In rectangle ANHG, whose perimeter is 100, OP, PQ, and QR are congruent and mutually perpendicular and O is the midpoint of AN. If GH = 40 which is PQ?

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:

21d +25 ≤ 115

Step-by-step explanation:

Jim's cost will be ...

... 21.00·d + 0.10·250 = 21d +25

He wants his cost not to exceed his budget, so ...

... 21d +25 ≤ 115

_____

The solution is ...

... 21d ≤ 90 . . . . subtract 25

... d ≤ 90/21 ≈ 4.3

so Jim can rent the car a maximum of 4 days.

The inequality that can be used to express this scenario is

115 ≤  21*d+ 25

Given data

Charges = $21 per day

Cost per driven distance = $0.10

Distance he will travel = 250 miles

Amount her has to spend = $115

Let the maximum number of days Jim can rent a car with $115 be "m"

Hence

Total amount = 21*d+ 0.1*250

Substituting and Simplifying we have

115 ≤  21*d+ 25

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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:

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.

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.

See the attachment for a labeled graph.

_____

I find it convenient to use "technology" to draw the graph. A spreadsheet, graphing calculator, or on-line graphing program can do this for you.

(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:

$6.40 because 32 ounces is the difference 128 and 96 hence 32 is 1/3 of 96 so divide $4.80 by 3 which is $1.60 then add $1.60 + $4.80 =$6.40

Step-by-step explanation:

Answer:

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Step-by-step explanation:

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]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:

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:

see attached

Step-by-step explanation:

Each of the pie slices can be cut in half different ways. An easy way to do it and to understand it is to draw another cut from the center to the middle of the edge.

The result of cutting these slices is that instead of 8 equal pieces (of which 3 are colored), there will be 16, of which 6 are colored.

The new fraction is 6/16.

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:

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

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:

19

Step-by-step explanation:

b - 7 = 12

b = 7 + 12

b = 19

Hi there! :)

Answer:

b=19

*The answer must have a positive sign.*

Step-by-step explanation:

First, you add by 7 from both sides of an equation.

[tex]b-7+7=12+7[/tex]

Then, you add by the numbers from left to right.

[tex]12+7=19[/tex]

Final answer is b=19

I hope this helps you!

Have a nice day! :)

:D

-Charlie

Thank you so much! :)

Answer:

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

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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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.

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.

[tex]\displaystyle x^{\frac{2}{3}}[/tex]

The rules of exponents tell you ...

... (a^b)(a^c) = a^(b+c) . . . . . . applies inside parentheses

... (a^b)^c = a^(b·c) . . . . . . . . applies to the overall expression

The Order of Operations tells you to evaluate inside parentheses first. Doing that, you have ...

... x^(4/3)·x^(2/3) = x^((4+2)/3) = x^2

Now, you have ...

... (x^2)^(1/3)

and the rule of exponents tells you to multiply the exponents.

... = x^(2·1/3) = x^(2/3)

Answer:

x^(2/3)

Step-by-step explanation:

(x^a.x^b)^c = x^[c*(a+b)]

using the above eqn, u can simplify the given expression to

x^[1/3*(4/3+2/3)]

=x^[1/3*(6/3)]

=x^(2/3)

ans is the 2nd choice

(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:

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

The volume of metal is the difference of the overall volume of the cylinder and the volume of the hole in it. The formula for the volume of a cylinder is ...

... V = π·r^2·h . . . . . radius r and height h

For the overall dimensions, the radius is half the diameter, so is 10 mm. The hole is said to have a radius of 6 mm. The overall "height" is 21 mm, so the volume in mm³ will be ...

... V_overall -V_hole = π(10)^2(21) -π(6)^2(21)

... = 21π·10^2 -21π·6^2 . . . . . . . matches the first selection

... = 2100π -756π . . . . . . . . . . . matches the third selection

... = 1344π . . . . . . . . . . . . . . . . doesnt' match any selection

The correct expressions for the volume of metal needed, in cubic millimeters, to make the pipe are,

⇒ 21π(10)² – 21π(6)²

⇒ 2,100π – 756π

To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

Given that;

A cylindrical metal pipe has a diameter of 20 millimeters and a height of 21 millimeters.

And,  A cylindrical hole cut out of the center has a radius of 6 millimeters.

Hence, The formula for the volume of a cylinder is,

V = π·r²·h

Where, radius r and height h.

Now, For the overall dimensions, the radius is half the diameter, so is 10 mm. The hole is said to have a radius of 6 mm. The overall "height" is 21 mm,

so the volume in mm³ will be;

V (overall) -V (hole) = π(10)²(21) -π(6)²(21)

= 21π·10² -21π·6²

= 2100π -756π

= 1344π

Thus, The correct expressions for the volume of metal needed, in cubic millimeters, to make the pipe are,

⇒ 21π(10)² – 21π(6)²

⇒ 2,100π – 756π

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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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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.

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.