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Mnenie [13.5K]
3 years ago
6

At Walmart, apples cost $0.50 each, and grapefruits cost $2 each. You want to spend exactly $15.

Mathematics
1 answer:
Papessa [141]3 years ago
6 0

Define the variables: a is the number of apples I can buy                                                                        g is the number of grapefruits I can buy.

Write the equation: 15 = 0.50a + 2g

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Answer:

(5x+2) = (7x-12)

=x=7

Step-by-step explanation:

the answer is 7

you gotta move the variables

collect like terms then divide

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Let f be the function defined by f(x) = e^(x) cos x.
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The average rate of change of f on the interval 0 ≤ x ≤ π is

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(b)

f(x) = e^{x} cos x \implies f'(x) = e^x \cos(x) - e^x \sin(x) \implies \\ \\&#10;f'\left(\frac{3\pi}{2} \right) = e^{3\pi/2} \cos(3\pi/2) - e^{3\pi/2} \sin(3\pi/2) \\ \\&#10;f'\left(\frac{3\pi}{2} \right) = 0 - e^{3\pi/2} (-1) = e^{3\pi/2}

The slope of the tangent line is e^{3\pi/2}.

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The absolute minimum value of f occurs at a critical point where f'(x) = 0 or at endpoints.

Solving f'(x) = 0

f'(x) = e^x \cos(x) - e^x \sin(x) \\ \\&#10;0 = e^x \big( \cos(x) - \sin(x)\big)

Use zero factor property to solve.

e^x \ \textgreater \  0\forall x \in \mathbb{R} so that factor will not generate solutions.
Set cos(x) - sin(x) = 0

\cos (x) - \sin (x) = 0 \\&#10;\cos(x) = \sin(x)

cos(x) = 0 when x = π/2, 3π/2, but x = π/2. 3π/2 are not solutions to the equation. Therefore, we are justified in dividing both sides by cos(x) to make tan(x):

\displaystyle\cos(x) = \sin(x) \implies 0 = \frac{\sin (x)}{\cos(x)} \implies 0 = \tan(x) \implies \\ \\&#10;x = \pi/4,\ 5\pi/4\ \forall\ x \in [0, 2\pi]

We check the values of f at the end points and these two critical numbers.

f(0) = e^1 \cos(0) = 1

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\displaystyle f(5\pi/4) = e^{5\pi/4} \cos(5\pi/4) = e^{5\pi/4}  \frac{-\sqrt{2}}{2} = -e^{\pi/4}  \frac{\sqrt{2}}{2}

f(2\pi) = e^{2\pi} \cos(2\pi) = e^{2\pi}

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The absolute minimum value of f <span>on the interval 0 ≤ x ≤ 2π is
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