if the perimeter of a triangle measures 42 feet and the three sides are consecutive numbers , what are the side lengths

The roots for the equation [tex]\(2x^2 + 7x = -2\)[/tex] are [tex]\(x = \frac{{-7 \pm \sqrt{65}}}{{4}}\).[/tex] So, option D is correct.

To find the roots of the quadratic equation [tex]\(2x^2 + 7x = -2\),[/tex] we can use the quadratic formula:

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

Here, [tex]\(a = 2\), \(b = 7\), and \(c = -2\).[/tex]

Substituting these values into the formula:

[tex]\[x = \frac{{-7 \pm \sqrt{{7^2 - 4 \cdot 2 \cdot (-2)}}}}{{2 \cdot 2}}\][/tex]

[tex]\[x = \frac{{-7 \pm \sqrt{{49 + 16}}}}{{4}}\][/tex]

[tex]\[x = \frac{{-7 \pm \sqrt{{65}}}}{{4}}\][/tex]

So, the correct answer is option D:

[tex]\[x = \frac{{-7 \pm \sqrt{{65}}}}{{4}}\][/tex]

Complete Question:

The speed of 70 miles per hour is approximately equivalent to 1.17 miles per minute. This conversion is done by dividing the speed in miles per hour by 60, the number of minutes in an hour.

To calculate the conversion from miles per hour to miles per minute, you divide the speed in mph by 60, as there are 60 minutes in an hour. So if we are given a speed limit of 70 miles per hour, that would convert to approximately 1.17 miles per minute.

This is calculated as 70 miles per hour ÷ 60 minutes per hour = 1.17 miles per minute

It's important to remember to use the correct conversion factor related to time, in this case that there are 60 minutes in an hour, to ensure the accuracy of the conversion.

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To construct a 95% confidence interval for the population proportion, calculate the sample proportion p' and its complement q', determine the Z-score for 95% confidence, calculate the margin of error using the formula E = Z*sqrt((p'q')/n), and add/subtract E from p' to get the lower and upper bounds.

To construct a 95 percent confidence interval for the population proportion p using the given sample data, we must first calculate the sample proportion (p') and its complement, the estimated proportion of failures (q'). Using the formula p' = x/n, we find that p' = 162/195. Next, we determine q' by calculating q' = 1 - p'.

With the sample proportion and its complement, we can use the standard formula for a confidence interval for a population proportion: p' ± Z*sqrt((p'q')/n), where Z* is the Z-score corresponding to the given degree of confidence. For a 95% confidence level, the Z-score is approximately 1.96.

By substituting the values of p', q', n, and the Z-score into the formula, we calculate the margin of error (E) and then the lower and upper bounds of the 95 percent confidence interval.

Suppose p' is 0.83 and q' is 0.17 for n = 195 and the Z-score for a 95% confidence interval is 1.96. The margin of error (E) would then be 1.96 * sqrt((0.83*0.17)/195), and the confidence interval would be p' ± E, resulting in a specific numerical range which would constitute our 95% confidence interval for the true population proportion.

The 95% confidence interval for the population proportion [tex]\( p \)[/tex] is [tex]\( (0.7783, 0.8833) \)[/tex].

To construct a confidence interval for the population proportion [tex]\( p \),[/tex] we will use the given information: sample size [tex]\( n = 195 \)[/tex], number of successes [tex]\( x = 162 \),[/tex] and a confidence level of 95%.

The formula for the confidence interval for a population proportion [tex]\( p \)[/tex] is:

[tex]\[ \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \][/tex]

where:

- [tex]\( \hat{p} \)[/tex] is the sample proportion [tex](\( \frac{x}{n} \)),[/tex]

- [tex]\( z^* \)[/tex] is the critical value from the standard normal distribution corresponding to the desired confidence level.

Calculate the sample proportion [tex]\( \hat{p} \):[/tex]

[tex]\[ \hat{p} = \frac{x}{n} = \frac{162}{195} \][/tex]

[tex]\[ \hat{p} \approx 0.8308 \][/tex]

For a 95% confidence level, the critical value [tex]\( z^* \)[/tex] can be found using the standard normal distribution table or a calculator. It corresponds to the middle 95% of the distribution, which leaves 2.5% in each tail.

The critical value [tex]\( z^* \)[/tex] for a 95% confidence level is approximately 1.96.

Calculate the standard error [tex]\( SE \):[/tex]

[tex]\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\ SE = \sqrt{\frac{0.8308 \cdot (1-0.8308)}{195}} \\ SE \approx \sqrt{\frac{0.8308 \cdot 0.1692}{195}} \\ SE \approx \sqrt{\frac{0.1405}{195}} \\ SE \approx \sqrt{0.0007205} \\ SE \approx 0.0268 \][/tex]

Now, we can construct the 95% confidence interval for [tex]\( p \):[/tex]

[tex]\[ \hat{p} \pm z^* \cdot SE \][/tex]

[tex]\[ 0.8308 \pm 1.96 \cdot 0.0268 \][/tex]

Calculate the margin of error:

[tex]\[ 1.96 \cdot 0.0268 \approx 0.0525 \][/tex]

So, the confidence interval is:

[tex]\[ 0.8308 \pm 0.0525 \][/tex]

Finalize the interval: [tex]\[ (0.7783, 0.8833) \][/tex]

The 95% confidence interval for the population proportion [tex]\( p \)[/tex] is approximately [tex]\( (0.7783, 0.8833) \)[/tex]. This means we are 95% confident that the true population proportion [tex]\( p \)[/tex] lies between 0.7783 and 0.8833.

To minimize unit cost, 160 machines must be made.

Step-by-step explanation:

Find the first derivative of the cost function:  C'(x) = 1.6x - 256.

Set the derivative equal to 0 and solve for x to find the critical point: 1.6x - 256 = 0. x = 160.

Check the nature of the critical point using the second derivative test to confirm that x = 160 gives the minimum cost.

Therefore, the number of machines that must be made to minimize the unit cost is 160 machines.

The number of machines that must be made to minimize the unit cost is 160.

To find the number of machines that must be made to minimize the unit cost, we need to find the minimum point of the function [tex]\(C(x) = 0.8x^2 - 256x + 25,939\).[/tex]

The function \(C(x)\) represents a quadratic equation, and the vertex of a quadratic equation represents its minimum or maximum point. The x-coordinate of the vertex of a quadratic function in the form [tex]\(ax^2 + bx + c\) is given by \(-\frac{b}{2a}\).[/tex]

[tex]For the function \(C(x) = 0.8x^2 - 256x + 25,939\), we have \(a = 0.8\) and \(b = -256\).Now, let's calculate the x-coordinate of the vertex:\[ x_{\text{vertex}} = -\frac{b}{2a} = -\frac{-256}{2 \times 0.8} = -\frac{-256}{1.6} = 160 \][/tex]

So, the number of machines that must be made to minimize the unit cost is 160.

Answer:

  (x +5)

Step-by-step explanation:

The problem statement is telling you that one factor of (x⁴ +5x³ -3x -15) is (x³ -3). It is asking for the other factor. Clearly, you can find the other factor by dividing the polynomial by the given factor.

That is ...

  (x⁴ +5x³ -3x -15) / (x³ -3) = (x +5)

so ...

  (x⁴ +5x³ -3x -15) / (x +5) = (x³ -3)

The divisor of interest is (x +5).

Answer:

(x+5)

The answer is c.

Final answer:

The algebraic expression for finding the average melting points of helium, hydrogen, and neon, using variables h, j, and k as their respective melting points, is (h + j + k) / 3.

Explanation:

The question asks for the algebraic expression that represents the average melting points of helium, hydrogen, and neon. The variables h, j, and k denote the individual melting points of these elements, respectively. To calculate the average melting point, you would add the melting points of each element and divide by the number of elements.

The algebraic expression for the average melting point is:

(h + j + k) / 3

multiply 6400 x 6%

6% = 0.06

6400 x 0.06 = 384

 her commission was $384

Answer:

The commission amount of the salesperson is $384.

Step-by-step explanation:

A salesperson sold a total of $6,400.00.

The rate of commission is 6% or 0.06. Commissions are based on sales. These are some percentage of the sales amount.

So, here the amount will be = [tex]0.06\times6400=384[/tex] dollars

So, the commission amount of the salesperson is $384.

16/7 = 12/y

84/16

84/16 = 5.25

5.25+7 = 12.25

 x = 12.25

Answer:

So the answer would be No fine is paid if the laptop is returned exactly at the time at which it is due

Answer: 1800°

Step-by-step explanation: In this problem, we're given that a polygon has 12 sides and we're asked find the sum of the measures of its interior angles.

The formula for finding the sum of the measures of the interior angles of a polygon is 180 (n - 2) where n represents the number of sides.

So here, since our polygon has 12 sides, we can plug a 12 in for the n in our formula and we have 180 (12 - 2) which is our equation.

Simplifying inside the parentheses first, 12 - 2 is 10 so we have 180 (10) which is 1800.

So if a polygon has 12 sides, then the sum of the measures of its interior angles is 1800°.

we know that

The x-intercept is the value of the coordinate x when the value of the function is equal to zero

so

In this problem we have that the x-intercepts of the graphs are the points

[tex](-2,0)\\(-1,0)\\(1,0)\\(2,0)[/tex]

therefore

the answer is the option

B)-1,0

The rational zero -1 is a root of f(x). Synthetic division yields [tex]\(x^3 + 2x^2 - 7x - 2\)[/tex]. Further factorization or testing other rational roots finds the remaining zeros.

To find a rational zero of the polynomial function [tex]\(f(x) = x^4 + 3x^3 - 5x^2 - 9x - 2\)[/tex], we can use the Rational Root Theorem. According to this theorem, any rational zero of the polynomial function must be of the form ±p/q, where p is a factor of the constant term (-2 in this case) and q is a factor of the leading coefficient (1 in this case).

The factors of -2 are ±1, ±2, and the factors of 1 are ±1. Therefore, the possible rational zeros are:

±1, ±2

We can try these values to see if they are roots of the polynomial.

Let's start by trying x = 1:

[tex]\[f(1) = (1)^4 + 3(1)^3 - 5(1)^2 - 9(1) - 2\]\[= 1 + 3 - 5 - 9 - 2\]\[= -12\][/tex]

So, x = 1 is not a root.

Next, let's try x = -1:

[tex]\[f(-1) = (-1)^4 + 3(-1)^3 - 5(-1)^2 - 9(-1) - 2\]\[= 1 - 3 - 5 + 9 - 2\]\[= 0\][/tex]

Therefore, x = -1 is a root of the polynomial.

To find the other zeros, we can perform polynomial division or synthetic division by dividing f(x) by (x + 1). Let's use synthetic division:

-1       1      3       -5       -9      -2  

         1      2       -7       -2      ↓

The result is [tex]\(x^3 + 2x^2 - 7x - 2\)[/tex]. Now, we can factor this cubic polynomial or continue using the Rational Root Theorem to find additional roots. Let's try x = 1 again:

[tex]\[f(1) = (1)^3 + 2(1)^2 - 7(1) - 2\]\[= 1 + 2 - 7 - 2\]\[= -6\][/tex]

x = 1 is not a root, so we continue to try the other possible rational zeros. However, to save time, let's check if any of the values of [tex]\(x = \pm 2\)[/tex] are roots using synthetic division:

For x = 2:

2     1      2        -7     -2

      1      4          1      ↓

For \(x = -2\):

-2     1      2      -7       -2

         1     0      -7        ↓

Since none of these values result in a remainder of 0, [tex]\(x = \pm 2\)[/tex] are not roots.

Therefore, the zeros of the polynomial function [tex]\(f(x) = x^4 + 3x^3 - 5x^2 - 9x - 2\) are \(x = -1\),[/tex] and the other zeros can be found by further factoring the reduced cubic polynomial.

To prove that there are no integers m and n that satisfy 2m + 4n = 7, we can assume the opposite and show that it leads to a contradiction. We can rearrange the equation and analyze the parity of the terms to prove there are no integer solutions.

To prove that there does not exist integers m and n such that 2m + 4n = 7, we can start by assuming that such integers do exist. Let's suppose m and n are integers that satisfy the equation.

Rearranging the equation, we have 2m = 7 - 4n. This means that 2m is an even number and 7 - 4n is an odd number. However, there is no way for an even number and an odd number to be equal. Therefore, our assumption was incorrect, and there are no integers m and n that satisfy the equation 2m + 4n = 7.

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There was 3 answers.

Answer one is It has 4 zeros and at most 3 relative maximums or minimums.
Answer two is It is a translation of the percent function 4 units to the right.
Answer three is Both ends of the graph of the function go up.

:)

it is a transition of the parent function 4 units to the right, it has 4 zeros and at most 3 relative maximums and minimums, both ends of the graph of the function go up this is for apex

David must deposit approximately $16,150.01 at the end of each year to accumulate $1 million by the end of 20 years at a 10 percent interest rate.

To calculate the equal annual end-of-year deposits that David must make to accumulate $1 million in 20 years at a 10 percent interest rate, we can use the formula for the future value of an ordinary annuity.

The formula for the future value of an ordinary annuity is given by:

[tex]FV = P * ((1 + r)^n - 1) / r[/tex]

where:

FV is the future value of the annuity (the desired $1 million in this case)

P is the annual deposit (what we need to find)

r is the annual interest rate (10% or 0.10 as a decimal)

n is the number of years (20 years in this case)

Substituting the known values:

[tex]$1,000,000 = P * ((1 + 0.10)^{20} - 1) / 0.10[/tex]

Now, we can solve for P:

$1,000,000 = P * (6.1917364224) / 0.10

$1,000,000 = P * 61.917364224

P = $1,000,000 / 61.917364224

P ≈ $16,150.01

So, David must deposit approximately $16,150.01 at the end of each year to accumulate $1 million by the end of 20 years at a 10 percent interest rate.

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Answer:

Given the statement:

Lines a and b are parallel and lines e and f are parallel.

if [tex]m \angle 1 = 89^{\circ}[/tex]

By supplementary angles:

[tex]m\angle 1+ m\angle 2 = 180^{\circ}[/tex]

⇒[tex]89^{\circ}+ m\angle 2 = 180^{\circ}[/tex]

Subtract 89 degree from both sides we have;

[tex]m \angle 2 = 91^{\circ}[/tex]

Since,

m∠4 = m∠5        [Vertically Opposite angles]           .....[1]

m∠4 = m∠3          [Alternate Interior angle]              .....[2]

By [1] and [2] we have;

m∠5  =m∠3                                   ....[3]

Also;

m∠2 = m∠3       [Alternate interior angle]                 ....[4]

by [3] and [4] we have;

m∠5  = m∠2

Substitute the given values we have;

[tex]m \angle 5 = 91^{\circ}[/tex]

Therefore, the measure of [tex]m \angle 5[/tex] is, [tex]91^{\circ}[/tex]

Answer:

x = 16.

Step-by-step explanation:

Given : Transverse line b and parallel line e and f.

To find : What is the value of x.

Solution : We have given Transverse line b and parallel line e and f.

Corresponding angles : When two lines are crossed by another line the angles in matching corners are called corresponding angles.

corresponding angles are always equal.

2x + 18 = 4x - 14.

On subtracting both sides by 4x

2x -4x + 18 = -14.

- 2x + 18 = - 14 .

On subtracting both sides by 18

- 2x = - 14 -18 .

- 2x = - 32 .

On dividing both sides by -2 .

x = 16.

Therefore, x = 16.

Answer:

Step-by-step explanation:

Given that A cone is placed inside a cylinder. The cone has half the radius of the cylinder, but the height of each figure is the same

Whatever position cone is placed, the space remaining will have volume as

volume of the cylinder - volume of the cone

Let radius of cylinder be r and height be h

Then volume of  cylinder  = [tex]\pi r^2 h[/tex]

The cone has height as h and radius as r/2

So volume of cone = [tex]\frac{1}{3} \pi (\frac{r}{2} )^2h\\=(\pi r^2 h)\frac{1}{24}[/tex]

the volume of the space remaining in the cylinder after the cone is placed inside it

=[tex]\pi r^2 h (1-\frac{1}{24} )\\=\frac{23 \pi r^2 h}{24}[/tex]

Answer:

11/12 pie r^2 h

Step-by-step explanation:

The correct answer to this is that:

Any irrational number when added to 2 / 5 still produces an irrational number.

For example, if we use π to add to 2/5 or 0.4. As far as we know the decimal digits for π just go on forever and do not have a repeating cycle hence making it an irrational number. Adding a rational number such as 0.4 to the value of π does not really greatly change the value of π. The decimal digits (hundredths place and so on) of the resulting number will still go on forever without a continual repeat.

So 0.4 + π is still irrational.

Answer:

5

Step-by-step explanation: