Answer:
7 to the power of 1
Step-by-step explanation:
HELP I WILL GIVE BRAINLIEST
Answer:
d :)
Step-by-step explanation:
A baseball manager is determining the batting order for the team.The team has 9 players,but the manager definitely wants the pitcher to bat last.How many batting orders are possible.
SOLUTION:
Step 1:
In this question, we are given the following:
A baseball manager is determining the batting order for the team.
The team has 9 players, but the manager definitely wants the pitcher to bat last. How many batting orders are possible.
Step 2:
The details of the solution are as follows:
Since the pitcher bats last, there is only one choice for the 9th slot.
For the 1st slot, there are 8 choices remaining.
For the 2nd slot, there are 7 choices remaining, etc.
So the number of possible batting orders is:
\(8\text{ x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 8}!=\text{ 40320 orders}\)
9. Use the graph of the function f(x) = x³ – 7x² + 10x to
x`
identify its relative maximum and minimum.
da
-4
8
8
2
K
T
2
da
y
8
+w
8
maximum = 4.1, minimum = -8.2
maximum = 0.9, minimum = 3.8
The extremas of the function f(x) = x³ - 7x² + 10x are given as follows:
Relative maximum: (0.88, 4.061).Relative minimum: (3.786, -8.209).What are the relative minimums and the relative maximums of a function?The relative minimums of a function are given by the points in which the function's behavior changes from decreasing to increasing, that is, where the function curves down.The relative maximums of a function, meanwhile, are given by the points in which the function's behavior changes from increasing to decreasing, that is, where the function curves up.More can be learned about extremas of a function at https://brainly.com/question/9839310
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hhhhhhhhhhhhhhhheeeeeeeeeeeeeeeeeeeeeeeeeelllllllllllllllllpppppppppppp
Answer:
yesss???
Step-by-step explanation:
Answer:yes
Step-by-step explanation:
Select the correct answer.
Each statement describes a transformation of the graph of f(x) = x. Which statement correctly describes the graph of g(x) if g(x) = f(x - 11)?
A. It is the graph of f(x) where the slope is increased by 11.
It is the graph of f(x) translated 11 units to the left.
It is the graph of f(x) translated 11 units up.
It is the graph of f(x) translated 11 units to the right.
B.
C.
OD.
The correct answer is C. It is the graph of f(x) translated 11 units to the left.
The correct answer is:
C. It is the graph of f(x) translated 11 units to the left.
When we have a function of the form g(x) = f(x - a), it represents a horizontal translation of the graph of f(x) by 'a' units to the right if 'a' is positive and to the left if 'a' is negative.
In this case, g(x) = f(x - 11), which means that the graph of f(x) is being translated 11 units to the right. However, the answer options do not include this specific transformation. The closest option is option C, which states that the graph of g(x) is translated 11 units to the left.
The graph of f(x) = x is a straight line passing through the origin with a slope of 1. If we apply the transformation g(x) = f(x - 11), it means that we are shifting the graph of f(x) 11 units to the right. This results in a new function g(x) that has the same shape and slope as f(x), but is shifted to the right by 11 units.
Therefore, the correct answer is C. It is the graph of f(x) translated 11 units to the left.
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On a coordinate plane, a point is 4 units to the left and 1 unit down.
For the point shown:
The x-coordinate is
.
The y-coordinate is
.
The point is in quadrant
.
.
The point has an x-coordinate of -4 and a y-coorindate of -1, and it is on the third quadrant.
In which quadrant is the point?First, remember that for a point (x, y):
if x >0, y > 0, then the point is in quadrant I.if x <0, y > 0, then the point is in quadrant II.if x < 0, y < 0, then the point is in quadrant III.if x >0, y < 0, then the point is in quadrant IV.For the point that is 4 units to the lefft and 1 unit down (of the origin) it is written as:
(-4, -1)
Then we can see that this point is on the quadrant III.
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"My friend here Justin, he's already taken, and he's cracked at Fornite my guy.
I've been contemplating this video a ton of late. I've been considering the video since it gives a provocative message and keen editorial about enslavement and fixation.
To start with, we should take this word by word. The initial segment of the primary sentence; the principal shimmer of gold, is "My companion here Justin" An everyday sentence from the outset, however kindly recollect what I said, to arrive at the pot of gold at the focal point of the cavern of incongruity is justified, despite any trouble. This little portion of the sentence addresses that the storyteller has a companion named Justin. The principal word is possessive; my, and companion here addresses the kid before the camera is indeed Justin, showed by the storyteller naming Justin just after saying companion here. how the storyteller we never see, and even the way that we never see the storyteller entreats us to embed ourselves into them, to place ourselves into their point of view.
"He's now taken" This passage is clear as crystal, aside from one section. "He" shows the subject of the sentence; Justin, is "now" demonstrating that Justin has as of now and is "taken". The virtuoso of this sentence is unpretentious yet significant pleasantry in "taken", as in the old style utilization of this sentence it would imply that Justin is associated with a sentimental or sexual relationship.
"Also, he's cracked at Fortnite" the biggest portion, consequently the one generally stuffed with significance, this selection is the zenith of the multitude of different passages of the sentence. "Also, expands on the last selection demonstrating this is the continuation of the story, and "He's cracked at Fortnite" is the star proclamation of this extract.
"My person". "My person" can allude to numerous things, the storyteller could be alluding to Justin or some obscure outsider. I would excuse the chance of the storyteller alluding to an outsider nonetheless, on the grounds that so far there has been no sign of any outsider in the sum of the video. All things being equal, I think "My person" is alluding to you, the watcher, audience, and peruser. The explanation the storyteller removes time from his day to effectively allude to the eyewitness is on the grounds that the storyteller makes a particular highlight incorporate you, in this manner demonstrating that you are, with regards to the story, in the room noticing Justin.
In general, this show-stopper addresses the narrative of a man's compulsion breaking him while fusing the absolute best incongruity and pleasantry I've ever seen, all done in one video, eight seconds, one sentence, and fourteen words. The incongruity choice and diverting, and the pleasantry smart, this is a provocative piece of media that I would never expect to completely examine, for this remarkable piece of media is worth undeniably more than 200 40,000 words.
Answer:
i dk what to say
Step-by-step explanation:
Answer:
hhahahha i need my points on my alt account
Step-by-step explanation:
let me answer!!!!!!!!!
NO LINKS!! URGENT HELP!!
Calculate the perimeter of the following figures.
Answer:
Quadrilateral: 20.5 units
Triangle: 8.6 units
Step-by-step explanation:
Quadrilateral:
We can use the following formula to find the perimeter of a quadrilateral.
Perimeter = AB + BC + CD + DA
where AB, BC, CD, and DA are the lengths of the four sides of the quadrilateral.
In your case, the coordinates of the vertices of the quadrilateral are A(-3,-1), B(-3,3), C(2,3), and D(4,-1).
Using the distance formula, we can find the lengths of the four sides of the quadrilateral as follows:
\(AB = \sqrt{(-3 - (-3))^2 + ((-1) - 3)^2} = \sqrt{16} = 4\)
\(BC = \sqrt{(-3 - 2)^2 + ((3) - 3)^2} = \sqrt{25}= 5\)
\(CD = \sqrt{(2 - 4)^2 + ((3) - (-1))^2} = \sqrt{4+16} = 2\sqrt{5}\)
\(DA = \sqrt{(4 - (-3))^2 + ((-1) - 1)^2} = \sqrt{49} =7\)
Therefore, the perimeter of the quadrilateral is:
Perimeter = AB + BC + CD + DA
= \(4+5+2\sqrt5+7=20.5\) units
Triangle
The perimeter of a triangle is the total length of all three sides of the triangle. To find the perimeter of a triangle, we can use the following formula:
Perimeter = AB + BC + CA
where AB, BC, and CA are the lengths of the three sides of the triangle.
In your case, the coordinates of the vertices of the triangle are
E(-4,1), F(-2,3), and G(-2,4). Using the distance formula, we can find the lengths of the three sides of the triangle as follows:
\(EF = \sqrt{(-4 - (-2))^2 + ((1) - (3))^2} = 2\sqrt{2}\)
\(FG = \sqrt{(-2 - (-2))^2 + ((3) - (4))^2} = 1\)
\(EG = \sqrt{(-4 - (-2))^2 + ((1) - (4))^2} = \sqrt{4 + 9} = \sqrt{13}\)
Therefore, the perimeter of the triangle is:
Perimeter = EF + FG + EG
= 4 + 1 + \(\sqrt{13}\)
=8.6 units
Therefore, the perimeter of the quadrilateral is 20.5 units and the perimeter of the triangle is 8.6 units
Assume the random variable X is normally distributed with mean = 50 and standard deviation = 7. Compute the probability, P(X > 35). Be sure to draw a normal curve with the area corresponding to the probability shaded.
Answer:
P [ X > 35 ] = 0,983 or P [ X > 35 ] = 98,3 %
Step-by-step explanation: See Annex valid region for 98,3 in red lines
P [ X > 35 ] = 1 - P [ X ≤ 35 ]
P[ X ≤ z ] = ( X - μ₀ ) / σ
P [ X ≤ 35 ] = ( 35 - 50 ) / 7
P [ X ≤ 35 ] = - 15 / 7
P [ X ≤ 35 ] = - 2,1428
We find for z(score) = - 2,14 in z-table the value of 0,01618
P [ X ≤ 35 ] = 0,01618 or P [ X ≤ 35 ] ≈ 1,62 %
And
P [ X > 35 ] = 1 - 0,01618
P [ X > 35 ] = 0,983 or P [ X > 35 ] = 98,3 %
Can you guys please help me?
Answer:
its 11
Step-by-step explanation:
(2*3)^2-5^2=11
36-25=11 hope it helps
Kate is making biscuits. She uses flour, butter, and sugar in the ratio 5:3:2 Kate has 800 g flour 550 g butter. Kate wants to make the maximum number of biscuits possible. She works out how much sugar she needs. (a) How much sugar does Kate need? You must show working.
Answer:
800 grams flour ÷ 5 grams flour/biscuit = 160 biscuits
160 biscuits × 3 grams butter/biscuit = 480 grams butter
160 biscuits × 2 grams sugar/biscuit = 320 grams sugar
Katie needs 320 grams sugar.
Kate needs 270g of sugar to make the maximum number of biscuits possible.
What is Proportional?Any relationship that is always in the same ratio and quantity which vary directly with each other is called the proportional.
Now, Based on the ratio of flour, butter, and sugar of 5:3:2, we can determine that Kate needs;
⇒ 800g + 550g
= 1350g of the mixture.
Hence, For find out the amount of sugar needed, we need to determine the proportion of sugar in the mixture.
We can do this by adding up the ratio numbers;
(5+3+2) = 10.
Then we can divide each ratio number by the total (10) to get the proportion of each ingredient.
So the proportion of sugar in the mixture is 2/10. To find out how much sugar Kate needs, we can set up a proportion:
2/10 = x/1350
We can solve for x by cross-multiplying:
2 × 1350 = 10x
2700 = 10x
x = 270
Therefore, Kate needs 270g of sugar to make the maximum number of biscuits possible.
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There are 3 types of PercentProblems.
a) Number compared to the Base is unknown: x/100 = ?/b
b) Base Number is unknown: a/?= x/100
c) Percent is unknown: a/b = ?/100
Solve the following percent problems using the appropriate percent equation. Describe
the steps used to calculate each answer.
1) What distance is 24% of 710 miles?
2) 62 hours is what percent of 3 days?
3) 16% of what number is 8?
Answer:A. x= .100
Step-by-step explanation:because of the distance in miles
Find the value of the trigonometric ratio. Simplify the ratio if possible.
(Hint: You may need to use some pythagorean theorem first!)
Step-by-step explanation:
Given that,
BC = 8
Ac = 15
We can find AB using the pythagoas theorem.
\(AB=\sqrt{AC^2+BC^2}\\\\=\sqrt{15^2+8^2}\\\\AB=17\)
We know that,
\(\cos\theta=\dfrac{B}{H}\), B is base and H is Hypotenuse
\(\cosB=\dfrac{8}{17}\)
Hence, this is the required solution.
Pls help me with this
Answer:
a=43
b=19.30263678
c=26.39302534
Step-by-step explanation:
a=90-47
since it is a right triangle and has a right angle, the other two angles add up to 90 degrees
b=
tan= opposite leg/adjacent leg
tan 47= b/16 (cross multiply)
b=16*tan47
plug in calculator
c=
cos= adjacent leg/hypothenuse
cos 47= 18/c (cross multiply)
c=18/cos 47
plug in calculator
On a trip, a student drove 42 miles per hour for 3 hours and then drove 26 miles per hour for 5 hours. What is the student’s average rate of speed, in miles per hour, for the whole trip
Answer:
34 mph Is correct answer
Step-by-step explanation:
Average speed = total distance / total time
Total distance = ( 40 × 2 + 30 × 3 )
= 170 miles
Total time = 5 h
So, avg speed = 170/5 = 34 mph
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Kiyana's grandmother has a collection of hats for special occasions. Each hat is in a cube-shaped box with edges 1 ft long. She stores the boxes in a rectangular closet with the dimensions shown. Length = 6 ft N Height = 9 ft Width= 5 ft 3 4 5 > 2 of 5 Completed What is the maximum number of boxes that Kiyana's grandmother can fit inside the closet? Choose one option from each drop-down menu to answe the question. Kiyana's grandmother can fit a maximum of [ ] boxes along the length,[. ] boxes along the wide and [ ]boxes along the height of the closet. This means she can fit a maximum of [ ]boxes inside the closet
The maximum number of cubical boxes of side 1 feet that can be placed in a rectangular closet of length 6 feet, height 9 feet and breadth 5 feet is 270 boxes. Maximum number of boxes that can be placed along the length, height and breadth will be 168 boxes, 198 boxes and 150 boxes respectively.
Since the cube shaped boxes of length 1 feet are to be placed in the rectangular closet, the maximum number of boxes that will fit completely would be given as a comparison of volume of rectangular closet being equal to product of number of boxes and volume of 1 cubical box.
Volume of rectangular closet = Number of boxes * Volume of 1 cubical box
6 * 9 * 5 = n * (1*1*1)
Therefore, n = 270 boxes.
To calculate maximum number of boxes that will fit along the length would be calculated as product of length and breadth along 2 faces of closet and length and height along another 2 faces of closet.
Thus, [(9*6) + (9*6) + (6*5) + (6*5)] = 168 boxes along the length.
Similarly, to calculate maximum number of boxes that will fit along the height would be calculated as product of height and breadth along 2 faces of closet and length and height along another 2 faces of closet.
Thus, [(9*6) + (9*6) + (9*5) + (9*5)] = 198 boxes along the height
Similarly, to calculate maximum number of boxes that will fit along the breadth would be calculated as product of height and breadth along 2 faces of closet and length and breadth along another 2 faces of closet.
Thus, [(5*6) + (5*6) + (9*5) + (9*5)] = 150 boxes along the breadth.
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A class has 26 students. Each student has to present a project to the class. To decide the order, the teacher writes every letter in the
alphabet on pieces of paper and puts it in a hat. What is the probability that the first letter drawn is not a vowel?
A. 7/26
B. 19/26
C. 21/26
D. 5/26
Answer:
Choice B.
19/ 20
Step-by-step explanation:
This is because there are 26 letters in the alphabet, 19 of which are non-vowels.
Answer:
21/26
Step-by-step explanation:
because there are 26 letter in the alphabet
and there are 5 vowels
so 26 - 5 = 21
21/26
0.02 X 100 = 0.02 X_____=_____
Answer:
0.02 × 100 = 2
Step-by-step explanation:
move the decimal 2 steps backward according to the numbers of zeros behind the multiplier(100).
0.02 × 100 = 2
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You stand at zero on a number line and flip a coin. When the coin is heads, you move one unit to the right. When the coin is tails, you move one unit to the left. After each flip, you record your position on the number like. Let g represent your position after the nth flip.
a.Explain why g is a function
b. What does g(5)=3 represent
c. What is the probability that g(3)=0
By using the concept of number line, it can be calculated that
a) g is a function
b) g(5) = 3 means after 5 flips, the person is at '3' on the number line
c) Probability that g(3) = 0 is zero
What is a number line?
All the real numbers, integers, fractions can be pictorially represented with the help of a line. The line is known as number line.
a) Here, g represents the position after the nth flip
After n flips, a person stands at only one position on the number line, not more than one. So every point on the domain has unique image. So g is a function.
b) g(5) = 3 means after 5 flips, the person is at '3' on the number line
c) The person is initially standing at zero. To move to zero position, even number of flips have to be made. But 3 is not an even number. So probability that g(3) = 0 is zero
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Point D is in the interior of
Answer:
Angle ABC,
Step-by-step explanation:
Please give more detail to your question, this doesn't help at all...
Negative 3 (8 minus 5) squared minus (negative 7) = negative 3 (3) squared minus (negative 7) = negative 3 (9) minus (negative 7) = 27 minus (negative 7) = 34.
What was Huda’s error?
Huda evaluated (3) squared incorrectly.
Huda found the product of –3 and 9 as positive.
Huda subtracted –7 from 27 incorrectly.
Huda did not follow the order of operations.
Huda's error in evaluating (3) squared incorrectly led to the incorrect final result.
The correct answer should be -20, not 34.
Huda's error was that she evaluated (3) squared incorrectly.
Instead of calculating 3 squared as 9, she mistakenly considered it as 3. This error led to incorrect subsequent calculations and the final result of 34, which is not the correct answer.
To evaluate the expression correctly, let's go through the steps:
Negative 3 (8 minus 5) squared minus (negative 7) \(= -3(3)^2 - (-7)\)
First, we simplify the expression within the parentheses:
\(-3(3)^2 - (-7) = -3(9) - (-7)\)
Next, we evaluate the exponent:
-3(9) - (-7) = -3(9) + 7
Now, we perform the multiplication and addition/subtraction:
-3(9) + 7 = -27 + 7 = -20
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(3x + 7)
(10x – 22)
What does x equal?
Answer:
3x+7=10x-22
We simplify the equation to the form, which is simple to understand
3x+7=10x-22
We move all terms containing x to the left and all other terms to the right.
+3x-10x=-22-7
We simplify left and right side of the equation.
-7x=-29
We divide both sides of the equation by -7 to get x.
x=4.14285714286
Answer:
x=15
Step-by-step explanation:
(10x-22)+(3x+7)=180
Find the slope (-1,9) and (5,-6)
Answer:
-5/2
Step-by-step explanation:
The figure shown is composed of two cubes.
Which equation shows the surface area
ne figure?
SA = 6(5-5)+6(3-3)-2(3-3)
What is the surface area of
the figure shown?
cm²
3cm
5c
186 cm³ will be the surface area of the given figure.
To calculate the surface area of the given figure, we need to add the surface area of two cubes and reduce the area of the hidden part.
Thus,
Total surface area = Surface area of bigger cube + surface area of the smaller cube - 2* surface area of the hidden part.
So,
Total surface area = 6*(5*5)+6*(3*3)-2*(3*3)
Total surface area = 150+54-18
Total surface area = 186
Therefore, the surface area of the given figure will be 186 cm³.
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7/15% of a quantity is a equal to what fraction of the quantity?
Answer:
\( \frac{7}{15\%} = \frac{7}{ \frac{15}{100} } \\ = \frac{7 \times 100}{15} \\ = 46 \frac{2}{3} \)
Step-by-step explanation:
\(\bold{\dfrac{7}{15\%} }\)
\(\bold{\dfrac{7}{\dfrac{15}{100}} }\)
\(\bold{\dfrac{7×100}{15} }\)
\(\bold{\dfrac{7×\cancel{100}}{\cancel{15}} }\)
\(\bold{\dfrac{7×20}{3} }\)
\(\bold{\dfrac{140}{3} }\)
\(\bold{46\dfrac{2}{3} }\)
First one to answer receives 100 points and brainiest!
Prove the value of the expression(s):
#1: (2^10+2^8)(2^5-2^3) is divisible by 60.
#2: (16^3-8^3)(4^3+2^3) is divisible by 63.
#3: (125^2+25^2)(5^2-1) is divisible by 39.
3 images will be provided if you do not understand what to do.
Answer and Step-by-step explanation:
116 b.
\(2^{10} = 1024\\2^8 = 256\\\\1024 + 256 = 1280\\\\\\2^5 = 32\\2^3 = 8\\\\32 - 8 = 24\)
1280 × 24 = 30720
30720 ÷ 60 = 512. 512 is the solution; the expression is divisible by 60.
116 c.
\(16^3 = 4096\\8^3 = 512\\\\4096 - 512 = 3584\\\\\\4^3 = 64\\2^3 = 8\\\\64 + 8 = 72\)
3584 × 72 = 258048
258048 ÷ 60 = 4096. 4096 is the solution; the expression is divisible by 63.
116 d.
\(125^2 = 15625\\25^2 = 625\\\\15625 + 625 = 16250\\\\\\5^2 = 25\\\\25 - 1 = 24\)
16250 × 24 = 390000
390,000 ÷ 39 = 10,000. 10,000 is the solution; the expression is divisible by 39.
#teamtrees #PAW (Plant And Water)
2
Select the correct answer from each drop-down menu.
Triangle ABC is shown in the coordinate plane. It is translated 6 units left and 4 units down. Then it is dilated by a scale factor of 3 centered about the
origin. Complete the statements.
-6 -4
v.
6
4-
2-
lo
-2-
-4-
-6-
The translation of triangle ABC
A
2
с
6
B
Reset
The dilation of triangle ABC
Next
For the transformations of the triangle, we have that:
1. The translation of triangle ABC preserves side lengths and angles.
2. The dilation of triangle ABC preserves angles but not side lengths.
Hence the correct options are, respectively, C and B.
A translation involves shift left, shift right, shift down, or shift up, meaning that just the coordinates of the figure change, not the angles or the side lengths,
1. The translation of triangle ABC preserves side lengths and angles.
For a dilation, the coordinates of the vertices are multiplied by a constant, hence the side lengths change while the angles remain constant, thus:
2. The dilation of triangle ABC preserves angles but not side lengths.
Hence the correct options are, respectively, C and B.
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Pls help me fast rapidly:
Answer:
\(\huge\boxed{\sf 1.8 \times 10^3}\)
Step-by-step explanation:
Quotient means "to divide".
Solution:\(\displaystyle = \frac{4.68 \times 10^6}{2.6 \times 10^3} \\\\\underline{\sf Exponent \ rule:}\frac{a^m}{a^n} = a^{m-n}\\\\= 1.8 \times 10^{6-3}\\\\= 1.8 \times 10^3\\\\\rule[225]{225}{2}\)
Select all the expressions that are equivalent to - 2/5 (15-20d + 5c).
-6 + 8d - 2c
-2c + 8d - 6
-2(3 - 4d + c)
-30 + 40d - 100
O 6-8d + 20
The National Assessment of Educational Progress (NAEP) is a nationwide assessment of students’ proficiency in nine subjects: mathematics, reading, writing, science, the arts, civics, economics, geography, and U.S. history. The main NAEP assessments are conducted annually on samples of students from grades 4, 8, and 12.
In 2002, the reading scores for female students had a mean of 269 with a standard deviation of 33. Assume that these scores are normally distributed with the given mean and standard deviation.
Identify the scores that are three standard deviationsabove and below the mean of the population. For this example, the limits will be 269 ± (33)(3). The lower limit is . The upper limit is . The probability that a female student will have a score between these limits is .
A score of 302 is above the mean. As a result, the percentage of female students with scores below 302 is .
You can infer that 97.72% of the female students have scores above .
"97.72% of the female students have scores above," seems to be a misinterpretation as the correct statement is that approximately 99.72% of the female students have scores between the lower and upper limits
To calculate the scores that are three standard deviations above and below the mean, we use the formula:
Lower limit = Mean - (Standard Deviation * 3)
Upper limit = Mean + (Standard Deviation * 3)
Given:
Mean = 269
Standard Deviation = 33
Using the formula, we can calculate the limits:
Lower limit = 269 - (33 * 3) = 269 - 99 = 170
Upper limit = 269 + (33 * 3) = 269 + 99 = 368
Therefore, the lower limit is 170 and the upper limit is 368.
To calculate the probability that a female student will have a score between these limits, we need to find the area under the normal distribution curve between the lower and upper limits. This can be calculated using a standard normal distribution table or calculator.
Since the distribution is assumed to be normal, approximately 99.72% of the scores will fall within three standard deviations from the mean. Therefore, the probability that a female student will have a score between these limits is approximately 99.72%.
For a score of 302, which is above the mean of 269, we can calculate the percentage of female students with scores below 302:
Percentage = (1 - Probability) * 100
= (1 - 0.9972) * 100
= 0.0028 * 100
= 0.28%
Therefore, approximately 0.28% of the female students have scores below 302.
It's important to note that the value mentioned, "97.72% of the female students have scores above," seems to be a misinterpretation as the correct statement is that approximately 99.72% of the female students have scores between the lower and upper limits
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