a) we reject the null hypothesis at the 5% significance level. The P-value is less than 0.01. b) This interval does not contain zero, we can conclude that the difference in means is statistically significant at the 5% level. c) The power of the test is 0.31. d) we need a sample size of 23 in each group to achieve a power of 0.95.
(a) To test the hypothesis H0: μ1 = μ2 against H1: μ1 ≠ μ2, we can use a two-sample t-test assuming equal variances. The test statistic is given by:
t = (X1 - X2) / [tex]\sqrt{s^{2}p*(1/n1 + 1/n2) }[/tex]
where X1 and X2 are the sample means, s²p is the pooled sample variance, n1 and n2 are the sample sizes, and t follows a t-distribution with degrees of freedom df = n1 + n2 - 2.
The pooled sample variance is given by:
s²p = [(n1 - 1) * s1² + (n2 - 1) * s2²] / (n1 + n2 - 2)
where s1² and s2² are the sample variances for the first and second sample, respectively.
Using the given values, we have:
X1 = 4.7, X2 = 7.8
s1² = 4, s2² = 6.25
n1 = n2 = 15
s²p = [(15 - 1) * 4 + (15 - 1) * 6.25] / (15 + 15 - 2) = 4.625
Substituting these values into the formula for t, we get:
t = (4.7 - 7.8) / [tex]\sqrt{4.625*(1/15 + 1/15)}[/tex] = -3.23
The P-value for this test is the probability of obtaining a t-value more extreme than -3.23 under the null hypothesis. This can be calculated using a t-distribution with 28 degrees of freedom (df = n1 + n2 - 2), or by using software or a t-table. For α = 0.05, the critical values are ±2.048. Since -3.23 is outside this range, we reject the null hypothesis at the 5% significance level. The P-value is less than 0.01.
(b) To construct a confidence interval for the difference in means, we can use the formula:
(X1 - X2) ± tα/2, [tex]\sqrt{s^{2}p*(1/n1 + 1/n2) }[/tex]
where tα/2 is the t-value with α/2 area to the right (or left) under the t-distribution with degrees of freedom df = n1 + n2 - 2, and s²p is the pooled sample variance as before.
Using the same values as before and α = 0.05, we have:
tα/2 = 2.048 (from the t-table)
(X1 - X2) ± 2.048 * [tex]\sqrt{4.625*(1/15 + 1/15)}[/tex]
= -3.126 to -0.474
We can interpret this interval as follows: we are 95% confident that the true difference in means falls between -3.126 and -0.474. Since this interval does not contain zero, we can conclude that the difference in means is statistically significant at the 5% level.
(c) The power of the test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. It depends on several factors, including the sample sizes, the level of significance, the effect size (i.e., the difference between the population means), and the variability of the data.
To calculate the power of the test for a true difference in means of 3, we need to specify the effect size and the sample sizes. Assuming equal variances and a two-sided test with a level of significance of 0.05, we can use a standard formula or a statistical software to calculate the power of the test. For example, using the R statistical software, we can use the power.t.test function. the power of the test is approximately 0.31, which means that there is a 31% chance of correctly rejecting the null hypothesis if the true difference in means is 3.
d) To determine the sample size needed to achieve a desired level of power, we need to specify the effect size, the level of significance, the desired power, and the variability of the data. Assuming equal variances and a two-sided test with a level of significance of 0.05, we can use a standard formula or a statistical software to calculate the sample size needed to achieve a desired power. For example, using the R statistical software, we can use the power.t.test function. we need a sample size of approximately 23 in each group to achieve a power of 0.95, assuming a true difference in means of -2 and a standard deviation of 4.5 (the average of the two sample standard deviations).
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How to find the answer?
i) The number of students that come to school by motorcycle only = 8
ii) The number of students that did not take all the three types of transportation = 9
iii) Number of students that did not come to school by car = 28
Solving questions on venn diagramThe total number of students = 40
The number of students that come to school by car alone = 9
The number of students that come to school by motorcycle only = 8
The number of students that come to school by car only = 11
The number of students that come to school by both car and motorcycle = 3
Number of students that did not take all the three types of transportation = 40 - (9 + 8 + 3 + 11)
= 40 - 31
Number of students that did not take all the three types of transportation = 9
Total students that come to school by car = 9+3 = 12
The total number of students that did not come to school by car = 40 - 12 = 28
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Rule 1: Multiply by 2 then add 1 starting from 1.
Rule 2: Divide by 2 then add 4 starting from 40.
I need to make an sequence for both of them and I need help
Rule 1:
1, 3, 7, 15, 31, 63, 127, 255, 511, 1023 etc
Rule 2:
40, 24, 16, 12, 10, 9, 8.5, 8.25, 8.125, 8.0625 etc
Hope this helps!
The first sequence is 1,3,7,15,31 and second sequence is 40,24,16,12,10.
Given that first rule is that multiply by 2 then add 1 starting from 1 and second rule is that divide by 2 then add 4 starting from 40.
We have to find the sequences for each rule.
We have to start from starting point and then apply the rule to find the sequence.
Sequence of rule 1:
Second term of sequence=2*1+1=3
Third term of the sequence=2*3+1=7
Fourth term of the sequence=2*7+1=15
Fifth term of the sequence=2*15+1=31.
Sequence will be 1,3,7,15,31.
Sequence of rule 2:
Second term of sequence=40/2+4=24
Third term of sequence=24/2+4=16
Fourth term of sequence=16/2+4=12
Fifth term of sequence=12/2+4=10
Sequence will be : 40,24,16,12,10.
Hence the first sequence is 1,3,7,15,31 and second sequence is 40,24,16,12,10.
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1. sonya's salary increases at a rate of 4% per year. her starting salary is $45,000. what is her annual salary, to the nearest $100, after 8 years of service?
Her annual salary after 8 years will be $57,600.
What is the percentage?A percentage is a figure or ratio stated as a fraction of 100 in mathematics. It is frequently symbolized with the percent sign, " %". A % is a number with no dimensions; it has no unit of measurement.To find what is her annual salary, to the nearest $100, after 8 years of service:
Find 4% of 45,000
[tex]\frac{x}{45000} *100=4\\\frac{x}{450} =4\\x =1800[/tex]
So, 4% of $45,000 of $1800.
1st year = $45000
2nd year = 45000 + 1800 = $46800
3rd year = 46800 + 1800 = $48600
4th year = 48600 + 1800 = $50400
5th year = 50400 + 1800 = $52200
6th year = 52200 + 1800 = $54000
7th year = 54000 + 1800 = $55800
8th year = 55800 + 1800 = $57600
Therefore, her annual salary after 8 years will be $57,600.
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The salary of Sonya after 8 years will be $61,600 (nearest $100).
What is compound interest?The interest on the a loan or deposit that is calculated using the both initial principal as well as the accrued interest from prior periods is known as compound interest (sometimes known as compounding interest).
Some features of compounding are-
Compound interest is calculated just on initial principal of a deposit or loan, which also takes into account all of the interest accrued from prior periods.The yearly interest rate is raised to a number of compounded periods minus one, and the starting principal amount is multiplied by both of these factors.The frequency plan for compounding interest can be set to be continuous, daily, or yearly.The quantity of compounding periods has a bigl impact on compound interest calculations.Calculation for the amount after 8 years-
The principal amount is $45,000
Rate of interest is 4% per year.
Time duration is 8 years.
The formula for total amount after compounding is -
[tex]A=P\left(1+\frac{r}{n}\right)^{n t}[/tex]
A = final amount
P = initial principal balance
r = interest rate
n = number of times interest applied per time period.
t = number of time periods elapsed.
As the rate of interest is applied only one in a year ; n = 1.
Substitute the values in the formula to get the value.
[tex]A=45,000\left(1+\frac{.04}{1}\right)^{8}\\A=45,000\left(1.04}\right)^{8}\\\\A=61585.60[/tex]
Therefore, the salary of Sonya will be $61600 (nearest $100) after 8 years.
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1. Drew ran the first 2 3/4 miles of a 5-mile race in 1/2 hour. What was his average rate, in miles per hour, for this first part of the race? Describe how you solved the problem.
Answer:
5 1/2
Step-by-step explanation:
Rate x Time = Distance. If you know 2 of the parts, you can figure out the 3rd part. You know the time and the distance, so plug those in and solve.
Let R = rate
R(1/2) = 2 3/4 Let's change 2 3/4 into a improper fraction.
R(1/2) = 11/4 Multiply 4x2 and then add 3 that gets you 11 and the bottom number stays the same. Next, multiply both sides of the equation by 2.
R = 22/4 or 11/2 or 5 1/2 miles per hour.
A tank of water contains 35 liters of water that leaks 6 liters of water an hour another container contains 29 liters of water that leaks 3 an hour if no the the tank and container start leaking at the same time how long does it takes for both the tank and container to have the same amount of water
Answer:
Step-by-step explanation:
Let the # of liters in both containers = y when equality is reached.
Container 1: loses 6 liters / hour
Container 2: loses 3 liters / hour
Equation
35 - 6x = 29 - 3x
Solution
35 - 6x = 29 - 3x Subtract 29 from both sides
35 - 29 - 6x = 29 - 29 - 3x Combine
6 - 6x = - 3x Add 6x to both sides
6 - 6x + 6x = - 3x + 6x Combine
6 = 3x Divide both sides by 3
6/3 = 3x/3
2 = x
After 2 hours they should have the same amount of water.
35 - 6(2) = 35 - 12 = 23
29 - 3*2 = 29 - 6 = 23
Answer
After 2 hours have passed, both containers should have 23 Liters each.
Use the quadratic model y=-4x^2-3x+4 to predict y if x equals 5
Answer:
y = -4x^2 - 3x + 4
x = 5
y = -4(5)^2 - 3(5) + 4
y = -4(25) - 15 + 4
y = -100 - 15 + 4
y = -114
Step-by-step explanation:
Determine the measure of the interior angle at vertex E. A. 135 B. 67.5 C. 60 D. 90
Answer:
90
Step-by-step explanation:
Why cant people just answer questions without guessing or answer them correctly.
22. If 1 m,m/19 - 118°, and m/16 - 48°, calculate m/12.
0 0
103"
98
109⁰
110"
18
17
19 20
14
16
13
15
11
10
12
9
63
5
m
Answer: Option 4
Step-by-step explanation:
[tex]m\angle 20=62^{\circ}[/tex] (linear pair)
[tex]m\angle 11=70^{\circ}[/tex] (angle sum in a triangle)
[tex]m\angle 12=110^{\circ}[/tex] (linear pair)
Find the gradient of the tangent to the curve y = (√x + 3)(3√x - 5) at the point where x = 1
Answer:
Gradient (slope) is 5.
Step-by-step explanation:
To find the gradient (slope) of a curve graph at x = x₁ can be done by steps following:
Differentiate the function - this is to find a gradient (slope) at any points (technically function of slope)Substitute x = x₁ in the derived function - you'll receive a slope at x = x₁ point.First, derive the given function which is:
[tex]\displaystyle{y = (\sqrt{x}+3)(3\sqrt{x}-5)[/tex]
Differentiation can be done two ways - go ahead and expand the expression then derive it or you can use the product rule where it states that [tex]\displaystyle{y'=u'v+uv'}[/tex]
I'll be using product rule:
[tex]\displaystyle{y' = (\sqrt{x}+3)'(3\sqrt{x}-5)+(\sqrt{x}+3)(3\sqrt{x}-5)'}[/tex]
Note that the following process will require you to have knowledge of Power Rules:
[tex]\displaystyle{y = ax^n \to y' = nax^{n-1}}[/tex]
Hence:
[tex]\displaystyle{y'=\dfrac{1}{2\sqrt{x}}(3\sqrt{x}-5) + (\sqrt{x}+3)\dfrac{3}{2\sqrt{x}}[/tex]
Now we know the derivative. Next, we find the slope at x = 1 which you substitute x = 1 in derived function:
[tex]\displaystyle{y'(1)=\dfrac{1}{2\sqrt{1}}(3\sqrt{1}-5) + (\sqrt{1}+3)\dfrac{3}{2\sqrt{1}}}\\\\\displaystyle{y'(1)=\dfrac{1}{2}(3-5) + (1+3)\dfrac{3}{2}}\\\\\displaystyle{y'(1)=\dfrac{1}{2}(-2) + (4)\dfrac{3}{2}}\\\\\displaystyle{y'(1)=-1 + 2(3)}\\\\\displaystyle{y'(1)=-1 + 6}\\\\\displaystyle{y'(1)=5}[/tex]
Finally, we have found the slope or gradient at x = 1 which is 5.
Please let me know if you have any questions!
Write an absolute value inequality for each of the following. Then graph the
solution set on a number line.
all numbers less than 7and greater than -7
An absolute value inequality for "all numbers less than 7 and greater than -7" is given by |x| < 7.
How to write an absolute value inequality?Based on the information provided, we can logically deduce that the statement describes an intersection of set because the word "and" was used.
This ultimately implies that, we would use the less than (<) symbol to write an absolute value inequality for the given statement:
|x| < 7
Therefore, this can be translated as follows:
-7 < x < 7 or x < 7 and x > -7.
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Find the length of BC¯¯¯¯¯¯¯¯
Answer:
B
Step-by-step explanation:
I love trig lol,
Assuming you know trig, Cos(17)=BC/42
Cos(17)*42=BC
Cos(17)~0.956
0.956*42=40.152=BC
Since this is a approximation, B is the closest to our estimate.
Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.
Answer:
45
Step-by-step explanation:
small brain
2x - y = 4
y = 2x + 4
Solution(s):
PLEASEEE
Can some solve for x please?
Answer:
x = 19
Step-by-step explanation:
You would use vertical angles to move the 8x - 14 so that we could use the Consecutive Interior angles theorem. Which says that consecutive angles equal 180. So, you would set up your equation: 8x-14+2x+4 = 180. Add the 8 and 2 = 10. Add the -14 and 4 = -10. You get 10x-10 = 180. 180+10 = 190. 190/10 = 19. X = 19
(-1/2 - 1/8) + (3/4 + 5/6) A.1/12
Answer:
23/24
Step-by-step explanation:
Let's convert all the fractions to a common denominator
(-12/24 - 3/24) + (18/24 + 20/24)
Combine like terms and you get 23/24
find (f•g) (k) equations in red letters
The answer is (f . g)(x) = - (34k + 17k⁷ + 17k⁵ + 40k³ + 20k² + 60).
Remember that :
(f . g)(x) = f(x) × g(x)
Hence :
(f . g)(x) = (2k³ + k² + 3)(-17k⁵ - 20)(f . g)(x) = -34k⁸ - 17k⁷ - 17k⁵ - 40k³ - 20k² - 60(f . g)(x) = - (34k + 17k⁷ + 17k⁵ + 40k³ + 20k² + 60)An airplane covers 3450 km in an hour. how much distance will it cover in 7 hours?
Answer:
24,150km
Step-by-step explanation:
The plane travels 3450km in 1 hr, since we need the distance traveled in 7 hrs we need to multiply 3450 by 7.
distance in 7hrs: 3450*7 = 24,150km
Use the ideal gas law (PV=nRT) to find P. Round answer to 1 d.p.
Given: V = 3 L, n = 2 mol, T = 300 K, R = 0.0821
Answer:
P = 16.4
Step-by-step explanation:
Given the equation PV = nRT and the values we know, we need to isolate P .
Thus, we have:
[tex]PV=nRT\\3P=2*0.0821*300\\P=49.26/3\\P=16.42=16.4[/tex]
Find the coefficient of t to the 4th power in the expansion of (-9t-9) to the 7th power select one: a. -25,515 b. -100,442,349 c. -167,403, 915 d. -4,782, 969
The coefficient of t to the 4th power in the given expansion is -167403915.
Option (c) is the correct answer.
What is an expansion?An expansion of a product of sums uses the fact that multiplication distributes across addition to represent it as a sum of products. A polynomial expression can be expanded by repeatedly substituting equivalent sum of products for subexpressions that multiply two other subexpressions, at least one of which is an addition, up until the expression becomes a sum of (repeated) products. Simplifications like grouping similar terms or canceling terms may also be used during the expansion.
Given expansion=[tex](-9t-9)^{7}[/tex]
Coefficient of the 4th power of t in the expansion,
=[tex]_{7} {C} _{7-4} (-9t)^{4}(-9)^{7-4}[/tex]
=[tex]_{7}C_{3}(-9)^{4}t^{4}(-9)^{3}[/tex]
=[tex]_{7}C_{3}(-9)^{7}t^{4}[/tex]
=[tex]35t^{4}(-9)^{7} \\[/tex]
=[tex]35t^{4}(4782969)[/tex]
= -167403915[tex]t^{4}[/tex]
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h(n) = 41 - 5n
Complete the recursive formula of h(n).
h(1) =
h(n) = h(n-1)
The recursive formula of h(n) is h(1) = 36 and h(n) = h(n -1) - 5
How to determine the recursive formula?The function is given as:
h(n) = 41- 5n
Calculate h(1) and h(2)
h(1) = 41- 5(1)
h(1) = 36
h(2) = 41- 5(2)
h(2) = 31
Calculate the difference between h(1) and h(2)
d = 31 - 36
d = -5
This means that:
h(1) = 36 and h(n) = h(n -1) - 5
Hence, the recursive formula of h(n) is h(1) = 36 and h(n) = h(n -1) - 5
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Triangle JKL is graphed on the coordinate plane. -5 -4 -3 y B MD N In L 2 4 5 X What are the coordinates of its image, JKL, if this triangle is dilated by a factor of 5 with respect to the origin? O J(2, 7), K (8, 7), and L'(5, 1) O J'(-5, 2), K'(5, 2), and L'(0, -5) O J'(-3, 2), K'(8, 2), and L'(0, -9) O J'(-15, 10), K'(15, 10), and L'(0, -20)
The coordinate of the image of JKL when dilated by a factor of 5 with respect to the origin is J'(-15, 10), K'(15, 10) and L'(0, 20).
What is transformation?Transformation is the movement of a point from its initial location to a new location. Types of transformations are reflection, rotation, translation and dilation.
Translation is the movement of a point either up, left, right or down in the coordinate plane.
Let us assume that triangle JKL has vertices at J(-3, 2), K(3, 2) and L(0, 4).
The coordinate of the image of JKL when dilated by a factor of 5 with respect to the origin is J'(-15, 10), K'(15, 10) and L'(0, 20).
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If t = 18 and r = 12, find S. Round to the nearest tenth.
Answer:
13.4 or √180
Step-by-step explanation:
Pythagorean theorem states
a² + b² = c²
C^2 is the longest side. The corner of the right angle points to it. The other sides are your a and b squared. They are called the legs or Opposite and Adjacent sides.
Answer: S ≈ 48.2°
Step-by-step explanation:
We can use trigonometry functions to solve.
Looking at angle S, t = 18 is the hypotenuse and r = 12 is the adjacent side. This means we can use the cosine function.
[tex]\displaystyle cos(S) = \frac{\text{adjacent side}}{\text{hypotenuse }}=\frac{12}{18}[/tex]
[tex]\displaystyle cos^{-1} (cos(S)) = (\frac{\text{adjacent side}}{\text{hypotenuse }}=\frac{12}{18})cos^{-1}[/tex]
[tex]S = 48.189685...\\S \;$\approx$\; 48.2^{\circ}[/tex]
what are the domain and range of the graph below.
O domain: [0,00)
range: (-00,00)
O domain: [0,00)
range: (-0,4]
domain: (0,4)
range:
(-00,00)
domain: (-∞,4]
range: [0,00)
Answer:
Domain: [0,∞)
Range: (-∞, 4]
Step-by-step explanation:
The required domain and range of the function shown in the graph are,
Domain = (0, ∞)
Range = (-∞, 4)
To calculate the domain and range of a function from its graph, you can follow these steps:
Identify the x-coordinates of all the points on the graph. The set of all possible x-coordinates is the domain of the function.Write the domain as an interval or as a set of values, depending on the format of the original problem. For example, if the graph is defined for all real numbers, the domain would be written as (-∞, ∞).Identify the y-coordinates of all the points on the graph. The set of all possible y-coordinates is the range of the function.Write the range as an interval or as a set of values, depending on the format of the original problem.So, from the above definition, from the graph we have,
Domain = (0, ∞)
Range = (-∞, 4)
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Complete the double number line to show the other values of fat and protein.
The double number line which shows the other values of fat and protein is: option B.
What is a number line?A number line is a type of graph with a graduated straight line which contain numerical values that are placed at equal intervals along its length.
For the fat component, we have:
Fat = 2n = 0, 2, 4, 6, 8.
For the protein component, we have:
Protein = 11n = 0, 11, 22, 33, 44.
Note: n represent integers such as 0, 1, 2, 3 and 4.
In conclusion, the double number line which represent the other values of fat and protein is shown in the image attached below.
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Complete Question:
One serving of beef je-rky contains 2 grams of fat and 11 grams of protein. Complete the double number line to show the other values of fat and protein.
What is the slope of a line that passes through the point (-3,0) and has a y-intercept of 12
The slope of a line that passes through the point (-3,0) and has a y-intercept of 12 is 4.
How to find the slope of a line?Using the point slope equation,
y = mx + b
where
m = slopeb = y-interceptTherefore, using (-3, 0)
0 = -3m + 12
-12 = -3m
divide both sides by -3
m = -12 / -3
m = 4
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I am thinking of a two digit number.it is smaller then 5 tens but greater then 4 tens. its ones digit is 3 more than its tens digit.what number am i thinking of?
The two digit number larger than 40 and lesser than 50, and the difference between the ones and tens digits gives the two digit number as 41
How can the two digit number be found?Let XY represent the two digit number;
From the question, we have the following inequalities and equations;
40 < XY < 50X = Y + 3Therefore;
4 ≤ X < 5Y > 0Which gives;
4 ≤ Y + 34 - 3 = 1 ≤ YY = 1
X = 4
Therefore;
XY = 41The two digit number, XY = 41
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hellooo please i need help on these question
( f - g )(x) = 10x - 17 and ( f ° g)(-6) = 60. The answers are option D and B
There are different type of FunctionDifferent types of function demands different ways of solving them. Different types are simple function, function by substitution and function of function.
Given that
f(x) = [tex]x^{2}[/tex] + 9x - 14 and g(x) = [tex]x^{2}[/tex] - x + 3
( f - g )(x) will be the difference between f(x) and g(x)
( f - g )(x) = [tex]x^{2}[/tex] + 9x - 14 - ( [tex]x^{2}[/tex] - x + 3)
( f - g )(x) = [tex]x^{2}[/tex] + 9x - 14 - [tex]x^{2}[/tex] + x - 3
( f - g )(x) = 10x - 17
Also, if f(x) = [tex]3x^{2}[/tex] - 15 and g(x) = 1 - 2/3 x
( f ° g)(-6) means that we will evaluate g at -6, then evaluate f at the result of g(-6)
g(-6) = 1 - 2/3 (-6)
g(-6) = 1 + 4
g(-6) = 5
f(g(-6)) = 3([tex]5^{2}[/tex]) - 15
f(g(-6)) = 3 x 25 - 15
f(g(-6)) = 75 - 15
f(g(-6)) = 60
Therefore, ( f - g )(x) = 10x - 17 and ( f ° g)(-6) = 60
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Instructions: Find the surface area of the figure below. Round your answers to the nearest tenth, if necessary.
The surface area of the cone is calculated to the nearest tenth as: 173.4 cm².
What is the Surface Area of a Cone?Surface area of a cone = πr(r + l),
Given the following:
Radius (r) = 4 cmSlant height (l) = 9.8 cmPlug in the values
Surface area of the cone = π × 4(4 + 9.8)
Surface area of the cone = 173.4 cm²
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2. Two rectangular lots are adjacent to each other, as shown in the diagram below. Using a function operation, determine how many times bigger the large lot is than the small lot. Clearly identify the individual component functions and simplified final equation.
The large rectangle is 8/3 times bigger than the small rectangle.
How to explain the area?Area of large rectangle will be:
= 4x(2x - 2)
= 8x² - 8x
Area of small rectangle will be:
= (3x - 3)x
= 3x² - 3x
The ratio will be:
= 8x² - 8x/3x² - 3x
= 8(x² - x)/3(x² - x)
= 8/3
The large rectangle is 8/3 times bigger than the small rectangle.
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Find the area of a triangle who's side lengths are 13, 14, and 15. (The questions doesn't tell the type of triangle.)
Answer:
84 square units.
Step-by-step explanation:
Area of scalene triangle:[tex]\sf \boxed{\bf Area = \sqrt{s*(s-a)(s-b)*(s-c)}}[/tex]
Here, a, b and c are the sides of the triangle. s is the semi perimeter.
a = 13
b = 14
c = 15
[tex]\sf s= \dfrac{a+b+c}{2}\\\\ =\dfrac{13+14+15}{2}\\\\=\dfrac{42}{2}\\\\s = 21[/tex]
s -a = 21 - 13 = 8
s -b = 21 - 14 = 7
s - c = 21 - 15 = 6
[tex]\sf Area = \sqrt{21*8*7*6}[/tex]
[tex]= \sqrt{ 7* 3 * 2 * 2 * 2 * 7 * 2 * 3}\\\\=7 * 3 *2*2\\\\= 84 \ square \ units[/tex]