Mathematics, 04.11.2020 19:00 chris3487

Find the equation of the line through the point (2, 5) that cuts off the least area from the first quadrant.

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The equation of the line through the point (2, 5) that cuts off the least area from the first quadrant is $y = -2.321\cdot x +9.642$ .

Step-by-step explanation:

From Analytical Geometry we get that the equation of the line is represented by:

$y = m\cdot x + b$ (Eq. 1)

Where:

- Independent variable, dimensionless.

- Dependent variable, dimensionless.

- Slope, dimensionless.

- y-Intercept, dimensionless.

From (Eq. 1) we find that intercepts are:

x-Intercept

$x_{i} = -\frac{b}{m}$

y-Intercept

$y_{i} = b$

And the area done by the line in the first quadrant is:

$A = \frac{1}{2}\cdot x_{i}\cdot y_{i}$

$A = \frac{1}{2}\cdot \left(-\frac{b}{m} \right) \cdot b$

$A = -\frac{b^{2}}{2\cdot m}$ (Eq. 2)

If we know that (x, y) = (2,5) , then the equation of the line is reduced into this:

$2\cdot m + b = 5$

$b = 5-2\cdot m$ (Eq. 3)

And we apply (Eq. 3) in (Eq. 2):

$A = -\frac{(5-2\cdot m)^{2}}{2\cdot m}$

$A = -\frac{25-20\cdot m +4\cdot m^{2}}{2\cdot m}$

$A = -\frac{25}{2}\cdot m^{-2}-10+2\cdot m$ (Eq. 4)

The first and second derivatives are, respectively:

$A' = 25\cdot m^{-3}+2$ (Eq. 5)

$A'' = -75\cdot m^{-4}$ (Eq. 6)

Then the first derivative is equalized to zero and solved:

$25\cdot m^{-3}+2=0$

$25\cdot m^{-3} = -2$

$25 = -2\cdot m^{3}$

$m^{3} = -\frac{25}{2}$

m = -2.321

And the second derivative is evaluated at critical value:

$A'' = -75\cdot (-2.321)^{-4}$

A'' = -2.584

The critical value is associated with a positive area. Then, the y-intercept is: ( m = -2.321 )

$b = 5-2\cdot (-2.321)$

b = 9.642

Therefore, the equation of the line through the point (2, 5) that cuts off the least area from the first quadrant is $y = -2.321\cdot x +9.642$ .

Find the equation of the line through the point (2, 5) that cuts off the least area from the first q

Answer from: Quest

a) y = -3x +2

b) y = 1/2x -6

these 2 can be combined into one equation.

-3x +2 = .5x -6

8 = 3.5x

x = 8 / 3.5 = 2.2857142857

y = -3 * (8 / 3.5) +2

y = (-24 / 3.5) + 2

y = -4.8571428571

step-by-step explanation:

Answer from: Quest

2 & 7

step-by-step explanation:

he climbs 300 ft/hr at a constant rate. the peak is at 1,500 ft.

when going up, he reaches 600 ft in 2 hours. remember, that he is going in a constant rate. divide 600 with 300:

600/300 = 2

assuming that he takes the same route back (meaning that it's a 1500 ft route back, you can solve.

note also that you are solving when he gets back to 600 ft after he reaches the peak.

first, find the amount of hours it takes to get up the peak. divide the total amount (1500), with the ft/her (300) to get the amount of hours (x):

1500/300 = x

x = 5

it takes jamie 5 hours to get up. now, he is returning down, and reaches 600 (his current destination). remember that he climbs at a constant 300 ft/hr. divide 600 with 300:

600/300 = x

x = 2

now add the two numbers together.

x = 5 + 2 = 7

jamie takes 7 hours to go up the mountain and reach his current position when descending.

(total amount/constant rate) = (time).

remember that it is a constant rate, so in total, when he gets back down the mountain, it will take the same amount of time as when he goes up. the total amount for the round trip is: 5 + 5 = 10.

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