Write the expression in words. three times, four plus two, to the second power
1 answer:
You might be interested in
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.
Answer:
A.
- X ~ N(6.2kg, 2.25kg²)
B. What is the median seedless watermelon weight?____ kg.
- 6.2 kg
C. What is the Z-score for a seedless watermelon weighing 8 kg?
- 1.2
D. What is the probability that a randomly selected watermelon will weigh more than 7 kg?
- 0.2981
E. What is the probability that a randomly selected seedless watermelon will weigh between 4 and 5 kg?
- 0.1411
F. The 80th percentile for the weight of seedless watermelons is _____ kg.
- 7.5 kg
Explanation:
<em>A. X ~ N(___ , ____ )</em>
<em></em>
<em></em>
The distribution of a random variable in a sample extracted from a population that follows a normal distribution is represented by the notation:
- X ~ N(μ, σ²)
Where:
- X is the random variable
- N stands for normal distribution function
- μ is the median of the population
- σ² is the variance of the population
Here, you have:
- μ = 6.2 kg
- σ² = (1.5kg)² = 2.25 kg²
- X ~ N(6.2kg, 2.25kg²)
<em>B. What is the median seedless watermelon weight?____ kg.</em>
<em></em>
<em></em>
The median of a random variable that follows a normal distribution is equal to the mean, thus it is 6.2 kg.
<em>C. What is the Z-score for a seedless watermelon weighing 8 kg?</em>
<em />
The z-score is the standardized value of the random variable. It measures how far away is the variable from the mean.
It is calcuated with the formula:
Thus for X=8:
<em>D. What is the probability that a randomly selected watermelon will weigh more than 7 kg?</em>
<em></em>
You want P(X>7)
You must use the tables for the standardized normal distribution.
Find the corresponding Z-score for X = 7
You must use a table for the standardized normal distribution which gives the cumulative distribution or area under the curve of the standard normal distribution and find P(Z>0.53).
That is the area to the right of Z=0.53. The table shows 0.2981.
Thus, the probability that a randomly selected seedless watermelon will weigh more than 7 kg is 0.2981
<em>E. What is the probability that a randomly selected seedless watermelon will weigh between 4 and 5 kg?</em>
<em></em>
For this case you must find the Z-scores for X=4 and X=5 and then find the area under the curve of the standardized normal distribution between those two Z-scores.
- Z(X=4) = (4 - 6.2)/1.5 ≈ -1.47
- Z(X=5) = (5 - 6.2)/1.5 = -0.8
<em></em>
In the table the area to the right of Z = - 1.47 is 1 - 0.0708 = 0.9292
<em></em>
And the area to the right of Z = - 0.8 is 1 - 0.2119 = 0.7881
Thus, the area in between is the difference 0.9292 - 0.7881 = 0.1411.
<em>F. The 80th percentile for the weight of seedless watermelons is _____ kg.</em>
<em></em>
The 80th percentile is the weigh of the top 20% seedless watermelons: this is 80% of the weighs are below that weight.
You must find the Z-score for which the area under the curve is less than 0.80.
The area less than 0.80 is 1 less the area that is less than 0.20.
From the table, the Zscore that defines the area less than 0.20 is 0.845 (interpolating).
Thus, the 80th percentile is the X value that makes the Z-score greater than or equal to 0.845:
- Z ≥ 0.845
- (X - 6.2)/1.5 ≥ 0.845
- X ≥ 0.845 × 1.5 + 6.2
- X ≥ 7.4675
- X ≥ 7.5 kg ← answer