A rectangle has an area of 30 square meters and a perimeter of 34 meters. What are the dimensions of the rectangle?
1 answer:
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The diagram of the triangle is missing, so i have attached it.
Answer:
Surface area = 9.425 sq.units
Step-by-step explanation:
When a triangle having an interior angle as 30°, 60° or 90° is rotated, then a cone is formed.
Thus, we need to find the surface area of the cone.
Surface area of a cone is πr ² + πrL
where;
r = radius of the cone
L = slant height of the cone
I have attached the cone formed as the second image.
Looking at the image of the cone, we need to find the radius and slant height.
From the triangle image, we can use trigonometric ratios;
tan 30 = BC/AB
In surd form, BC/AB = 1/√3
Thus, BC is equivalent to 1 and AB equivalent to √3
Using Pythagoras theorem, we can find AC.
(AC)² = (AB)² + (BC)²
AC = √[(AB)² + (BC)²]
AC =√[(√3)² + (1)²]
AC = √[3 + 1]
AC = √4
AC = 2
So, radius, r corresponds to BC = 1
And slant height L corresponds to AC = 2.
Thus,r = 1 and L = 2
So,
Surface area of cone = π(1)² + π(1)(2) = π + 2π = 9.425 sq.units
Answer:
a) 30.08% probability the sample mean will be within $100 of the population mean.
b) 0% probability the sample mean will be greater than $12,600
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the Central Limit Theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question, we have that:
a. What is the probability the sample mean will be within ±$100 of the population mean?
This is the pvalue of Z when X = 100 divided by s subtracted by the pvalue of Z when X = -100 divided by s. So
Z = 0.4 has a pvalue of 0.6554, Z = -0.4 has a pvalue of 0.3556
0.6554 - 0.3546 = 0.3008
30.08% probability the sample mean will be within $100 of the population mean.
b. What is the probability the sample mean will be greater than $12,600?
This is 1 subtracted by the pvalue of Z when X = 12600. So
By the Central Limit Theorem
has a pvalue of 1.
1 - 1 = 0
0% probability the sample mean will be greater than $12,600