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valentinak56 [21]
3 years ago
12

Question #7

Mathematics
1 answer:
Sever21 [200]3 years ago
7 0

Answer:

The area is 56.7 square inches

Step-by-step explanation:

Given

See attachment for corner shelf

Required

The area

The corner shelf is a quadrant with radius;

r = 8.5in

The area of a quadrant is:

Area = \frac{\pi r^2}{4}

So, the area of the corner shelf is:

Area = \frac{\pi * 8.5^2}{4}

Area = \frac{\pi * 72.25}{4}

Area = \pi * 18.0625

Take \pi as 3.14

Area = 3.14 * 18.0625

Area = 56.7 -- approximated

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The general form of a solution of the differential equation is already provided for us:

y(x) = c_1 \textrm{e}^x + c_2\textrm{e}^{-x},

where c_1, c_2 \in \mathbb{R}. We now want to find a solution y such that y(-1)=3 and y'(-1)=-3. Therefore, all we need to do is find the constants c_1 and c_2 that satisfy the initial conditions. For the first condition, we have:y(-1)=3 \iff c_1 \textrm{e}^{-1} + c_2 \textrm{e}^{-(-1)} = 3 \iff c_1\textrm{e}^{-1} + c_2\textrm{e} = 3.

For the second condition, we need to find the derivative y' first. In this case, we have:

y'(x) = \left(c_1\textrm{e}^x + c_2\textrm{e}^{-x}\right)' = c_1\textrm{e}^x - c_2\textrm{e}^{-x}.

Therefore:

y'(-1) = -3 \iff c_1\textrm{e}^{-1} - c_2\textrm{e}^{-(-1)} = -3 \iff c_1\textrm{e}^{-1} - c_2\textrm{e} = -3.

This means that we must solve the following system of equations:

\begin{cases}c_1\textrm{e}^{-1} + c_2\textrm{e} = 3 \\ c_1\textrm{e}^{-1} - c_2\textrm{e} = -3\end{cases}.

If we add the equations above, we get:

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If we now substitute c_1 = 0 into either of the equations in the system, we get:

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which do correspond to the desired initial conditions.

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