Answer:
40 minutes
Step-by-step explanation:
Math Problems 3 15
---------------------- ----- = ------
Minutes 8 x
3x = 15 * 8
x = 15 * 8 / 3
x = 120 / 3
x = 40
40 Minutes
Leon verified that the side lengths 21, 28, 35 form a Pythagorean triple using this procedure.
Step 1: Find the greatest common factor of the given lengths: 7
Step 2: Divide the given lengths by the greatest common factor: 3, 4, 5
Step 3: Verify that the lengths found in step 2 form a Pythagorean triple: 3 squared + 4 squared = 9 + 16 = 25 = 5 squared
Leon states that 21, 28, 35 is a Pythagorean triple because the lengths found in step 2 form a Pythagorean triple. Which explains whether or not Leon is correct?
Yes, multiplying every length of a Pythagorean triple by the same whole number results in a Pythagorean triple.
Yes, any set of lengths with a common factor is a Pythagorean triple.
No, the lengths of Pythagorean triples cannot have any common factors.
No, the given side lengths can form a Pythagorean triple even if the lengths found in step 2 do not
Answer:
Yes, multiplying every length of a Pythagorean triple by the same whole number results in a Pythagorean triple.
Step-by-step explanation:
Another example of multiplying a Pythagorean triple is
Take the known Pythagorean triple 5, 12 and 13:
5 12 13 * 2 = 10 24 26
and 26^2 = 10^2 + 24^2
676 = 100 + 576 = 676
How does the figure help verify the triangle inequality theorem?
The line is drawn from a length of 15, two lines constructing a triangle and intersecting the arc from a length of the lines 7, 4
A.
by showing that a triangle cannot be formed when the sum of the lengths of two sides is less than the length of the third side
B.
by showing that only one triangle can be formed when the sum of the lengths of two sides equals the length of the third
C.
by showing that only one triangle can be formed when the sum of the lengths of two sides is less than the length of the third
D.
by showing that a triangle cannot be formed when the sum of the lengths of two sides equals the length of the third
The figure description helps; Choice A; by showing that a triangle cannot be formed when the sum of the lengths of two sides is less than the length of the third side.
What is the triangle inequality theorem?The triangle inequalities theorem postulates that the sum of lengths of two sides of a triangle must be greater than the length of the third side.
On this note, it follows that the since the sum of sides given in the description 7 and 4 is less than 15, the segments cannot be used to form a triangle.
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The table displays the scores of students on a recent exam. Find the mean of the scores to the nearest 10th.
The mean of the scores to the nearest 10th digit will be 74.9.
What is Mean?The mean is the straightforward meaning of the normal of a lot of numbers. In measurements, one of the markers of focal propensity is the mean.
Then the mean is given as,
Mean = (Total score) / (total student)
The total score will be
Total score = 65 x 8 + 70 x 7 + 75 x 3 + 80 x 8 + 85 x 3 + 90 x 2 + 95 x 2
Total score = 520 + 490 + 225 + 640 + 225 + 180 + 190
Total score = 2470
The total number of students will be
Total student = 8 + 7 + 3 + 8 + 3 + 2 + 2
Total student = 33
Then the mean will be
Mean = 2470 / 33
Mean = 74.85 ≈ 74.9
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ANSWER ASAP!
Tim is playing table tennis with a friend. the ball accidentally falls off the table and each bounce reaches a third of the height of the previous bounce. The ball starts bouncing with a height of 3 feet. Which sequence describes the balls bounce height?
A. 3, 6, 9, 12,...
B. 1, 3, 9, 27,...
C. 1/9, 1/3, 1, 3,...
D. 3, 1, 1/3, 1/9,...
The sequence of the ball is 3, 1, 1/3, 1/9,...
How to find terms of a sequence?The ball starts bouncing with a height of 3 feet.
Each bounce reaches a third of the height of the previous bounce.
Therefore,
a = first term = 3
r = common ratio = 1 / 3
Therefore,
nth term = arⁿ⁻¹
where
a = first termr = common ration = number of termsFirst term = 3
2nd term = 3 × (1 / 3) = 1
Third term = 3 × (1 / 3)² = 1 / 3
Fourth term = 3 × (1 / 3)³ = 1 / 9
Therefore, the sequence of the ball is 3, 1, 1/3, 1/9,...
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0.10x how much is the bill for a person who uses 600 kWh in a month
The total bill paid for a person who uses 600 kWh in a month is $60.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let us assume that the utility company charges $0.10 per kWh, hence:
Total charge in a month = 0.10 * 600 = $60
The total bill paid for a person who uses 600 kWh in a month is $60.
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The 1997 Red River flood was considered a 200-year flood. Why is it now considered only to be a 65-year flood
Large floods in 2007 and 2010 changed the flood recurrence curve.
What caused the Red River flood 1997?
A extremely unusual thawing of winter snow and river ice after a winter season that saw far above-normal precipitation across the Northern Plains was the main contributor to the flooding in 1997.
Why is the Red River called the Red River?
After it was explored in 1732–33 by the French voyageur Pierre Gaultier de Varennes et de La Vérendrye, the river, called Red because of the reddish brown silt it carries, served as a transportation link between Lake Winnipeg and the Mississippi River system.
How did the Red River flood affect people?The floodway was designed to manage a flow of 1,700 m3/s, but 1,800 m3/s was instead used. 28,000 people were evacuated due to the 1997 flood, which also caused $500 million in damage. Additionally, parts of Minnesota, North Dakota, and southern Manitoba were inundated by the river.Learn more about red river flood
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HELP!!!
A bag contains 1 white (W), 3 blue (B1, B2, B3), and 2 red (R1, R2) marbles. (4 points)
-Use a tree diagram to list all the possible outcomes for tossing a coin and then drawing a marble from the bag.-
SHOW ALL WORK
1. Are these independent or dependent events?
2. Find the probability of tossing a head and drawing a red marble. Explain your reasoning.
3. Find the probability of drawing two blue marbles if the first marble is not replaced. Explain your reasoning.
These are independent events
and probability of drawing red marble is 1/3
and probability of drawing two blue marble is 1/2 ,1/2.
According to the statement
A bag contains 1 white (W), 3 blue (B1, B2, B3), and 2 red (R1, R2) marbles.
And we have to find the probability of tossing a head and drawing a red marble. and probability of drawing two blue marbles if the first marble is not replaced.
here we solve this problem by Probability
So, Probability of drawing red marble = Number of red marble / total number of marbles.
Probability = 2/6
Probability = 1/3.
Then
Probability of drawing blue marbles = number of blue marbles /total marbles
Probability = 3/6
Probability = 1/2.
1st blue marble Probability =1/2
2nd blue marble Probability = 1-(1/2) = 1/2.
These are non dependent events because they do no affect the probability of each others.
So, These are independent events
and probability of drawing red marble is 1/3
and probability of drawing two blue marble is 1/2 ,1/2.
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Pls help me!! Due today:(
Answer:
Khan Academy.
Step-by-step explanation:
You can go on khan academy and search up "percentages and numbers." The videos should immediately pop up. there are also lessons if you don't want to risk getting your problem wrong. If you are still confused and do not understand please just comment and I will respond.
Condense log₂ 4 + log₂ 5
Answer: [tex]\log_2(20)[/tex]
Work Shown:
[tex]\log_2(4) + \log_2(5)\\\\\log_2(4*5)\\\\\log_2(20)\\\\[/tex]
The rule used is [tex]\log_{b}(\text{x}) + \log_{b}(\text{y}) = \log_{b}(\text{xy})[/tex]
[tex]\large{\textbf{Heya !}}[/tex]
✏[tex]\large\sf{\bigstar Given:-}[/tex] ✏
㏒ expression = [tex]\sf{\log_24+\log_25}[/tex]✏[tex]\large{\sf{\bigstar To\quad Find:-}[/tex] ✏
Condense the ㏒ into ONE expression - ?✏[tex]\large{\sf{\bigstar Solution\quad steps:-}[/tex]
`To find what we want we need to apply ㏒ rules
[tex]\large{\sf{\longmapsto{log_ba+log_bc=log_b(ac)}[/tex]
using formula,
[tex]\large{\sf{\longmapsto{log_24+log_25=log_2(4\cdot5)}[/tex]
simplifying,
[tex]\large{\sf{\longmapsto{log_220}[/tex]
`hope it was helpful ! ~
Can someone please help me answer this question?
Answer:
(c) 0.18
Step-by-step explanation:
If the x-values are exhaustive, then the p(X=x) values must total 1.00.
Probability total0.11 +? +0.62 +0.09 = 1.00
0.82 +? = 1.00 . . . . . . . simplify
? = 0.18 . . . . . . . . . subtract 0.82
P(X=3) = 0.18
Determined to save money and eat healthier, John and his family decide to build a vegetable garden. They order fencing
divide the bottom part of the garden from the top and also to go around the vegetable plot, but not on the outside border
plot. There also needs to be 2 gates measuring 2.5ft that are not fenced.
Use the conversion factors
1 foot = 30.48cm
1 foot = 12 inches
Available fencing panes measure 7 inches in width. How many panels will they need to
buy? Round your answer to the nearest whole number.
panes
10m
Pond
*13m
Gate
6.5m
10m
1.3m
Lawn
Fencing
The answers are:
The perimeter of vegetable plot = 19mArea = 67.6m²Fencing = 32.5mWhat is the perimeter about?Given that:
The perimeter of vegetable plot:
= 4m + 3.5m + 4m + 2m + (3.5 + 2m)
= 19m
Area of vegetable plot:
(1.3 x 5.5) + ( (4 - 1.3) x 3.5)
rounding up to one decimal place = 67.6m²
To know the numbers of meter fence that one can buy, the equation will be:
(perimeter of vegetable plot + the left plot) - gates.
(19m + (6.5 + 4 + (6.5 - 2) ) )m - (2.5ft x 2)
= 19m + 15m - 1.5424m
=32.476
=32.5m
Therefore, the answers are:
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If a trip to Andover is 866 miles, and your car avg miles per gallon is 13.6. With the average cost of gas being 4$, how much would it cost?
Answer:
[tex]254 \frac{12}{17}[/tex] or approximately 254.71
Step-by-step explanation:
First, you need to find out how many gallons it will take to travel 866 miles. Because the car's avg miles/gallon is 13.6, you divide 866 by 13.6. However, this gets you an irrational number. To avoid this, make 13.6 into a fraction. Now divide 866 by [tex]13 \frac{3}{5}[/tex]. This becomes 866 ÷ [tex]\frac{68}{5}[/tex]. Convert this into a multiplication problem by converting [tex]\frac{68}{5}[/tex] into its reciprocal. This gets you 866 × [tex]\frac{5}{68}[/tex]. This gets you [tex]\frac{2165}{34}[/tex]. Now that you have the amount of gallons it will take to get to your destination, multiply it by 4. [tex]\frac{2165}{34}[/tex] × [tex]4[/tex][tex]= \frac{4330}{17}[/tex]. This becomes [tex]254 \frac{12}{17}[/tex] or approximately 254.71.
Complete the proof of the Pythagorean theorem.
Given: abc is a right triangle, with a right angle at Prove: a^2+ b^2=c^2
(Please see image for the fill in the blank questions and please answer correctly or don't just guess for points. I only need the ones shown in the image, I was able to get 12 through 17.)
The fill up are:
5. ∠B ≅ ∠B
6. AA Similarity Postulate
7. a/c = c/a
8. a² = cx
9. ∠CDA ≅ ∠BCA
10. ∠A ≅ ∠A
11. ΔCBA ~ ΔDBA
What is Similarity Postulate?The above answers shows that:
5. Reflexive Property
6. AA Similarity Postulate
7. Polygon Similarity Postulate
8. Cross Multiply and Simplify
9. All right angles are Congruent
10. Reflexive Property
11. AA similarity Postulate
To prove the above:
Note that the full proof of the Pythagorean theorem is that:
We are given: Δ ABC as a right triangle, with a right angle at ∠C.
Then the Prove: A²+B² =C²
Therefore, If you follow the theories above, you will see that the answers given are correct.
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The total value of a collection of nickels can be described by this sequence: 5, 10, 15, 20, 25, 30, ...
Type of sequence:
Explicit formula:
Recursive formula:
Type of sequence: arithmetic progression
Explicit formula: 5n
Recursive formula: [tex]5_{n-1}[/tex] + 5
What does the word Sequence mean ?A Sequence is a series of numbers in a regular pattern of increment. A sequence can be arithmetic or geometric in nature.
Given that a total value of a collection of nickels can be described by this sequence: 5, 10, 15, 20, 25, 30, .....
The difference in the sequence above is 10 - 5 = 5. There is an increment of 5 in each term.
Therefore, the type of sequence is arithmetic progression
The difference d = 5 and the first term is also 5
An explicit formula will be
a + (n - 1)d
substitute a and d into the formula
5 + (n - 1)5
5 + 5n - 5
5n
Therefore, 5n gives the value of a specific term based on the position
A recursive formula will be
[tex]a_{n-1}[/tex] + d
substitute a and d into the formula
[tex]5_{n-1}[/tex] + 5
Therefore, [tex]5_{n-1}[/tex] + 5 gives the value of a specific term based on the previous term
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1-10) Matching. Use OA below to match each part to its corresponding label.
PLEASE HELP DUE IN A FEW
The matched answers are:
1. semicircle
2. central angle
3. Radius
4. Secant
5. center
6. tangent
7. diameter
8. minor arc
9. chord
10. major arc
What is a circle?A circle is the locus of points equidistant from a given point called the center of the circle.
A circle can be described if its radius and the center of the circle is known.
Analysis:
1. RCH is called a semicircle when a diameter divides a circle into two equal parts, a part is called the semicircle.
2. ∠RAH is the central angle, because it is the angle subtended at the Center of the circle.
3. CA is the radius of the circle, a line drawn from the center to any point on the circumference of the circle.
4. MR is a secant, a line that cuts the circle into two parts.
5. A is the center of the circle.
6. MT is the tangent, because line MT touches the circle at a point T.
7. CH is the diameter of the circle, a line that divides the circle into two semi circles.
8. CT is the minor arc, the curved line that is found in the smaller part of the divided circle.
9. HT is chord, it is a line that divides the circle into two unequal parts.
10. CRH is major arc, the curved line of the bigger portion of the divided circle.
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Which of the following expressions is equal to 3x² +27?
Answer:
B. (3x - 9) (x + 3i)
Step-by-step explanation:
[tex]\bf 3 {x}^{2} = + 27[/tex]
[tex]\bf (3x - 9)(x + 3i)[/tex][tex]\bf 3 {x}^{2} +9 xi - 9xi - 27 {i}^{2} [/tex][tex]\bf 3 {x}^{2} - 27( {i}^{2} )[/tex][tex]\bf 3 {x}^{2} - 27( - 1)[/tex][tex]\bf 3 {x}^{2} + 27[/tex]Therefore, B is your answer!!!!
suppose sin(A) = 1/4. use the trig identity sin^2(A)+cos^2(A)=1 to find cos(A) in quadrant II. around to ten-thousandth.
a. 0.1397
b. 0.4630
c. -0.8572
d. -0.9682
In quadrant II, [tex]\cos(A)[/tex] will be negative. So
[tex]\sin^2(A) + \cos^2(A) = 1 \implies \cos(A) = -\sqrt{1 - \sin^2(A)} = -\dfrac{\sqrt{15}}4 \approx \boxed{-0.9682}[/tex]
The requried, rounded to four decimal places, cos(A) is approximately -0.9682. The correct answer is option d. -0.9682.
To find cos(A) in quadrant II, we can use the given information and the trigonometric identity [tex]sin^2(A) + cos^2(A) = 1[/tex].
Given [tex]sin(A) = 1/4[/tex], we can find [tex]cos(A)[/tex] as follows:
[tex]sin^2(A) + cos^2(A) = 1[/tex]
[tex](1/4)^2 + cos^2(A) = 1[/tex]
[tex]1/16 + cos^2(A) = 1[/tex]
Now, let's solve for [tex]cos^2(A)[/tex]:
[tex]cos^2(A) = 1 - 1/16[/tex]
[tex]cos^2(A) = 16/16 - 1/16[/tex]
[tex]cos^2(A) = 15/16[/tex]
Next, find the value of [tex]cos(A)[/tex] by taking the square root:
[tex]cos(A) = \pm \sqrt{(15/16)[/tex]
Since we are in quadrant II, where cosine is negative, we take the negative value:
[tex]cos(A) \approx -\sqrt{(15/16)[/tex]
Now, calculate the approximate value of cos(A):
[tex]cos(A) \approx -0.9682[/tex]
Rounded to four decimal places, cos(A) is approximately -0.9682. The correct answer is option d. -0.9682.
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the length of a rectangle exceeds its width by 17 inches, and the area is 18 square inches. what are the length and width of the rectangle
The length and width of the rectangle are 18 in and 1 in, respectively.
QuadrilateralsThere are different types of quadrilaterals, for example, square, rectangle, rhombus, trapezoid, and parallelogram. Each type is defined accordingly to its length of sides and angles. For example, in a rectangle, the opposite sides are equal and parallel and their interior angles are equal to 90°.
The area of a rectangle can be found for the formula : b*h, where b = base and h =height.
For this question, the length exceeds its width by 17 inches - L=W+17. Here, the length is the base and the width is the height. Thus, from the value of area given, you can find the values of the length and width of the rectangle.
A=b*h
18=(W+17)*W
18=W²+17W
W²+17W-18=0
Solving this quadratic equation, you have:
[tex]w_{1,\:2}=\frac{-17\pm \sqrt{17^2-4\cdot \:1\cdot \left(-18\right)}}{2\cdot \:1}[/tex]
[tex]w_{1,\:2}=\frac{-17\pm \:19}{2\cdot \:1}[/tex]
w1=1 and w2= -18
For dimensions, only positive numbers must be used. Then, the width is equal to 1 inch.
As, the area (l*w) is 18 in², see.
18=l*w
18=l*1
l=18 in
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Answer:
9
——- =
-2.1
Answer:
-9/2
In alternate forms -4•5,-4 1/2
Determine which graph represents this system of equations to complete the statements that follow. y = -x2 + x + 5, y = -x + 2.
The solution to the system of equations is (-1, 3) and (3, -1)
How to graph the equations?The system of equation is given as:
y = -x^2 + x + 5
y = -x + 2
Next, we plot the graph of the system using a graphing tool
From the graph, both inequalities intersect at
(-1, 3) and (3, -1)
Hence, the solution to the system of equations is (-1, 3) and (3, -1)
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Answer:
graph 1,
3,-1 and -1,3
Step-by-step explanation:
See screenshot
In a certain clothing store, 6 shirts and 3 ties cost $79.50, and 3 shirts and 2 ties cost $41. Determine the cost of each shirt.
a) $10.00
b) $11.00
c) $12.00
The price of shirt is $12 and the price of tie is $2.5.
According to the statement
We have given that:
Let money spent on shirt = X and money spent on tie is Y.
So, 6X + 3Y = $79.5 and 3X + 2Y = $41 then
We have to find the the cost of each shirts.
Here we use the Elimination method.
Now,
6X + 3Y = $79.5 -(1)
3X + 2Y = $41 -(2)
By elimination method
Multiply (1) by 3 and (2) by 6 then
18X + 9Y = $238.5
18X + 12Y = $246
Then solve it and equation become
-3Y = -$7.5
Y = 2.5
It means the cost of tie is $2.5 and now the cost of shirt is
6X + 3Y = $79.5
6X + 3(2.5) = $79.5
6X + 7.5 = $79.5
6X = $72
X = $12.
So, The price of shirt is $12 and the price of tie is $2.5.
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Pure beeswax has a density of 0.555 ounce per cubic inch. An online company sells pure beeswax at a price of $8.00 per ounce. What is the selling price, in dollars per cubic inch, for pure beeswax purchased from this company
4.44 is the selling price, in dollars per cubic inch, for pure beeswax purchased from this company given that pure beeswax has a density of 0.555 ounce per cubic inch and an online company sells pure beeswax at a price of $8.00 per ounce. This can be obtained by multiplying the values.
What is the selling price?Given that,
Density of pure beeswax = 0.555 ounce per cubic inch
Cost of beeswax per ounce = $8.00 per ounce
Selling price in dollars per cubic inch will be,
⇒0.555×$8.00 = $4.44
Hence 4.44 is the selling price, in dollars per cubic inch, for pure beeswax purchased from this company given that pure beeswax has a density of 0.555 ounce per cubic inch and an online company sells pure beeswax at a price of $8.00 per ounce.
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4(2a+3b)+5(2a=4b) need help
Answer:
18a-8b
Step-by-step explanation:
i think this right but i think u messed up putting an equal sign instead of a subtraction.
sharon mixed meat with potatoes in the ratio 7 : 3 to make 4kg of meat loaf. how much meat did she use
Answer:
2.8kg
Step-by-step explanation:
There are 7 parts of meat for every 3 parts so there are 7+3=10 parts in total.
4/10=1 "part"
there are 7 parts of meat so (4/10)*7=28/10=2.8kg of meat.
The lateral surface area of cone A is exactly 1/2 the lateral surface area of the cylinder B.
A. True
B. False
The statement, "the lateral surface area of cone A is exactly half of that of cylinder B" is: A. True.
What is the Lateral Surface Area of a Cone and a Cylinder?Lateral surface area of cone = πrl
Lateral surface area of cylinder = 2πrh
Lateral surface area of cone A = πrl = πrh
Lateral surface area of cylinder B = 2πrh
This means the lateral surface area of cylinder B is twice that of cone A. Thus, lateral surface area of cone A is exactly half of that of cylinder B.
The answer is: A. True.
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Gelatin Methacryloyl Hydrogels in the Absence of a Crosslinker as 3D Glioblastoma Multiforme (GBM)-Mimetic Microenvironment
For the eradication of malignancies like the aggressive type of human brain tumour known as glioblastoma multiforme (GBM), 3D platforms are crucial for tracking tumour growth and assessing treatment candidates.
What is Glioblastoma Multiforme (GBM)?Glioblastoma (GBM), commonly known as a grade IV astrocytoma, is an aggressive brain tumour that grows quickly. The local brain tissue gets invaded, but it typically does not spread to distant organs. GBMs can develop from lower-grade astrocytomas or develop from new brain tumours.
Some characteristics of GBM are-
We explore cellular behavior, expression profiles of malignancy, and apoptosis-related genes within this novel network using suspension and spheroid-based models of GBM. In-depth research is also done on the sensitivity to the anticancer medication Digitoxigenin. According to the research, GelMA hydrogels contain high levels of prosurvival Bcl-2 gene expression and sparse or spheroid U373 cells, which have much lower expressions of apoptosis-activating genes like Bad, Puma, and Caspase-3.The up regulation of matrix-metalloproteinase genes within GelMA suggests that gels have a beneficial effect on the extracellular remodelling of cancer cells. Through the investigation of 3D malignant cancer cell behavior, this special photo curable gelatin shows significant promise for the practical translation of cancer research and, consequently, for more effective treatment techniques for GBM.To know more about brain tumour, here
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Helpppppppp pleaseeee
Answer:
Vertical Angles TheoremTransitive Property of CongruenceAt a certain fast food restaurant, 77.5% of the customers order items from the value menu. If 14 customers are randomly selected, what is the probability that at least 9 customers ordered an item from the value menu
The probability that at least 9 customers ordered an item from the value menu is 0.927.
What is binomial probability?In an experiment with two possible outcomes, the likelihood of precisely x successes on n repeated trials is known as the binomial probability (generally refereed as binomial experiment).
As per the given problem-
This likelihood is a binomial one, so take note. In this instance, a success is defined as a consumer placing an order from the value menu, hence we are looking for such probability between 9 to 14 success, inclusive.
The probability is indeed the complements of the likelihood of any success, ranging from 0 to 8. The procedures below can be used to figure out the probability from such a binomial distribution utilizing Excel.-
Press FORMULAS first, followed by INSERT FUNCTION.Secondly, choose the BINOM.DIST function.Next, input the figures for the quantity of successes, the quantity of trials, the likelihood of success, and the quantity of successes. Insert 8, 14, then 0.775 in this case, in that order. Because it is a cumulative probability, enter 1 for Cumulative.Click OK. The probability should then be shown in Excel. The probability that results in this case is 0.072766, or 0.073 scaled to three decimals.Subtract the preceding probability from 1 to get the likelihood of 9 to 14 people ordering out from value menu.
The likelihood is 1 x 0.073 = 0.927.
Therefore, the probability that at least 9 customers ordered an item from the value menu is 0.927.
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How many different ways are there to choose a dozen donuts from the 4 varities at a donut shop so that you get atleast one donut of every variety g
There are 330 different ways for choosing a dozen donuts from the 4 varieties at a donut shop. (at least one donut of every variety must be selected)
How to calculate the number of ways to select items?There are 'r' items from 'n' different varieties (repetition allowed). Then, the number of ways to select items is given by
The number of ways = [tex]_{n+r-1}C_r[/tex]
Where the combination [tex]_{n}C_r[/tex] is calculated as
[tex]_{n}C_r=\frac{n!}{r!(n-r)!}[/tex]
Calculation:It is given that there are 4 varieties of donuts in a shop. I.e., n = 4
Number of donuts to be selected r = 12 (one dozen)
And also given that at least one donut of every variety has to be selected.
Since there are 4 varieties, at least one from each of these means the count is 4.
So, the remaining number of donuts to be selected is 12 - 4 = 8.
So, r becomes 8 i.e., r = 8
On substituting,
the number of ways of selecting the remaining 8 donuts = [tex]_{4+8-1}C_4[/tex]
⇒ [tex]_{11}C_4[/tex]
⇒ [tex]\frac{11!}{4!(11-4)!}[/tex]
⇒ [tex]\frac{11!}{(4!)(7!)}[/tex]
⇒ 330
Therefore, there are 330 different ways for choosing a dozen donuts from the 4 varieties at a donut shop.
Learn more about combinations here:
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A triangle has vertices at (4, 4), (-6, 2) and (2, 0).
a. Find the coordinates of the mid-points of eachside
b. Find the lengths of the sides of the triangle
formed by joining the mid-points.
each side.
Answer:
A. (-1, 3) (3, 2) (-2, 1)
B. [tex]\sqrt{17}[/tex] [tex]\sqrt{5}[/tex] [tex]\sqrt{26}[/tex]
Step-by-step explanation:
The formula for finding the midpoints is [tex](\frac{x_{1}+x_{2} }2}, \frac{y_{1}+y_{2} }2} )[/tex].
MIDPOINTS
[tex](\frac{x_{1}+x_{2} }2}, \frac{y_{1}+y_{2} }2} )[/tex] = [tex](\frac{4-6}2,\frac{4+2 }2} )[/tex] = (-1, 3)
[tex](\frac{x_{1}+x_{2} }2}, \frac{y_{1}+y_{2} }2} )[/tex] = [tex](\frac{4+2}2,\frac{4+0 }2} )[/tex] = (3, 2)
[tex](\frac{x_{1}+x_{2} }2}, \frac{y_{1}+y_{2} }2} )[/tex] = [tex](\frac{-6+2}2,\frac{2+0 }2} )[/tex] = (-2, 1)
Next, we will use the distance formula, [tex]\sqrt{(x_{1}-x_{2})^{2} + (y_{1}-y_{2})^{2} }[/tex].
DISTANCE
[tex]\sqrt{(x_{1}-x_{2})^{2} + (y_{1}-y_{2})^{2} }[/tex] = [tex]\sqrt{(-1-3)^{2} + (3-2)^{2} }[/tex] = [tex]\sqrt{17}[/tex]
[tex]\sqrt{(x_{1}-x_{2})^{2} + (y_{1}-y_{2})^{2} }[/tex] = [tex]\sqrt{(-1+2)^{2} + (3-1)^{2} }[/tex] = [tex]\sqrt{5}[/tex]
[tex]\sqrt{(x_{1}-x_{2})^{2} + (y_{1}-y_{2})^{2} }[/tex] = [tex]\sqrt{(3+2)^{2} + (2-1)^{2} }[/tex] = [tex]\sqrt{26}[/tex]
a. To find the coordinates of endpoints we must add two x values and divide by 2 and then add 2 y- values and divide by 2.
(4-6)/2=-1 (4+2)/2=3
Repeat for other sides.
(2-6)/2=-2 (2+0)/2=1
(2+4)/2=3 (4+0)/2=2
Coordinates of midpoints are (-1,3), (-2,1), (3,2)
b. Now we use the distance formula for each midpoint to find the length of the inner- triangle.
sqrt((-1+2)^2 + (3-1)^2)
Sqrt(5)
Repeat.
Sqrt(17)
Sqrt(26)
The lengths of the inner triangle are as follows:
(-1,3), (-2,1) = Sqrt(5)
(-1,3), (3,2) = Sqrt(17)
(-2,1), (3,2) = Sqrt(26)