The table below shows the budget for the Smith family.
2 answers:
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Answer:
Verified
Area = 13.12 square units.
Step-by-step explanation:
Let the given points / vertices of the parallelogram be represented as follows:
A(2,-1,1),
B(5, 1,4),
C(0,1,1),
D(3,3,4)
In vector notation, we can have;
A = 2i - j + k
B = 5i + j + 4k
C = 0i + j + k
D = 3i + 3j +4k
One of the ways to prove that a quadrilateral is a parallelogram is to show that both pairs of opposite sides are parallel.
(i) Now, let's find the various sides of the assumed parallelogram. These sides are:
AB = B - A = [5i + j + 4k] - [2i - j + k] <em>open the brackets</em>
AB = 5i + j + 4k - 2i + j - k <em>collect like terms and solve</em>
AB = 5i - 2i + j + j - k + 4k
AB = 3i + 2j+ 3k
BC = C - B = [0i + j + k] - [5i + j + 4k] <em>open the brackets</em>
BC = 0i + j + k - 5i - j - 4k <em>collect like terms and solve</em>
BC = 0i - 5i + j - j + k - 4k
BC = -5i + 0j - 3k
CD = D - C = [3i + 3j +4k] - [0i + j + k] <em>open the brackets</em>
CD = 3i + 3j + 4k - 0i - j - k <em>collect like terms and solve</em>
CD = 3i - 0i + 3j - j + 4k - k
CD = 3i + 2j + 3k
DA = A - D = [2i - j + k] - [3i + 3j +4k] <em>open the brackets</em>
DA = 2i - j + k - 3i - 3j - 4k <em>collect like terms and solve</em>
DA = 2i - 3i - j - 3j + k - 4k
DA = - i - 4j - 3k
AC = C - A = [0i + j + k] - [2i - j + k] <em>open the brackets</em>
AC = 0i + j + k - 2i + j - k <em>collect like terms and solve</em>
AC = 0i - 2i + j + j + k - k
AC = - 2i + 2j +0k
BD = D - B = [3i + 3j + 4k] - [5i + j + 4k] <em>open the brackets</em>
BD = 3i + 3j + 4k - 5i - j - 4k <em>collect like terms and solve</em>
BD = 3i - 5i + 3j - j + 4k - 4k
BD = - 2i + 2j + 0k
(ii) From the results in (i) above, it has been shown that;
AB is equal to CD, and that implies that AB is parallel to CD. i.e
AB = CD => AB || CD
<em>Also,</em>
AC is equal to BD, and that implies that AC is parallel to BD. i.e
AC = BD => AC || BD
(iii) Therefore, ABDC is a parallelogram since its opposite sides are equal and parallel.
(B) Now let's calculate the area of the parallelogram.
To calculate the area, we find the magnitude of the cross product between any two adjacent sides.
In this case, we choose sides AC and AB.
Area = | AC x AB |
Where;
AC X AB = i(6 - 0) - j(-6 - 0) + k(-4 -6)
AC X AB = 6i + 6j - 10k
|AC X AB| =
|AC X AB| =
|AC X AB| =
|AC X AB| = 13.12
Therefore the area is 13.12 square units.
PS: The diagram showing this parallelogram has been attached to this response.
Answer:
The probability that the next two balls are blue is
Step-by-step explanation:
There is a total number of 12 balls in the jar (7 Red and 5 blue), if the first marble drawn is a given, that is it is a Red marble, then following the instruction the Red marble and two more Red marbles are added. Therefore making the total number of marbles in the jar to be 14 marbles (9 red and 5 blue).
<u>FIRST EVENT:</u> So the probability of drawing the second marble to be a blue marble becomes = 5/14 [<em>this is the number of blue marbles/ total number of marbles in the jar</em>].
Thereafter the blue marble is returned and two more blue marbles are added to the jar making a total of 16 marbles (9 red and 7 blue}.
<u>SECOND EVENT: </u>So the probability of drawing the third marble to be a blue marble becomes = 7/16 [<em>this is the number of blue marbles/ total number of marbles in the jar</em>].
However, this is not the final answer because the first event and the second event must occur together, therefore, one has to multiply the first events probability and the second events probability to get the probability we are looking for.
So, the probability of picking the next two to be blue = 5/14*7/16 (seven can reduce the 14 to a 2 based on division)
Therefore the probability = 5/32 (the 2 multiplies the 16)