For this item, enter the answer in the space provided. A concession stand at school is selling cups of tea and hot chocolate aft
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Answer:
Area = 13.15 square units
Step-by-step explanation:
Let the given vertices be represented as follows:
A(2, -1, 1) = 2i - j + k
B(5, 1, 4) = 5i + j + 4k
C(0, 1, 1) = 0i + j + k
D(3, 3, 4) = 3i + 3j + 4k
(i) Let's calculate the vectors of all the sides:
AB = B - A = (5i + j + 4k) - (2i - j + k)
AB = 5i + j + 4k - 2i + j - k [Collect like terms]
AB = 3i + 2j + 3k
BC = C - B = (0i + j + k) - (5i + j + 4k)
BC = 0i + j + k - 5i - j - 4k [Collect like terms]
BC = -5i + 0j - 3k
CD = D - C = (3i + 3j + 4k) - (0i + j + k)
CD = 3i + 3j + 4k - 0i - j - k [Collect like terms]
CD = 3i + 2j + 3k
DA = A - D = (2i - j + k) - (3i + 3j + 4k)
DA = 2i - j + k - 3i - 3j - 4k [Collect like terms]
DA = -i - 4j - 3k
AC = C - A = (0i + j + k) - (2i - j + k)
AC = 0i + j + k - 2i + j - k [Collect like terms]
AC = -2i + 2j
BD = D - B = (3i + 3j + 4k) - (5i + j + 4k)
BD = 3i + 3j + 4k - 5i - j - 4k [Collect like terms]
BD = -2i + 2j
(ii) From the results in (i) above, we can deduce that;
AB = CD This implies that AB || CD [AB is parallel to CD]
AC = BD This implies that AC || BD [AC is parallel to BD]
(iii) Therefore, ABDC is a parallelogram since opposite sides (AB and CD) are parallel. Hence, the points are vertices of a parallelogram
<u>Now let's calculate the area</u>
To find the area of the parallelogram, we find the magnitude of the cross product of any two adjacent sides.
In this case, we'll choose AB and AC
Area = |AB X AC|
Where;
<u></u>
AB X AC = i(0 - 6) - j(0 + 6) + k(6 + 4)
AB X AC = - 6i - 6j + 10k
|AB X AC| =
|AB X AC| =
|AB X AC| = 13.15
Area = 13.15 square units.
<u></u>
<u></u>
<u>PS: </u> ACBD is also a parallelogram. The diagram has also been attached to this response.
The elements are :
<span> 2,3,5,8,2.03,2.20,2.6,4.5
n = 8
</span>μ = ( <span>2 + 3 + 5 + 8 + 2.03 + 2.20 + 2.6 + 4.5 )/8 = 3.66625
sum = (</span> 2 - <span>3.66625)² + ( 3 </span><span>- 3.66625)² + ( 5 </span><span>- 3.66625)² + ( 8 </span><span>- 3.66625)² +
( 2.03 </span><span>- 3.66625)² + ( 2.20 </span><span>- 3.66625)² + ( 2.6 </span><span>- 3.66625)² + ( 4.5 </span>- <span>3.66625)²
= 30.44
∴ </span>The standard deviation = √ (30.44 / 8)
≈ 1.95 ( <span>to the nearest hundredth )</span>