The sum of my digits is an even number. My first digit is 6 less than my second digit. 12 is the number you get when you add my digits. What number am I?

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.

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]

(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

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:

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

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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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:

$1.25

Step-by-step explanation:

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.

To find the number of hours Liam volunteered during the first month, we can set up an equation using the given information. By solving the equation, we find that Liam volunteered for 15 hours during the first month.

To solve this problem, we can start by representing the number of hours Liam volunteered in the first month as x. The number of hours volunteered in the next month would be 1 2/3 times x, which can be written as (5/3)x. The total number of hours volunteered is given as 40, so we can write the equation: x + (5/3)x = 40. To solve for x, we can combine the like terms and solve the resulting equation.

Combining the x terms, we get (8/3)x = 40. Solving for x, we divide both sides of the equation by 8/3: x = 40 / (8/3).

Simplifying, we have x = 40 * (3/8) = 15. Therefore, Liam volunteered for 15 hours during the first month.

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By representing the number of hours in the first month as x, we can find the value of x by using the given information and solving the equation. The solution shows that Liam volunteered for 15 hours during the first month.

To solve this problem, we can set up an equation representing the total number of hours Liam volunteered. Let's represent the number of hours he volunteered in the first month as x. According to the problem, he volunteered for 1 2/3 times as many hours in the next month. So, the number of hours he volunteered in the next month is 1 2/3 x. The total number of hours is given as 40. Therefore, we have the equation:

x + 1 2/3 x = 40

To solve for x, we can simplify the equation by converting 1 2/3 to an improper fraction. This gives us:

5/3 x + 1 2/3 x = 40

Combining like terms, we get:

8/3 x = 40

To solve for x, we can multiply both sides of the equation by the reciprocal of 8/3, which is 3/8. This gives us:

x = (40)(3/8) = 15

Therefore, Liam volunteered for 15 hours during the first month.

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None of the above. All the answers refer to price. The given equation refers to value. We imagine price may stay the same or actually increase year-to-year. The problem statement gives insufficient information to conclude anything about price.

Math is about attention to detail. The equation tells you ...

... The value of the boat decreases by 12% every year.

The value of the boat is multiplied by 0.88 each time X increases by 1. That means the value is 88% of what it was the year before, a decrease of 12%.

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

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.

(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.

Answer:

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

Step-by-step explanation:

Answer:

A)55

B)110

Step-by-step explanation:500 /11%=55

55*2=110

The interest is $110 and the amount paid to the bank is $610.

Simple interest is calculated just on the loan's principal amount, whereas compound interest is calculated on both the principal and the cumulative interest.

It is given that you decide to borrow $500 to go on a trip to whatever land. You go to the bank and borrow the money at 11% for 2 years.

A. The simple interest is,

SI = ( Principle x rate x time ) / 100

SI = ( 500 x 11 x 2 ) / 100

SI = $110

B. The amount paid to the bank after 2 years is,

Amount = P + SI

Amount = $500 + $110 = $610

Therefore, the interest is $110 and the amount paid to the bank is $610.

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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:

  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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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

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]

Answer:

D) 91 °F

Step-by-step explanation:

        C = 5/9(F - 32)

      33 = 5/9 (F – 32)     Multiply each side by 9

  9×33 = 5(F- 32)           Divide each side by 5

9×33/5 = F – 32             Add 32 to each side

        F = 9×33/5 + 32

        F = 59.4 + 32

        F = 91 °F

Answer:

Its D

Step-by-step explanation:

USA TP

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. :)

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:

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: [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]

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

See the attachment

The expected value of store card transactions will be the product of their average value ($58) and their probability of occurrence (48%).

... $58 × 0.48 = $27.84

Transactions are either "store card" or "without a store card", so the probability of a transaction being made without a store card is

... 1 - 0.48 = 0.52

Using the same computatation as for store card expected value, the expected value of transactions made without a store card is ...

... $74 × 0.52 = $38.48

The expected value of the store's transactions is the sum of the expected values of the different ways transactions can be made:

... $27.84 +38.48 = $66.32

_____

The average value of transactions made without a store card is the weighted average of those made with another card and those made with cash or a gift card. Using c to represent the latter, this weighted sum is ...

... 74 = 0.80 × 70 + 0.20 × c

... 74 -56 = 0.20c . . . . . subtract 56

... 18/0.2 = c = 90 . . . . . divide by the coefficient of c

The average transaction made with cash or a gift card is $90.

Each expected value per transaction should be matched to the correct location on the tree diagram as shown in the image below.

In this exercise, we would use probability to make each of the decisions. The expected value for transactions made with a store card can be determined by multiplying its probability of occurrence (48%) by the average value ($58) as follows;

Expected value per transaction = 58 × 0.48

Expected value per transaction = $27.84

Since transactions are either made with a store card or without a store card, the probability of a transaction being made without a store card is given by;

Probability (without a store card) = 1 - 0.48 = 0.52

In this context, the expected value per transactions made without a store card is given by;

Expected value per transaction = 74 × 0.52

Expected value per transaction = $38.48

... $74 × 0.52 = $

For the store's expected value transactions, we would add each of the expected values calculated above;

Store's expected value transactions = 27.84 + 38.48

Store's expected value transactions = $66.32.

The average transaction value for transactions that are made with cash or a gift card can be calculated by finding the weighted average of all transactions made without a card;

74 = (0.80 × 70) + 0.20 × m

74 = 56 + 0.20m

0.20m = 74 - 56

0.20m = 18

m = 18/0.20

m = $90.

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