Find the equation of the line through P=(8,7) such that the triangle bounded by this line and the axes in the first quadrant has the minimal area.
Answers
The area of the triangle is given by the product of 0.5 and the distances of the x and y intercepts from the origin.
[tex]\mathrm{The \ equation \ of \ the \ line \ is, }\displaystyle \ \underline{ y = 14 - \frac{7}{8} \cdot x}[/tex]Reasons:
The area of the triangle, A = 0.5·x·y
Where;
x and y are the values of the line at the intercepts
The equation of the line is; (y - 7) = m·(x - 8)
At the y-intercept, x = 0, therefore;
(y - 7) = m·(0 - 8) = -8·m
At the y-intercept, y = 7 - 8·m
At the x-intercept, y = 0, therefore;
(0 - 7) = m·(0 - 8) = -8·m
At the x-intercept, -7 = m·x - 8·m
8·m - 7 = m·x
[tex]\displaystyle x = \mathbf{8 - \frac{7}{m}}[/tex]
Therefore;
[tex]\displaystyle A = 0.5 \times \left(7 - 8 \cdot m\right) \times \left(8 - \frac{7}{m} \right) = \mathbf{\frac{-32 \cdot m^2 +56 \cdot m - 24.5}{m}}[/tex]
When the area is minimal, we have;
[tex]\displaystyle \frac{dA}{dm} =0 = \frac{d}{dm} \left(\frac{-32 \cdot m^2 +56 \cdot m - 24.5}{m}\right) = \mathbf{-\frac{32 \cdot m^2 - 24.5}{m^2}}[/tex]
m² × 0 = 24.5 - 32·m²
[tex]\displaystyle m^2 = \frac{24.5}{32} = \frac{49}{64}[/tex]
[tex]\displaystyle m = \sqrt{\frac{49}{64}} = \frac{7}{8}[/tex]
[tex]\displaystyle m = \pm \frac{7}{8}[/tex]
The equation of the line when [tex]\displaystyle m = + \frac{7}{8}[/tex] is therefore;
(y - 7) = m·(x - 8)
[tex]\displaystyle (y - 7) = \mathbf{\frac{7}{8} \cdot (x - 8)}[/tex]
[tex]\displaystyle (y - 7) = \frac{7}{8} \cdot (x - 8) = \frac{7}{8} \cdot x - \frac{7}{8} \times 8 = \frac{7}{8} \cdot x - 7[/tex]
[tex]\displaystyle y = \frac{7}{8} \cdot x - 7 + 7 = \frac{7}{8} \cdot x[/tex]
[tex]\displaystyle y = \mathbf{\frac{7}{8} \cdot x}[/tex]
The x and y intercept of the above line are 0
When [tex]\displaystyle m = - \frac{7}{8}[/tex], we have;
[tex]\displaystyle (y - 7) = -\frac{7}{8} \cdot (x - 8) = -\frac{7}{8} \cdot x + \frac{7}{8} \times 8 = -\frac{7}{8} \cdot x + 7[/tex]
Which gives;
[tex]\displaystyle y = \frac{7}{8} \cdot x + 7 + 7 = \frac{7}{8} \cdot x + 14[/tex]
[tex]\mathrm{The \ equation \ of \ the \ line \ is, }\displaystyle \ y = \mathbf{14 - \frac{7}{8} \cdot x}[/tex]
The equation of the line through P = (8, 7) such that the triangle bounded by the line and the axes in the first quadrant has minimal area is therefore;
[tex]\displaystyle \ \underline{ y = 14 - \frac{7}{8} \cdot x}[/tex]
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Answer:
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Answers
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Step-by-step explanation:
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[tex] \: \: \: \: \: \: [/tex]
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Answer:
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Step-by-step explanation:
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Answers
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[tex]\mathbf{P(t) = 12e^{-0.1507t}}[/tex]
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Substitute 36 for t in P(t).
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[tex]\mathbf{P(36) = 12e^{-0.1507 \times 36}}[/tex]
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Multiply
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Answers
Answer:
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Step-by-step explanation: I used the formula y2-y1 over x2-x1 will give you the answer hope this helped!
Answer:
[tex]\boxed {\boxed {\sf m=3}}[/tex]
Step-by-step explanation:
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Answers
Answer:
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Answers
Answer:
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Step-by-step explanation:
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The boundary line of the inequality y <1/3x+ 1 is represented by the graph C.
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Step-by-step explanation:
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More info
Step-by-step explanation:
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(0,5)
4 3
2
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-1
-2
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TO
de
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The slope is m
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Answers
Answer:
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Step-by-step explanation:
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Answer:
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