how many roots exist for this polynomial function? F(x) = 4x5 – 3x​

Number of days Elizabeth spend making jam if she jarred 9 liters of jam is 4 days

Solution:

Given that, Elizabeth has already jarred 1 liter of jam

She will jar an additional 2 liters of jam every day

To find: Number of days Elizabeth spend making jam if she jarred 9 liters of jam

Let "x" be the number of days Elizabeth spend making jam

Then, by given information, we frame a equation as,

9 liters of jam = 1 liter of jam + 2 liters of jam( "x" days )

[tex]9 = 1 + 2x\\\\9 - 1 = 2x\\\\2x = 8\\\\x = 4[/tex]

Thus she spend 4 days in making jam

The domain of g(x) contains the domain of f(x) as a subset.

The given functions are f(x) = -log(x-2)-3 and g(x) = log(x-2). To determine which function has its domain contained within the domain of the other, we need to compare the two domains. The domain of f(x) consists of all real numbers greater than 2, while the domain of g(x) also consists of all real numbers greater than 2. Therefore, the domain of g(x) contains the domain of f(x) as a subset.

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To find (g * f)(x), compute f(x) and substitute it into g(x), resulting in g(f(x)) = (√{x}+3)² + 2/√(x+3). Expand and simplify as needed.

To find (g * f)(x) for the given functions f(x)=√{x}+3 and g(x)=x²+2/x, we first need to compute f(x) and then substitute the result into function g.

First, we compute f(x):
f(x)=√x+3

Now, we substitute f(x) into g(x):
g(f(x)) = (√x+3)² + 2/√(x+3)

Step-by-step calculation:

Answer:

Step-by-step explanation:

P(heads) = 1/2

P(blue) = 1/4

P (heads and blue) = 1/2 * 1/4 = 1/8

Answer:

x = 215.83654587...

Answer:

Exact Form:

x=279940/1297

Decimal Form:

x=215.83654587…

Mixed Number Form:

x=215[tex]\frac{1085}{1297}[/tex]

hope this helps:)

Final answer:

To solve the inequality, simplify the expression and isolate the variable y. The solution is y > 43/37, so the correct answer is option (c).

Explanation:

To solve the inequality 2y - 1/5 - 2 + 7y/15 > 2/3, we can simplify and isolate the variable y. Combining like terms, we have 2y + 7y/15 - 1/5 - 2 > 2/3. Multiplying all terms by 15, we get 30y + 7y - 3 - 30 > 10. Simplifying further, we have 37y - 33 > 10. Adding 33 to both sides, we have 37y > 43. Finally, dividing both sides by 37, we find that y > 43/37.

So the correct answer is option (c), y > 43/37.

Answer:

Since you didn't provide any choices, a possible equations would be h = (m - 10) / 5

Step-by-step Solution:

Since we know that the flat-rate 10 doesn't have anything to do with the hourly rate, we first subtract that. Then, we divide that number by 5 to get rid of it, so we're only left with h.

This process of removing things from the equation by reversing their methods can be applied all over math and is a strategy vary commonly used.

The value of x is [tex]x=0[/tex] or [tex]x=-8[/tex]

Step-by-step explanation:

The expression is [tex]x^{2} +8x=0[/tex]

To complete the square, the equation is of the form [tex]ax^{2} +bx+c=0[/tex]

The constant term c can be determined using, [tex]c=\left(\frac{\frac{b}{a}}{2}\right)^{2}[/tex]

[tex]\begin{aligned}c &=\left(\frac{8}{2}\right)^{2} \\&=\left(\frac{8}{2}\right)^{2} \\c &=4^{2} \\c &=16\end{aligned}[/tex]

Rewriting the expression [tex]x^{2} +8x=0[/tex] and factoring the trinomial, we have,

[tex]\begin{array}{r}{x^{2}+8 x+16=16} \\{(x+4)^{2}=16}\end{array}[/tex]

Taking square root on both sides, we get,

[tex]\begin{aligned}&x+4=\sqrt{16}\\&x+4=\pm 4\end{aligned}[/tex]

Either,

[tex]\begin{array}{r}{x+4=4} \\{x=0}\end{array}[/tex]  or [tex]\begin{array}{r}{x+4=-4} \\{x=-8}\end{array}[/tex]

Thus, the value of x is  [tex]x=0[/tex] or [tex]x=-8[/tex]

Answer:

Step-by-step explanation:

1)y=-6x-14   ---------(i)

-x-3y=-9  ---------- (ii)

Substitution method:

Substitute y  value in equ (i)

-x -3*(-6x-14) = -9

-x - 3*-6x -3*(-14)= -9

-x + 18x + 42 = -9

17x = -9 -42

17x = -51

x = -51/17

x = -3

Substitute x value in equation (i)

y = -6*-3 -14

y =18-14

y = 4

Step-by-step explanation:

I'm using substitute in the first and something I don't know the name of in the rest.

Answer:

The range and mid-range are equal

Step-by-step explanation:

the range is 75-25=50

the mid-range is (75+25)/2 = 100/2 = 50

50 = 50

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

30, 25, 50, 75, 75, 60

The range is the difference between the highest value and the lowest values in the given set of numbers.

Midrange = Range ÷ 2

Option A is the correct answer.

The range of the set of numbers is 50

The midrange of the set of numbers is 25

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

30, 25, 50, 75, 75, 60

The range is the difference between the highest value and the lowest values in the given set of numbers.

Now,

The lowest value is 25

The highest value is 75

Range = 75 - 25 = 50

Midrange = 50/2 = 25

Thus,

The range of the set of numbers is 50

The midrange of the set of numbers is 25

Learn more about expressions here:

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Answer:

Shorter leg: 8 units

Longer leg: 15 units

Hypotenuse: 17 units

Step-by-step explanation:

We are given [tex]h=1+2s[/tex] where [tex]h[/tex] is the hypotenuse and [tex]s[/tex] is the length of the shorter leg.

We got this equation from reading that the "hypotenuse of a triangle is 1 foot more than twice the length of the shorter leg". I replaced the "hypotenuse of a triangle" with [tex]h[/tex], "is" with [tex]=[/tex], "1 foot more than" with [tex]1+[/tex] and finally "twice the length of the shorter leg" with [tex]2s[/tex].

We also have "longer leg is 7 feet longer than the shorter leg".

I'm going to replace "longer leg" with [tex]L[/tex].

I'm going to replace "is" with [tex]=[/tex].

I'm going to replace "7 feet longer than the shorter leg" with [tex]7+s[/tex].

So we have the equation [tex]L=7+s[/tex].

So we have a right triangle since something there is a side being referred to as the hypotenuse. We can use Pythagorean Theorem to find a relation between all these sides.

So by Pythagorean Theorem, we have: [tex]s^2+L^2=h^2[/tex].

Let's make some substitutions from above:

[tex]s^2+(7+s)^2=(1+2s)^2[/tex]

Let's expand the powers using:

[tex](a+b)^2=a^2+2ab+b^2[/tex]

Applying this now:

[tex]s^2+(49+2(7)s+s^2)=(1+2(1)(2s)+(2s)^2)[/tex]

[tex]s^2+49+14s+s^2=1+4s+4s^2[/tex]

Combine like terms on right hand side:

[tex]2s^2+49+14s=1+4s+4s^2[/tex]

Subtract everything on left hand side to get 0 on that side:

[tex]0=(1-49)+(4s-14s)+(4s^2-2s^2)[/tex]

Simplify:

[tex]0=(-48)+(-10s)+(2s^2)[/tex]

Reorder into standard form for a quadratic:

[tex]0=2s^2-10s-48[/tex]

Every term is even and therefore divisible by 2. I will divide both sides by 2:

[tex]0=s^2-5s-24[/tex]

I'm going to see if this is factoroable.

We need to see if we can come up with two numbers that multiply -24 and add up to be -5.

Those numbers are -8 and 3.

So the factored form is:

[tex]0=(s-8)(s+3)[/tex]

This implies that either [tex]s-8=0[/tex] os [tex]s+3=0[/tex].

The first equation can be solved by adding 8 on both sides: [tex]s=8[/tex].

The second equation can be solved by subtracting 3 on both sides: [tex]s=-3[/tex].

The only solution that makes sense for [tex]s[/tex] is 8 since it can't the shorter length cannot be a negative number.

[tex]s=8[/tex]

[tex]L=7+s=7+8=15[/tex]

[tex]h=1+2s=1+2(8)=1+16=17[/tex]

So the dimensions of the right triangle are:

Shorter leg: 8 units

Longer leg: 15 units

Hypotenuse: 17 units

Answer:

[tex]A=x^2+12x+27\ units^2[/tex]

Step-by-step explanation:

we know that

The area of the model is equal to the area of a rectangle

The area of a rectangle is equal to

[tex]A=LW[/tex]

we have

[tex]L=x+9\ units[/tex]

[tex]W=x+3\ units[/tex]

substitute

[tex]A=(x+9)(x+3)[/tex]

Apply distributive property

[tex]A=x^2+3x+9x+27[/tex]

Combine like terms

[tex]A=x^2+12x+27\ units^2[/tex]

Answer:

The perimeter of ABCD will be 14 units.

Step-by-step explanation:

Points A(-5,-1) and B(-1,-1) lies on the same line which is parallel to the x-axis.

So, length of line segment AB will be |- 5 - (- 1)| = 4 units.

Points B(-1,-1) and C(-1,-4) lies on the line which is parallel to the y-axis.

So, length of line segment BC will be |- 4 - (- 1)| = 3 units.

Points C(-1,-4) and D(-5,-4) lies on the same line which is parallel to the x-axis.

So, length of line segment CD will be |- 5 - (- 1)| = 4 units.

Points D(-5,-4) and A(-5,-1) lies on the line which is parallel to the y-axis.

So, length of line segment DA will be |- 4 - (- 1)| = 3 units.

Therefore, the perimeter of ABCD will be (4 + 3 + 4 + 3) = 14 units. (Answer)

Jason has a total of 43 stamps, and we have two different stamps involved.

x = 15 cents

y = 20 cents

We will need to set up two equations and then use substitution:

43 = x + y (represents total stamps)

7.50 = 0.15x + 0.20y (represents total value)

To use substitution, we'll use the first equation and get either x or y on its own. In this case I will choose to get y on its own:

43 - x = x - x + y (subtract x to get y alone)

43 - x = y

Now we will use this in the second equation and simplify:

7.50 = 0.15x + 0.20(43 - x)

7.50 = 0.15x + 8.6 - 0.20x

7.50 = 8.6 - 0.05x

-1.1 = -0.05x (divide by -1.1 to get x alone)

22 = x

Now that we know Jason has 22 of the stamps that are worth 15 cents, we need to find y by plugging x, 22, into the first equation:

43 = 22 + y

21 = y

Jason has 22 stamps that are worth 15 cents and 21 stamps that are worth 20 cents.

Answer:

Each crest takes 5 seconds to pass the observer

Step-by-step explanation:

First, we need to convert all the units to be the same.

Speed is given in meters per second and length in kilometers, so we will convert kilometers to meters:

One kilometer has 1000 meters, so that means that 0.02 km has 20 meters.

Now, we can solve the problem:

speed equals to length over time, or in equation v = s/t

That means that t = s/v

t = 20m / 4 m/s

t = 5 s

Alisia and Luis are both at the gym every 12 days. After the 12th day, the next day they will both be at the gym is the 24th day.

In this math problem, we figure out when Alisia and Luis will both be at the gym at the same time again. Alisia goes every 3 days, and Luis goes every 4 days. The days when they're both at the gym are multiples of the least common multiple (LCM) of 3 and 4. The LCM of 3 and 4 is 12, so they're both at the gym every 12 days.

They both are at the gym on the 12th day. To find out when they'll be there together next, we simply add 12 to the current day: 12 + 12 = 24. So, the next day they will both be at the gym is the 24th day.

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The statement m∠1 = m∠2 within a geometric context implies that the angles are congruent. If these angles are part of a triangle, the triangle is likely isosceles, which aligns with the theorem that equal angles in a triangle indicate equal opposite sides.

If m∠1 = m∠2, this equation suggests we are dealing with congruent angles, potentially within a geometric context. In the realm of geometry, congruency implies that both angles have the same measure. Thus, if these are angles within a triangle, according to the given theorem, the triangle is isosceles.

According to the property that the sum of angles in a triangle is equal to two right angles or 180 degrees (THEOREM 20), we can understand that in an isosceles triangle, the angles opposite to the equal sides are also equal. This would mean that if m∠1 equals m∠2, these could be the angles opposite the two equal sides in an isosceles triangle.

Furthermore, when interpreting equations such as m₁v₁ = 1 / 012₂ cos 0₂, we are likely dealing with more advanced mathematical or physical concepts, such as vectors or trigonometry. Yet, these equations do not apply directly to the statement m∠1 = m∠2 unless more context is provided about the relationship between these angles and the variables in those equations.

Answer:

The amount for social security for all 3 months will $558 and the contribution for medicare will be $130.5

Step-by-step explanation:

Given that the salary is paid in US dollars, we can assume that Renae Walters falls under the social security and medicare tax rates for the United States. These rates, as of 2020, fall at 6.2% for social security and 1.45% for medicare.

By formula:

-Social security contribution= Social security tax rate * Income earned

January for example Social Security= 0.062* $9000= $558                      

-Medicare contribution= Medicare tax rate * Income earned

  January for example Medicare = 0.0145* $9000= $130.5

The above method will apply through the months without changes.

Remember: The social security wage base for 2020 is fixed at $137,700. Any income above this amount is not subject to social security tax. This implies that if the total yearly income is above $137,700, the individual is only liable to pay for the amount below. In this case, this is not possible as the yearly income reaches $108,000 and hence payment has to be made for all 12 months. There is no wage base for medicare, so payments remain unaffected.

Answer:

Social security:558, Medicare: 130.50 for all three months

Step-by-step explanation:

A. 9000X.062=558 social security

9000x.0145=130.50

b. Same answer as a.

c. Same answer as a.

the amounts are the same for all 3 months.

Answer:

5.43×10^14 or

[tex]5 .43 \times 10^{14} [/tex]

Step-by-step explanation:

Scientific notation, the number must always be less than 10, in this case 5.43. The exponent represents how much times I moved the decimal point to the left.

Answer:

x = 61

Step-by-step explanation:

Angles A and D are consecutive interior angles of a parallelogram.

Consecutive interior angles of a parallelogram are supplementary.

m<A + m<D = 180

x + 2x - 3 = 180

3x - 3 = 180

3x = 183

x = 61

For this case we must propose a rule of three:

20 people ----------------> 24 days

32 people ----------------> x

Where the variable "x" represents the number of days it takes 32 people to build the barn.

[tex]x = \frac {32 * 24} {20}\\x = \frac {768} {20}\\x = 38.4[/tex]

Thus, it takes 32 people approximately 39 days to build the barn.

Answer:

It takes 32 people about 39 days to build the barn.

To find how many days it will take 32 people to build the barn, calculate the total man-hours (20 × 24 = 480 man-hours) and divide that by the number of workers (480 \/ 32 = 15). Thus, it takes 32 people 15 days to build the barn.

The student's question involves solving a problem by understanding the concept of work rate and man-hours. To find out how many days it would take for 32 people to build a barn if it takes 20 people 24 days, we first need to calculate the total man-hours required to build the barn. The total man-hours is the product of the number of workers and the number of days they work, which in this case is 20 people × 24 days = 480 man-hours.

Once we have the total number of man-hours, we can then calculate how many days it will take for 32 people to complete the same amount of work. This is done by dividing the total man-hours by the number of workers, resulting in 480 man-hours / 32 people = 15 days.

Therefore, it will take 32 people 15 days to build the barn.

The value of m is [tex]-\frac{1}{5}[/tex]

Step-by-step explanation:

The relation between the slopes of two perpendicular line is

∵ The equation of the perpendicular line is y = mx + b

∴ The slope of the perpendicular line is m

∵ The equation of the given line is y = 5x - 2

- The slope of the given line is the coefficient of x

∴ The slope of the given line = 5

- To find the slope of the perpendicular line to the given line

   reciprocal the slope of the given line and change its sign

∵ The reciprocal of 5 is [tex]\frac{1}{5}[/tex]

∴ The slope of the perpendicular line = [tex]-\frac{1}{5}[/tex]

∵ m is the slope of the perpendicular line

∴ m =  [tex]-\frac{1}{5}[/tex]

The value of m is [tex]-\frac{1}{5}[/tex]

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The percent of change is 42 %

Solution:

Given that, Hollister boys shirts for 26 dollars and sells them for 37 dollars

We have to find the percent of change

The percent change is given by formula:

[tex]Percent\ Change = \frac{\text{final value - initial value}}{\text{Initial value}} \times 100[/tex]

Here given that,

Initial value = 26 dollars

Final value = 37 dollars

Substituting the values we get,

[tex]Percent\ Change = \frac{37-26}{26} \times 100\\\\Percent\ Change = \frac{11}{26} \times 100\\\\Percent\ Change =0.423 \times 100\\\\Percent\ Change =42.3 \approx 42[/tex]

Thus percent of change is 42 %

Answer:               m<1 is 62°

Step-by-step explanation:

Alright, lets get started.

The two angles are given as 56° and 62°.

We know the sum of the angles of a triangle is 180°

So,

[tex]x+56+62=180[/tex]

[tex]x+118=180[/tex]

Subtracting 118 in both sides

[tex]x+118-118=180-118[/tex]

[tex]x=62[/tex]

Hence the desired angle 1 is 62°  ...........   Answer

Hope it will help :)

Answer:

B. 62°

Step-by-step explanation:

Hope this helps

Here is the pattern of Lily's towers in the form of x, y:

1, 1

2, 4

3, 9

4, 16

The equation x^2 fits for this problem, so 100^2 would mean it would take 10,000 LEGOS to build the 100th tower.

The solution to given system of equations is x = -7 and y = 5

Solution:

Given that, we have to solve the system of equations by linear combination method

Given system of equations are:

2x + 3y = 1 ---------- eqn 1

y = -2x - 9 ----------- eqn 2

We can use substitution method to solve the system of equations

Substitute eqn 2 in eqn 1

2x + 3(-2x - 9) = 1

Add the terms inside the bracket with constant outside the bracket

2x -6x - 27 = 1

Combine the like terms

-4x = 1 + 27

-4x = 28

Divide both sides of equation by -4

Substitute x = -7 in eqn 2

y = -2(-7) - 9

Simplify the above equation

y = 14 - 9

Thus solution to given system of equations is x = -7 and y = 5

The original price of one keyboard was $20.

Let's denote the original price of one keyboard as x.

The shop owner spent $540 to purchase a stock of computer keyboards. Therefore, the number of keyboards he purchased can be calculated as [tex]\( \frac{540}{x} \)[/tex].

If the price of each keyboard had been reduced by $2, the new price of one keyboard would be x - 2.

With this reduced price, the shop owner could have bought 3 more keyboards, so the total number of keyboards he could have purchased is [tex]\( \frac{540}{x-2} \)[/tex].

We are given that this value is 3 more keyboards than the original purchase, so we can set up the equation:

[tex]\[\frac{540}{x-2} = \frac{540}{x} + 3\][/tex]

To solve this equation, we first clear the fractions by multiplying both sides by (x(x-2):

[tex]\[540x = 540(x-2) + 3x(x-2)\][/tex]

Expanding and simplifying:

[tex]\[540x = 540x - 1080 + 3x^2 - 6x\][/tex]

[tex]\[0 = 3x^2 - 6x - 1080\][/tex]

Now, let's solve this quadratic equation for x.

Dividing both sides by 3:

[tex]\[x^2 - 2x - 360 = 0\][/tex]

Now, we can use the quadratic formula to solve for x:

[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]

Where a = 1, b = -2, and c = -360:

[tex]\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \times 1 \times (-360)}}{2 \times 1}\][/tex]

[tex]\[x = \frac{2 \pm \sqrt{4 + 1440}}{2}\][/tex]

[tex]\[x = \frac{2 \pm \sqrt{1444}}{2}\][/tex]

[tex]\[x = \frac{2 \pm 38}{2}\][/tex]

So, x can be either:

[tex]\[x = \frac{2 + 38}{2} = \frac{40}{2} = 20\][/tex]

or

[tex]\[x = \frac{2 - 38}{2} = \frac{-36}{2} = -18\][/tex]

Since the price of a keyboard cannot be negative, the only valid solution is x = 20.

Therefore, the original price of one keyboard is $20.

Answer:

The original length was 41 inches and the original width was 16 inches

Step-by-step explanation:

Let

x ----> the original length of the piece of​ metal

y ----> the original width of the piece of​ metal

we know that

When squares with sides 5 in long are cut from the four corners and the flaps are folded upward to form an open box

The dimensions of the box are

[tex]L=(x-10)\ in\\W=(y-10)\ in\\H=5\ in[/tex]

The volume of the box is equal to

[tex]V=(x-10)(y-10)5[/tex]

[tex]V=930\ in^3[/tex]

so

[tex]930=(x-10)(y-10)5[/tex]

simplify

[tex]186=(x-10)(y-10)[/tex] -----> equation A

Remember that

The piece of metal is 25 in longer than it is wide

so

[tex]x=y+25[/tex] ----> equation B

substitute equation B in equation A

[tex]186=(y+25-10)(y-10)[/tex]

solve for y

[tex]186=(y+15)(y-10)\\186=y^2-10y+15y-150\\y^2+5y-336=0[/tex]

Solve the quadratic equation by graphing

using a graphing tool

The solution is y=16

see the attached figure

Find the value of x

[tex]x=16+25=41[/tex]

therefore

The original length was 41 inches and the original width was 16 inches

Final answer:

calculate the length as x + 25, and solve the volume equation to find the dimensions as 30 inches by 55 inches.

Explanation:

The original dimensions of the piece of metal can be calculated as follows:

Let x be the width of the metal.

Then, the length would be x + 25.

After cutting out the squares and folding, the volume of the box would be (x-10)(x-10)(30) = 930.

Solving this equation, we get x = 30, so the original dimensions were 30 inches by 55 inches.