What is the slope of this line? Enter your answer as a fraction in simplest term in
the box.

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Answer 1

Answer:

The slope of the line is 1/4.

What is slope?

A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively. Therefore, it is possible to write the change in y coordinate with respect to the change in x coordinate as,

m = Δy/Δx

where, m is the slope

Given points:

(0, 6) and (8, 8)

Now, slope = $(y_2 - y_1)/ ( x_2- x_1)$

slope= ( 8- 6)/ (8-0)

slope = 2/8

slope= 1/4

Hence, the slope of the line is 1/4.

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Answer 2

Answer:

Answer:

$slope=(1)/(4)$

Step-by-step explanation:

The slope of a line can be seen as:

$(rise)/(run)$

Rise over run is the change in the y values over the change in x values. For example, in this graph, you would start on one of the points given. From there, you would move up first. After moving up a certain number of spaces, you would move to the side until you reach the other point.

In the graph, you would move up until you are in line with one of the other points. Starting at (-4,5), move up one space, then to the left 4 spaces to reach the point (0,6). Using the spaces moved in the rise over run:

$(1)/(4) =(rise)/(run)=slope$

Therefore, the slope is $(1)/(4)$ .

This is true for any two points on the line.

:Done

*When you move up, the number will be positive $((+y)/(x) )$ . If you move down, the number will be negative $((-y)/(x) )$ . If you move left, the number will be positive $((y)/(+x))$ . If you move right, the number will be negative $((y)/(-x) )$ . Keep this in mind. It is very important.

**Always move along the y-axis first, then move along the x-axis. If you do it the other way, the slope will be wrong.

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For the characteristic polynomialp(s) =s5+ 2s4+ 24s3+ 48s2−25s−50(a) Use the Routh-Hurwitz Criterion to determine the number of roots ofp(s) in the right-half plane, in the left-half plane, and on thejω-axis.(b) Use Matlab to determine the roots ofp(s), and verify your results in part 2a.

Answers

Answer:

1 root in the right half-plane
1 conjugate pair on the imaginary axis
2 roots in the left half-plane

Step-by-step explanation:

Without using the Routh-Hurwitz criterion at all, you know there is one positive real root. Descartes' rule of signs tells you the number of positive real roots is equal to the number of sign changes in the coefficients (perhaps less a multiple of 2). There is one sign change in + + + + - - , so there is one positive real root.

_____

(a) The Routh array starts as two rows of the polynomial's coefficients, alternate coefficients on each row. For this odd-degree polynomial, the number of coefficients is even, so no zero-padding is necessary at the right end of the second row. That is, we start with ...

$\begin{array}{cccc}s^5&1&24&-25\ns^4&2&48&-50\end{array}$

The next row is formed from combinations of coefficients in the two rows above. The computation is similar to that of a determinant. By matching the numbers to those in the array, you can see the pattern of the computation.

The next row values are ...

$\begin{array}{ccc}s^3&((2)(24)-(1)(48))/(2)&((2)(-25)-(1)(-50))/(2)\end{array}$

Simplifying, we find this row to be ...

$\begin{array}{ccc}s^3&0&0\end{array}$

The zero row is a special case that requires we proceed as follows. The row above (identified with s⁴) represents an "auxiliary polynomial":

$2s^4 +48s^2 -50$

To continue the process, we replace the zero row by the coefficients of the derivative of this auxiliary polynomial. Proceeding as before, the array now becomes ...

$\begin{array}{cccc}s^5&1&24&-25\ns^4&2&48&-50\ns^3&8&96\ns^2&24&-50\ns^1&112(2)/(3)&0\ns^0&-50\end{array}$

The number of sign changes in the first column (1) tells the number of roots in the right half-plane. The auxiliary polynomial will give us the remaining two pairs of roots:

$2s^4+48s^2-50=0\n\n2(s^2+25)(s^2-1)=0\n\ns=\pm 5i,\ s=\pm 1$

So, we have determined there to be ...

1 root in the right half-plane
2 roots on the jω axis
2 roots in the left half-plane

(b) The original polynomial can be factored as ...

p(s) = (s +2)(s² +25)(s +1)(s -1)

p(s) = (s +2)(s +1)(s -5i)(s +5i)(s -1)

This verifies our result from part (a).

_____

Additional comments

Any row can be multiplied by a convenient factor to simplify the arithmetic. Here, it would be convenient to divide the second row by 2 and the third row by 8.

A zero element (not row) in the first column is replaced by "epsilon" (a small positive number) and the rest of the arithmetic is continued as normal. That row is not counted (it is ignored) when counting sign changes in the first column.

a plane ticket cost b dollars for an adult ticket and k dollars for a child. Find an expression that represents the total cost for 9 adults and 3 children.

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Let C be the total cost. Then the required expression is:

C = 9b + 3k.

HELP!!! find the value of x

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Answer:

don't know

.....................

Answer:

x = 12

Step-by-step explanation:

Consider three orthogonal triangles (see picture)

1. The smallest triangle

2. The medium triangle

3. The big triangle that holds both triangles.

All are orthogonal (or right triangles) so you can use Pythagoras Theorem:

"The area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides"

Use Pythagoras theorem on triangle 1.

$a^(2) = 3^(2) + 6^(2) = 45$

Now user Pythagoras on triangle 2.

$b^(2) = x^(2) + 6^(2) = x^(2) + 36$

Now use Pythagoras on triangle 3 (the big triangle).

$(3 + x )^(2) = a^(2) + b^(2)\nx^(2) + 6x + 9 = a^(2) + b^(2)$

Now replace the values for $a^(2)$ and $b^(2)$ from the two equations derived from triangle 1 and 2:

$x^(2) + 6x + 9 = 45 + x^(2) + 36$

The two $x^(2)$ get cancelled out, so:

$6x + 9 = 45 + 36\nx = 12$

TA DA!

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) π/2 0 3 1 + cos(x) dx, n = 4

Answers

Split up the integration interval into 4 subintervals:

$\left[0,\frac\pi8\right],\left[\frac\pi8,\frac\pi4\right],\left[\frac\pi4,\frac{3\pi}8\right],\left[\frac{3\pi}8,\frac\pi2\right]$

The left and right endpoints of the $i$ -th subinterval, respectively, are

$\ell_i=\frac{i-1}4\left(\frac\pi2-0\right)=\frac{(i-1)\pi}8$

$r_i=\frac i4\left(\frac\pi2-0\right)=\frac{i\pi}8$

for $1\le i\le4$ , and the respective midpoints are

$m_i=\frac{\ell_i+r_i}2=\frac{(2i-1)\pi}8$

Trapezoidal rule

We approximate the (signed) area under the curve over each subinterval by

$T_i=\frac{f(\ell_i)+f(r_i)}2(\ell_i-r_i)$

so that

$\displaystyle\int_0^(\pi/2)\frac3{1+\cos x}\,\mathrm dx\approx\sum_(i=1)^4T_i\approx\boxed{3.038078}$

Midpoint rule

We approximate the area for each subinterval by

$M_i=f(m_i)(\ell_i-r_i)$

so that

$\displaystyle\int_0^(\pi/2)\frac3{1+\cos x}\,\mathrm dx\approx\sum_(i=1)^4M_i\approx\boxed{2.981137}$

Simpson's rule

We first interpolate the integrand over each subinterval by a quadratic polynomial $p_i(x)$ , where

$p_i(x)=f(\ell_i)((x-m_i)(x-r_i))/((\ell_i-m_i)(\ell_i-r_i))+f(m)((x-\ell_i)(x-r_i))/((m_i-\ell_i)(m_i-r_i))+f(r_i)((x-\ell_i)(x-m_i))/((r_i-\ell_i)(r_i-m_i))$

so that

$\displaystyle\int_0^(\pi/2)\frac3{1+\cos x}\,\mathrm dx\approx\sum_(i=1)^4\int_(\ell_i)^(r_i)p_i(x)\,\mathrm dx$

It so happens that the integral of $p_i(x)$ reduces nicely to the form you're probably more familiar with,

$S_i=\displaystyle\int_(\ell_i)^(r_i)p_i(x)\,\mathrm dx=\frac{r_i-\ell_i}6(f(\ell_i)+4f(m_i)+f(r_i))$

Then the integral is approximately

$\displaystyle\int_0^(\pi/2)\frac3{1+\cos x}\,\mathrm dx\approx\sum_(i=1)^4S_i\approx\boxed{3.000117}$

Compare these to the actual value of the integral, 3. I've included plots of the approximations below.

Final answer:

The question is asking to approximate the definite integral of 1 + cos(x) from 0 to π/2 using the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule for n=4. These are numerical methods used for approximating integrals by estimating the area under the curve as simpler shapes.

Explanation:

This question asks to use several mathematical rules, specifically the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule, to approximate the given integral with a specified value of n which is 4. The integral given is the function 1 + cos(x) dx from 0 to π/2. Each of these rules are techniques for approximating the definite integral of a function. They work by estimating the region under the graph of the function and above the x-axis as a series of simpler shapes, such as trapezoids or parabolas, and then calculating the area of these shapes. The 'dx' component represents a small change in x, the variable of integration. The cosine function in this integral is a trigonometric function that oscillates between -1 and 1, mapping the unit circle to the x-axis. The exact solution would require calculus, but these numerical methods provide a close approximation.

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What is the system of checks and balances designed to ensure?

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Answer:

ensure that no single branch of government would have too much power

Step-by-step explanation:

Solve for y. y/6 = 3/9 simplify your answer as much as possible

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Answer: y=2

Step-by-step explanation:

y/6 = 3/9 multiply both sides by 6/1 (since 6/1 times y/6 equals 1y or just y)

y = 18/9 now that y is isolated, just simplify 18/9

y=2 18 divided by 9 is 2

What is the slope of this line? Enter your answer as a fraction in simplest term in the box.

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What is slope?

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Final answer:

Explanation:

Learn more about Numerical Integration Rules here:

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What is the slope of this line? Enter your answer as a fraction in simplest term in
the box.