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miskamm [114]
3 years ago
5

Find the area of the polygon. Figure ABCDE is shown. A is at 6, 0. B is at negative 4, 5. C is at 2, 5. D is at 2, 9. E is at 6,

9. 41 square units 44 square units 52 square units 56 square units

Mathematics
1 answer:
kondaur [170]3 years ago
7 0

This question is incomplete. The attached image was obtained online

Answer:

41 square units

Step-by-step explanation:

Looking at diagram that was attached, we can see that the Polygon s the combination of a square and a triangle.

In square CDEF we have:

The lengths of the side of the square= 4 units

Hence, the area of square is: length ²

Area of square= 4²

=16 square units

Also, in ΔBFA we have:

The base of the triangle is calculated as

=6 + 4= 10 units

height of the triangle is: FA= 5 units

The formula for Area of a triangle = 1/2 × Base × Height

Hence, we have area of ΔBFA as:

= 1/2 × 10 × 5

Area= 25 square units

The Area of the Polygon = Area of the square + Area of the triangle

16 square units + 25 square units

= 41 square units

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Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x2 1 y2 − 4 and the plane x 1 y 1
charle [14.2K]

<u>Answer-</u> Length of the curve of intersection is 13.5191 sq.units

<u>Solution-</u>

As the equation of the cylinder is in rectangular for, so we have to convert it into parametric form with

x = cos t, y = 2 sin t   (∵ 4x² + y² = 4 ⇒ 4cos²t + 4sin²t = 4, then it will satisfy the equation)

Then, substituting these values in the plane equation to get the z parameter,

cos t + 2sin t + z = 2

⇒ z = 2 - cos t - 2sin t

∴ \frac{dx}{dt} = -\sin t

  \frac{dy}{dt} = 2 \cos t

  \frac{dz}{dt} = \sin t-2cos t

As it is a full revolution around the original cylinder is from 0 to 2π, so we have to integrate from 0 to 2π

∴ Arc length

= \int_{0}^{2\pi}\sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}+(\frac{dz}{dt})^{2}

=\int_{0}^{2\pi}\sqrt{(-\sin t)^{2}+(2\cos t)^{2}+(\sin t-2\cos t)^{2}

=\int_{0}^{2\pi}\sqrt{(2\sin t)^{2}+(8\cos t)^{2}-(4\sin t\cos t)

Now evaluating the integral using calculator,

=\int_{0}^{2\pi}\sqrt{(2\sin t)^{2}+(8\cos t)^{2}-(4\sin t\cos t) = 13.5191




8 0
3 years ago
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