Mathematics, 02.01.2022 14:00, brandoncarter2002

Write an equation of the line passing through the point (5, 1) that is perpendicular to the line 5x+3y=15. An equation of the line is y= x+ .

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answered: ogpuhl

$\displaystyle y = \frac{3}{5}\, x + (-2)$ .

Step-by-step explanation:

The slope of a line in a plane would be if the equation of that line could be written in the slope-intercept form $y = m\, x + b$ for some constant .

Find the slope of the given line by rearranging its equation into the slope-intercept form.

$3\, y = (-5)\, x + 15$ .

$\displaystyle y = \frac{(-5)}{3} \, x + 5$ .

Thus, the slope of the given line would be (-5) / 3 .

Two lines in a plane are perpendicular to one another if and only if the product of their slopes is (-1) .

Let $m_{1}$ and $m_{2}$ denote the slope of the given line and the slope of the line in question, respectively.

Since the two lines are perpendicular to each other, $m_{1}\, m_{2} = (-1)$ . Apply the fact that the slope of the given line is $m_{1} = (-5) / 3$ and solve for $m_{2}$ , the slope of the line in question.

$\begin{aligned}m_{2} &= \frac{(-1)}{m_{1}} \\ &= \frac{(-1)}{(-5) / 3} = \frac{3}{5}\end{aligned}$ .

In other words, the slope of the line perpendicular to $5\, x + 3\, y = 15$ would be (3 / 5) .

If the slope of a line in a plane is , and that line goes through the point $(x_{0},\, y_{0})$ , the equation of that line in point-slope form would be:

$(y - y_{0}) = m\, (x - x_{0})$ .

Since the slope of the line in question is (3 / 5) and that line goes through the point $(5,\, 1)$ , the equation of that line in point-slope form would be:

$\begin{aligned} (y - 1) = \frac{3}{5}\, (x - 5) \end{aligned}$ .

Rearrange this equation as the question requested:

$\begin{aligned} y - 1 = \frac{3}{5}\, x - 3 \end{aligned}$ .

$\begin{aligned} y = \frac{3}{5}\, x - 2 \end{aligned}$ .

$\begin{aligned} y = \frac{3}{5}\, x + (-2) \end{aligned}$ .

answered: Guest

part a

answer: a reasonable domain is $0 \le n \le 10$ where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

=====================================

part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

=====================================

part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).

answered: Guest

the simplified form is

$\frac{x^2(2x+3)}{2(4x-3)}$