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Anarel [89]
3 years ago
10

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

ing the form below. y − A = B(x − C)

Mathematics
2 answers:
taurus [48]3 years ago
5 0

y - 5 = - ⁵/₂ (x - 2)

<h3>Further explanation</h3>

This case asking the end result in the form of a point-slope form. Before that what has to be done is the differential process for the stationary state.

<u>Step-1:</u> prepare the point-slope form

\boxed{ \ y - y_1 = m(x - x_1) \ }

The equation of the line goes through points (2, 5) and has a gradient m.

\boxed{ \ y - 5 = m(x - 2) \ }

<u>Step-2:</u> the x-intercept and the y-intercept

For y = 0, we get the x-intercept.

\boxed{ \ 0 - 5 = m(x - 2) \ }

\boxed{ \ - 5 = mx - 2m \ }

\boxed{ \ mx = 2m - 5 \ }

Divide both sides by m.

Hence, \boxed{\boxed{ \ x = 2 - \frac{5}{m} \ }}

For x = 0, we get the y-intercept.

\boxed{ \ y - 5 = m(0 - 2) \ }

\boxed{ \ y - 5 = - 2m \ }

Add 5 form both sides.

Hence, \boxed{\boxed{ \ y = 5 - 2m \ }}

<u>Step-3:</u> prepare the area cut off in the first quadrant

We see a right triangle with:

  • \boxed{ \ base \ (x-intercept) = 2 - \frac{5}{m} \ }
  • \boxed{ \ height \ (y-intercept) = 5 - 2m \ }

Area cut off in the first quadrant = \boxed{ \ \frac{1}{2} \times base \times height \ }

\boxed{ \ A(m) = \frac{1}{2} \times (2 - \frac{5}{m}) \times (5 - 2m) \ }

\boxed{ \ A(m) = \frac{1}{2} \times (10 - 4m - \frac{25}{m} + 10) \ }

\boxed{ \ A(m) = \frac{1}{2} \times (20 - 4m - 25m^{-1}) \ }

Thus, \boxed{\boxed{ \ A(m) = 10 - 2m - \frac{25}{2}m^{-1} \ }}

<u>Step-4:</u> the stationary state for the least area

Let us differentiate for A(m).

\boxed{ \ A'(m) = -2 + \frac{25}{2}m^{-2} \ }

\boxed{ \ A'(m) = -2 + \frac{25}{2m^2} \ }

The stationary state for A(m).

\boxed{ \ A'(m) = 0 \ }

\boxed{ \ -2 + \frac{25}{2m^2} = 0 \ }

Multiply both sides by 2m².

\boxed{ \ -4m^2 + 25 = 0 \ }

\boxed{ \ -(4m^2 - 25) = 0 \ }

\boxed{ \ -(2m - 5)(2m + 5)) = 0 \ }

We get \boxed{m = \frac{5}{2}} and \boxed{m = -\frac{5}{2}}.

Because the line equation is in the first quadrant, with the line slopes downwards, this line has a negative gradien, i.e., \boxed{m = -\frac{5}{2}}.

<u>Final step:</u> the equation of the line in the point-slope form

Let us recall the equation in step-1 above.

\boxed{ \ y - 5 = m(x - 2) \ }

Substitute the value of m, so that the form y - A = B (x - C) is obtained, namely:

\boxed{\boxed{ \ y - 5 = -\frac{5}{2}(x - 2) \ }}

- - - - - - - - - -

<u>Notes:</u>

Let us find out the least area from the first quadrant.

\boxed{ \ m = -\frac{5}{2} \rightarrow A = 10 - 2\bigg(-\frac{5}{2}\bigg) - \frac{25}{2}\bigg(-\frac{2}{5}\bigg)} \ }

A = 10 + 5 + 5

Thus, the least area is 20 square units.

- - - - - - - - - -

The x-intercept

x = 2 - \frac{5}{-\frac{5}{2}} \rightarrow \boxed{ \ x = 4 \ }

The y-intercept

y = 5 - 2(-\frac{5}{2}) \rightarrow \boxed{ \ y = 10 \ }

<h3>Learn more</h3>
  1. Finding the equation, in slope-intercept form, of the line that is parallel to the given line and passes through a point brainly.com/question/1473992
  2. Determine the equation represents Nolan’s line  brainly.com/question/2657284  
  3. The derivatives of the composite function  brainly.com/question/6013189#

Keywords: the equation of the line, through the point, (2, 5), cuts off, the least area, the first quadrant, y − A = B(x − C), point-slop, gradient, slope

Amanda [17]3 years ago
4 0

The equation of the line would be

y = mx+b 

where m is the slope, b is the y intercept. 

<span>The line will form a triangular region in the first quadrant. Its area would be 1/2 base times height. The height is the y intercept and the base is y intercept divided by slope. Therefore,</span>

A = b^2/2m

At point (2,5)

5 = 2m+b

Substitute that in the area

A = b^2/5-b to find the least area, differentiate the area with respect to the height and equate it to 0 

dA/db = 0

<span>find b and use that to find m. Then, you can have the equation of the line.</span>


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