft will be

... time = distance/speed = (1200 ft)/(200 ft/h) = (1200/200) · ft · (h/ft)

... time = 6 h

Answer:

Right part: [tex]\dfrac{4}{5}x+a,[/tex] where [tex]a\neq -8.[/tex]

Step-by-step explanation:

Linear equation have no solution if it is impossible for the equation to be true no matter what value we assign to the variable x.

The left part of the equation Tomas wrote is

[tex]\dfrac{4}{5}x-8.[/tex]

The right part of the equation should be of the form

[tex]\dfrac{4}{5}x+a,[/tex]

where [tex]a\neq -8.[/tex]

In this case, the equation will take look

[tex]\dfrac{4}{5}x-8=\dfrac{4}{5}x+a,\\ \\-8=a.[/tex]

Since [tex]a\neq -8,[/tex] this equality is always false.

Remark: 1) If a=8, the equation [tex]\dfrac{4}{5}x-8=\dfrac{4}{5}x+a[/tex] is equivalent to the equation

[tex]\dfrac{4}{5}x-8=\dfrac{4}{5}x-8,\\ \\0=0[/tex]

and the lest equation has infinitely many solutions.

2) When coefficient at x differs from [tex]\dfrac{4}{5},[/tex] the equation has unique solution.

Part A: The exponential function is the best model for the given data. The data shows that every month the number of visitor approximately doubles from previous month. In other words the number is some base value (2, in this case) with the number of month (t) being in the exponent. This the a key characteristic of an exponential growth.  

Part B: (also explained above). Every month the number of visitors approximately doubles. Starting with 5.74 at month 8, the numbers goes up by about the same amount to 12.0 at month 9. At month 10, a similar increase occurs (doubling would be 24, and the data shows 25, so this is all "aproximate"). This trend continues throughout the table.

Part C:

Month 7 will be estimated as half of month 8 (going backward):

Month 7: 5.74/2=2.87

Estimate for month 7 = 2.87 thousand visitors.

a ≤ 3

Undo what is done to the variable. The variable is multiplied by 3 and 4 is subtracted from that result. To undo these operations, you add 4, then divide by 3.

... 3a ≤ 9 . . . . 4 is added to both sides of the inequality

... a ≤ 3 . . . . . both sides of the inequality are divided by 3

_____

Comment on solving inequalities

These are the same steps you would use to solve the 2-step equation ...

... 3a -4 = 5

The only difference between solving equations and solving inequalities is that when you multiply or divide by a negative number, you reverse the sense of the comparison (> becomes <, and vice versa; ≥ becomes ≤, and vice versa). You can see this if you consider numbers: 2 > 1. Multiplying by -1 gives -2 < -1.

Answer:

[tex]AS=18\ units,\ DS=\dfrac{54}{7}\ units.[/tex]

Step-by-step explanation:

Consider trapezoid ABCD with bases BC=11 units and AD=18 units.  The lengths of legs are AB=7 units and CD=3 units. Point S is the point of intersection of the extensions of the legs AB and CD.

Let BS=x units and CS=y units.

Consider triangles BSC and ASD. By AAA theorem these triangles are similar (because ∠SAD≅∠SBC, ∠ADS≅∠BCS and ∠S is common).

Then

[tex]\dfrac{BS}{AS}=\dfrac{CS}{DS}=\dfrac{BC}{AD},\\ \\\dfrac{x}{x+7}=\dfrac{y}{y+3}=\dfrac{11}{18}.[/tex]

Therefore,

[tex]\dfrac{x}{x+7}=\dfrac{11}{18},\\ \\18x=11(x+7),\\ \\18x=11x+77,\\ \\7x=77,\\ \\x=11\ units.[/tex]

[tex]\dfrac{y}{y+3}=\dfrac{11}{18},\\ \\18y=11(y+3),\\ \\18y=11y+33,\\ \\7y=33,\\ \\y=\dfrac{33}{7}\ units.[/tex]

The lengths of segments between point S and the vertices of the greater base are

[tex]AS=AB+BS=7+11=18\ units,\ DS=DC+CS=3+\dfrac{33}{7}=\dfrac{54}{7}\ units.[/tex]

Answer:

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Step-by-step explanation:

Answer:

D.

Step-by-step explanation:

The coefficient of the leading term tells you how "stretchy" a graph is. Since g(x)'s leading coefficient is 3, and f(x)'s leading coefficient is 9, and they're both [tex]x^{2}[/tex], g is just 3 times f. Then g's constant term is -4, and f's constant term is 0, then g is just f but translated 4 down.

Answer:

Option 4

Step-by-step explanation:

The 3 changing to the 9 stretches the graph vertically by a factor of 3.  The - 4

translates the graph 4 units down .

Its Option 4

Answer:

10 tables are needed to seat 65 people.

Step-by-step explanation:

Given :

There are total 65 people.

7 people can be seated on each table .

To Find : No. of tables required to seat 65 people.

Solution :

Since 7 people can sit on no. of tables = 1

1 people sit on no. of tables = 1/7

65 people sit on no. of tables

= [tex]\frac{1}{7} *65[/tex]

⇒ [tex]9.28[/tex]

⇒ 9.28 ≈ 10 tables are needed to seat 65 people.

Thus 10 tables are needed to seat 65 people.

Answer:

See below for a graph

Step-by-step explanation:

Let's call the parent function g(x) = log₅(x), which has a vertical asymptote at x=0, and goes through the points (1, 0), (5, 1), (25, 2).

Then this function is

... f(x) = g(2(x+6)) +2

That is, the function is horizontally compressed by a factor of 2, shifted left 6 units, and shifted up 2 units. This moves the points listed above to ...

... (-5.5, 2), (-3.5, 3), (6.5, 4)

The graph shows the parent function and the listed points in blue, then the transformed function and the corresponding points in red.

Answer: [tex]QS=11.8cm[/tex]

Step-by-step explanation:

1. You have the following information:

- The lenght of [tex]RS[/tex] is 2 centimeters.

- The angle m∠S=80°.

- Cos80°=0.17

2. Therefore, you can calculate the lenght of the segment [tex]QS[/tex] as following:

[tex]cos80=\frac{adjacent}{hypotenuse}[/tex]

[tex]adjacent=2cm\\hypotenuse=QS\\cos80=0.17[/tex]

3. Substitute values and solve for [tex]QS[/tex]:

[tex]0.17=\frac{2cm}{QS}\\QS=\frac{2cm}{0.17} \\QS=11.76cm[/tex]

4. To the nearest tenth:

[tex]QS=11.8cm[/tex]

21 miles

Big Bird's nest is halfway between Bert's place and Ernie's place, so the distance from the nest to either apartment is the same. Equating those distances, we have ...

... 5x +4 = 15x -30

... 34 = 10x . . . . . . add 30-5x

... 3.4 = x . . . . . . . .divide by 10

The value of x is not the distance. The distance is given by either of the two expressions

Using the first of these, the distance from either apartment to Big Bird's nest is ...

... 5·3.4 +4 = 17 +4 = 21 . . . . miles

Of course, using the second gives the same answer:

... 15·3.4 -30 = 51 -30 = 21

Answer:

206 Cornish hens

169 turkeys

Step-by-step explanation:

Let c represent the number of Cornish hens sold. Then c-37 is the number of turkeys sold. The total sales would be ...

... c + (c-37) = 375

... 2c = 412 . . . . collect terms, add 37

... c = 206 . . . . Cornish hens sold

... (c-37) = 169 . . . turkeys sold

_____

Comment on this type of problem

Note that the solution to this problem is that the larger number (number of cornish hens) is half the sum of the total and the difference: (375+37)/2 = 206. This is the general solution for this type of "sum and difference" problem.

The larger contributor is half the total plus half the difference; the smaller contributor is half the total minus half the difference.

Cornish hens = (1/2)(375 +37) = 206

Turkeys = (1/2)(375 -37) = 169

(2.8×10¹)x⁶y⁴

Multiplication can be done in the usual way, then the number converted to scientific notation. Scientific notation does not apply to the variables.

[tex](7x^3y^2)(4x^3y^2)=(7\cdot 4)x^{3+3}y^{2+2}=28x^6y^4\\\\=2.8\cdot 10^1x^6y^4[/tex]

_____

If you write the numbers in scientific notation to start, then do the multiplication, the effect is virtually the same: an adjustment is needed in the product to get it back to scientific notation.

(7)(4) = (7×10⁰)(4×10⁰) = 7·4×10⁰⁺⁰ = 28×10⁰ = 2.8×10¹

Answer:

31,250 ft²

Step-by-step explanation:

Let x represent the length of fence parallel to the river. Then the ends of the rectangle will have dimesion (500-x)/2, and the total area will be ...

... A = x(500-x)/2

This function describes a parabola with zeros at x=0 and x=500. The vertex (maximum) will be located halfway between these, at x=250.

The maximum size pen has area ...

... A = 250(500 -250)/2 = 31,250 . . . . sq ft

Answer:

216 cupcakes

Step-by-step explanation:

81 = 0.375 × order

81/0.375 = order = 216 . . . . . divide by the coefficient of the variable

_____

About percentages

% means /100

37.5% = 37.5/100 = 375/1000 = 0.375

Answer:

561 because if you multiple 300 /8 times the number of the before populaiton u got it right Step-by-step explanation:

The first line.

If we let x represent "a number", then "a number divided by 3" is x/3. The denominator is missing on the first line.

Further, this "is 4 more than -11", so is = 4 + (-11). The constant 4 is on the wrong side of the equal sign in line 1.

_____

The solution should be ...

... x/3 = 4 + (-11)

... x = 3·(-7)

... x = -21

___

Check

-21 divided by 3 is -7. That is 4 more than -11.

A) The problem statement tells you it is a geometric sequence, so you know each term is some multiple of the one before. The first terms of the sequence are given, so you know the first term. The common ratio (the multiplier of interest) is the ratio of the second term to the first (or any term to the one before), -6/2 = -3.

So, the recursive definition is ...

... a_1 = 2

... a_n = -3·a_(n-1)

B) The explicit formula is, in general, ...

... a_n = a_1 · r^(n -1)

where r is the common ratio and a_1 is the first term. Filling in the known values, this is ...

... a_n = 2·(-3)^(n-1)

Answer:

a1=2

an=(an-1)(-3)

Step-by-step explanation:

Hope it helps. :)

18

Put 3 where x is, then do the arithmetic.

... f(x) = x(5x -9)

... f(3) = 3(5·3 -9) = 3(15 -9) = 3·6

... f(3) = 18

_____

Your calculator can do this, too.

Answer:

  B)  Both functions have exactly three x-intercepts.

  C)  The function f(x) is an odd degree function.

  E)  The x intercepts for g(x) are (1, 0); (2, 0) and (-3/2, 0)

Step-by-step explanation:

The function f is a function of degree 3 (there are 3 x-terms in the product contributing to the highest-degree term), so is of odd degree. The leading coefficient in f(x) is the product of the coefficients of x: 1·1·2 = 2, a positive number.

For a polynomial function of odd degree, the general shape of the graph will be "/" if the leading coefficient is positive, and "\" if the leading coefficient is negative. That is, end behaviors will be opposites of each other. Function f has a positive leading coefficient, so its end behavior will be (-∞, -∞) and (+∞, +∞). Function g has end behavior that is (-∞, +∞) and (+∞, -∞), so is not the same. Apparently, g(x) has a negative leading coefficient.

The graph of g(x) shows it to have 3 x-intercepts. They can be read from the graph as (-3/2, 0), (1, 0) and (2, 0). The factoring of f(x) shows it to have 3 x-intercepts, the same number. There is an x-intercept of f(x) for each factor. (The x-intercept value is the value of x that makes the factor zero.)

The function f(x) is of odd degree and has three x-intercepts at -2, 1, and -3/2. The statements regarding the function g(x) cannot be confirmed without further information.

The function given is f(x) = (x+2)(x-1)(2x+3). The degree of the function is 3 as there are three terms multiplied together. Therefore, f(x) is an odd degree function, which makes option C correct.

The x-intercepts of the function occur where y = 0, i.e. where f(x) = 0. Solving this will give the roots or intercepts for x which are -2, 1, and -3/2. This means that the function f(x) has three x-intercepts, which makes option B correct.

Without information or a graph of function g(x), we cannot accurately answer options A, D and E. We cannot know the end behavior, the leading coefficient or the x intercepts for g(x). Therefore, only options B and C are true statements.

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Part A.

We're only told that the product of the mystery factor and the number shown is even. This is true when the number shown is odd as well as even, hence the mystery factor is even.

There are no other conditions on the product that might limit the value of the mystery factor (non-zero, or < 10, for example).

Hence, the mystery factor could be any single-digit even number. In our base-10 number system, there are 5 of them: {0, 2, 4, 6, 8}.

___

Part B.

There are only 5 even single-digit numbers in our base-10 number system, so by choosing 5, I know I have all of them.

Answer:

$300

Step-by-step explanation:

Let x represent the side length of the square base in feet. Then the height of each side is ...

... h = (50 ft³)/(x ft)² = (50/x²) ft

The cost of the sides of the box is then ...

... (4 sides) × (x ft)(50/x² ft)/side × $2/ft² = $400/x

The cost of the bottom is ...

... (x ft)² × $25/ft² = $25x²

So, the total dollar cost is

... C = 400/x + 25x²

This will be a minimum where its derivative with respect to x is zero.

... 0 = -400/x² +50x

... 400/50 = 8 = x³ . . . . . add 400/x²; multiply by x²/50

... x = ∛8 = 2

For this value of x, the minimum cost is ...

... C = 400/2 + 25·2² = 300

The minimum possible cost is $300.

_____

Comments on the problem

1) Cardboard boxes are usually made of cardboard. They are rarely made of wood and alumninum.

2) The cost of the bottom is half the cost of the sides. When the dimensions are unconstrained, you will find (as here) the cost is shared equally between the bottom and pairs of opposite sides—each being 1/3 the total cost.

as you recall from the pythagorean theorem, which applies only to right-triangles, c² = a² + b².

so, a = 6, b = 8, c = 9...... if that triangle is indeed a right triangle, then 9² = 6² + 8², let's check if that's true.

[tex]\bf \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \qquad \begin{cases} c=\stackrel{hypotenuse}{9}\\ a=\stackrel{adjacent}{6}\\ b=\stackrel{opposite}{8}\\ \end{cases} \\\\\\ 9=\sqrt{6^2+8^2}\implies 9=\sqrt{36+64}\implies 9=\sqrt{100}\implies 9\ne 10~~\bigotimes~~nope[/tex]

(2a^4+5a^3+4) -2a^2

There are 4 terms. Your trinomial could consist of any three of them. Here, we've chosen to use the one with the negative sign as the monomial, because the question asks for a difference.

I like to look at a graph of the function to see where the zeros might be. Here, there are x-intercepts at x=-2 and x=3. These can be factored out using synthetic division to find the factorization to be ...

... (x +2)(x -3)(x² +x +1) = 0

By completing the square, using the quadratic formula, or by looking at the graph of it, the complex roots of the quadratic factor can be found to be ...

... x = (-1 ±i√3)/2

_____

The second attachment shows my synthetic division. The first division takes out the root x=3 to give a quotient of x³ +3x² +3x +2. The second division takes out the root -2 to give the quotient of x² +x +1. (You can see that I tried -1 as a root first.)

The graph shows both the quartic and the quadratic factor of it. The latter has a leading coefficient of 1 and a vertex at (-1/2, 3/4), so you know the complex roots are -1/2 ±i√(3/4).

_____

From the beginning

There is only a very complicated formula for the roots of a quartic equation, so these are usually solved by machine or by some form of trial and error (iteration). There are some helps, like Descarte's Rule of Signs, and the Rational Root theorem.

Here, the former looks at the one sign change in the coefficients to tell you there will be 1 positive real root. Changing the sign of the odd-degree terms makes there be 3 sign changes, so there will be 3 or 1 negative real roots. Thus, we're assured at least two real roots, one of each sign.

We can look at the constant term to find the y-intercept to be -6. We can add the coefficients to find the value of the function is -18 for x=1, so the positive real root is larger than 1.

The Rational Root theorem says any rational roots will be factors of 6, the constant term. Choices are 1, 2, 3, 6. We have already eliminated 1 as a possibility, and we consider it unlikely that 6 will be a root. (The 4th power overwhelms the other terms in the function.) We tried 2 and found it doesn't work (this was before we graphed the function). The attached division result shows that 3 is a root, as does the graph.

Once you get down to a quadratic, you can find the remaining roots in the usual way. Because it is so simple to read them from the graph, we decided to graph the quadratic factor.

_____

Comment on terminology

"root" and "zero" are essentially the same thing when the function is equated to zero, as here. The terms refer to the value(s) of x that make the polynomial function evaluate to zero.

To find the zeros of the equation x^4-6x^2-7x-6=0, we can apply the rational root theorem and synthetic division. By trying different factors of the constant term -6, we can determine that one of the roots is x = -2. After dividing the equation by (x + 2), we obtain a quadratic equation x^3 - 2x^2 - 11x + 3 = 0. We can then solve this equation by factoring or using a graphing calculator to find the remaining two roots.

To find the zeros of the equation x^4-6x^2-7x-6=0, we can apply the rational root theorem and synthetic division. By trying different factors of the constant term -6, we can determine that one of the roots is x = -2. After dividing the equation by (x + 2), we obtain a quadratic equation x^3 - 2x^2 - 11x + 3 = 0. We can then solve this equation by factoring or using a graphing calculator to find the remaining two roots.

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Answer:

1/3 3/1 5/6 6/5 8/9 9/8

Step-by-step explanation: