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2 years ago

Line m passes through point (5,- 8) and has a slope - 3. What is the equation for the line point slope form

Mathematics

1 answer:

skad [1K]2 years ago

7 0

Answer:

$y + 8 = -3(x-5)$

Step-by-step explanation:

Point Slope form is:

$y - y_1 = m(x-x_1)\\\rule{150}{0.5}\\\text{ m - slope}\\(x_1,y_1) \text{ - point}\\\rule{150}{0.5}$

We are given the point (5, -8) and the slope of -3.

$y-y_1=m(x-x_1)\rightarrow y - (-8) - -3(x-5)\rightarrow \boxed{y+8=-3(x-5)}$

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Consider the initial value problem y′+5y=⎧⎩⎨⎪⎪0110 if 0≤t<3 if 3≤t<5 if 5≤t<[infinity],y(0)=4. y′+5y={0 if 0≤t<311 i

rosijanka [135]

It looks like the ODE is

$y'+5y=\begin{cases}0&\text{for }0\le t$

with the initial condition of $y(0)=4$ .

Rewrite the right side in terms of the unit step function,

$u(t-c)=\begin{cases}1&\text{for }t\ge c\\0&\text{for }t$

In this case, we have

$\begin{cases}0&\text{for }0\le t$

The Laplace transform of the step function is easy to compute:

$\displaystyle\int_0^\infty u(t-c)e^{-st}\,\mathrm dt=\int_c^\infty e^{-st}\,\mathrm dt=\frac{e^{-cs}}s$

So, taking the Laplace transform of both sides of the ODE, we get

$sY(s)-y(0)+5Y(s)=\dfrac{e^{-3s}-e^{-5s}}s$

Solve for $Y(s)$ :

$(s+5)Y(s)-4=\dfrac{e^{-3s}-e^{-5s}}s\implies Y(s)=\dfrac{e^{-3s}-e^{-5s}}{s(s+5)}+\dfrac4{s+5}$

We can split the first term into partial fractions:

$\dfrac1{s(s+5)}=\dfrac as+\dfrac b{s+5}\implies1=a(s+5)+bs$

If $s=0$ , then $1=5a\implies a=\frac15$ .

If $s=-5$ , then $1=-5b\implies b=-\frac15$ .

$\implies Y(s)=\dfrac{e^{-3s}-e^{-5s}}5\left(\frac1s-\frac1{s+5}\right)+\dfrac4{s+5}$

$\implies Y(s)=\dfrac15\left(\dfrac{e^{-3s}}s-\dfrac{e^{-3s}}{s+5}-\dfrac{e^{-5s}}s+\dfrac{e^{-5s}}{s+5}\right)+\dfrac4{s+5}$

Take the inverse transform of both sides, recalling that

$Y(s)=e^{-cs}F(s)\implies y(t)=u(t-c)f(t-c)$

where $F(s)$ is the Laplace transform of the function $f(t)$ . We have

$F(s)=\dfrac1s\implies f(t)=1$

$F(s)=\dfrac1{s+5}\implies f(t)=e^{-5t}$

We then end up with

$y(t)=\dfrac{u(t-3)(1-e^{-5t})-u(t-5)(1-e^{-5t})}5+5e^{-5t}$

3 0

3 years ago

f(x) = −16x2 + 24x + 16 Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 points) WILL GIVE BRAINLIEST

castortr0y [4]

Answer:

Step-by-step explanation:

f(x) = −16x2 + 24x + 16 = 0 can be reduced to -2x^2 + 3x + 2 = 0 and then to 2x^2 - 3x - 2 = 0. Solve this for x using the quadratic formula:

The discriminant is b^2 - 4ac = (-3)^2 - 4(2)(-2) = 9 + 16 = 25.

Therefore the roots are:

-(-3) ± √25 3 ± 5

x = ----------------- = ------------ => x = 2 and x = -1/2

2(2) 4

The x-intercepts are points: (2, 0) and (-1/2, 0)

Because the coefficient of the x^2 term is negative, this graph opens down and the vertex represents a maximum.

To graph this function, find and plot the vertex. It is exactly halfway between the x-intercepts, that is, at x = 1 1/4, for which the y value is 21:

vertex and maximum at (5/4, 21)

Finally, find the y-intercept. Let x = 0; we find that y = 16. The y-intercept is 0, 16)

We now have four points on the graph and know where the maximum is. Plot this max (5/4, 21) and the x-intercepts (2, 0) and (-1/2, 0), and finally the y-intercept. Draw a smooth curve through these points, remembering that the graph is symmetrical about the axis of symmetry x = 5/4.

3 0

2 years ago

Write the equation of the line that passes through the points (8,-1) and (2,-5) in standard from, given that the point-slope fo

Gnesinka [82]

Answer: 2/3x-y=19/3

Step-by-step explanation:

Points are useless since we already know the point slope form and we can just simply that

y+1=2/3(x-8)

y+1=2/3x-16/3

y=2/3x-19/3

2/3x-y=19/3

4 0

2 years ago

(10 points if you answer) A recipe has a ratio of 2 cups of soy sauce to 3 cups of light soy sauce. How many cups of dark soy sa

tekilochka [14]

Answer:

5 cups of dark soy sauce

Step-by-step explanation:

2 + 3 = 5

So if there is 1 cup of dark soy sauce for every cup of soy sauce then it 5.

4 0

2 years ago

A pilot can see the runway she is about to land on directly ahead. when looking at each end of the runway the angles of depressi

pogonyaev

We will use trigonometry:
tan 45° = 1500 / x
1 = 1500 / x
x = 1500 ft
tan 42° = 1500 / y
0.9 = 1500 / y
y = 1500 : 0.9
y = 1666.67 ft
L = y - x ( where L is the length of the runway )
L = 1666.67 - 1500 = 166.67 ft.

8 0

3 years ago