Based on the figure, select all true equations

Mathematics

1 answer:

Lady bird [3.3K]3 years ago

3 0

Answer:

a) no

b) true

c) true

d) no

e) true

f) true

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The back of Jake's property is a creek. Jake would like to enclose a rectangular area, using the creek as one side and fencing f

vekshin1

Answer:

125000 square feet

Step-by-step explanation:

Since there are only three sides of the rectangle, the perimeter of the fence is:

Let x and y be the sides of the rectangle, we are left with:

2 * x + y = 1000

solving for and:

y = 1000 - 2 * x

The area of the corral is:

A = x * y

replacing

A = x * (1000 - 2*x)

A = 1000 * x - 2*x^2

to find the maximum for the parabolic function A = 1000 * x - 2*x^2

The function has a maximum since the quotient before x ^ 2 is negative: -2 <0

Amax = c - b^2 /4*a

where a = -2, b = 1000, c = 0

A max = 0 - 1000^2/(4 * (- 2))

A max = 125000 ft^2

The maximum possible area of the pen is 125000 square feet.

4 0

4 years ago

Solve the equation 2 sec² x = 3 - tan x for the domain 0° ≤ x ≤ 360°

vodka [1.7K]

$2 \sec^2 x = 3- \tan x \\\\\implies 2(1 +\tan^2 x) = 3- \tan x \\\\\implies 2 + 2 \tan^2 x +\tan x -3 =0\\\\\implies 2 \tan^2 x + \tan x -1 =0\\\\\implies 2u^2 + u -1 =0~~~ ;[\text{set} \tan x = u]\\\\\implies u = \dfrac{-1\pm \sqrt{1-4\cdot 2 \cdot (-1)}}{2(2)}\\\\\implies u =\dfrac{-1 \pm\sqrt{9}}{4}\\\\\implies u = \dfrac{-1 \pm 3}{4}\\\\\implies u = \dfrac{2}{4}=\dfrac 12~~ \text{or} ~~u =\dfrac{-4}{4} =-1\\\\$

$\implies \tan x = \dfrac 12 ~~ \text{or} ~~ \tan x =-1~~~ ;[\text{Substitute back u =tan x}]\\\\\text{Now,}~ \\\\\tan x = -1,\\\\\implies x = n\pi - \dfrac{\pi}4\\\\\\\text{For interval,}~ [0,2\pi) \\\\x = \dfrac{3\pi}4, \dfrac{7\pi}4\\\\\text{In degrees,}~ x = 135^{\circ}, x =315^{\circ}\\\\\tan x = \dfrac 12\\\\\implies x = n\pi + \tan^{-1} \left(\dfrac 12 \right)\\\\\text{For interval} ~[0,2\pi),\\\\$

$x=\tan^{-1} \left(\dfrac 12 \right), \left[\pi +\tan^{-1} \left( \dfrac 12 \right)\right]\\\\\text{in degrees,} ~~x=\tan^{-1} \left(\dfrac 12 \right), \left[180^{\circ} +\tan^{-1} \left( \dfrac 12 \right)\right]\\\\\\\\\text{Combine all solutions:}\\\\x=\dfrac{3\pi}4, \dfrac{7\pi}4, \tan^{-1} \left(\dfrac 12 \right), \left[\pi +\tan^{-1} \left( \dfrac 12 \right)\right]\\\\\\$