The Greatest Common Factor
The greatest common factor (or divisor) (GCF or GCD or HCF) of a set of whole numbers is the largest positive integer that divides evenly into all numbers of that set.
For example, we have the numbers 16 and 24. One common divisor is 2 because 16/2=8 and 24/2=12. Both divisions are exact.
But 2 is not the greatest common divisor. To find the GCF, the procedure is:
* Write the prime factor of each number:
16 = 2*2*2*2
24 = 2*2*2*3
Select the maximum number of common prime factors from the numbers, even if they repeat.
We can select the product of 2*2*2 = 8 as the GCF, thus:
Answer = 8
here is a net of right triangles and rectangles all measurements are given in centimeters.
Problem
Solution
For this case we we can find the area on this way:
[tex]A=\frac{5\cdot4}{2}+6\cdot4+6\cdot5+6\cdot3+\frac{4\cdot3}{2}[/tex]And solving we got:
[tex]A=10+24+30+18+6=88[/tex]The area for this case is 88 unit^2
1. The graph shown represents the altitude of a hikerduring a period of time. Write a possible situationrepresented by the graph.Altitude (feet)Time (hours)2. Use the vertical line test to determine if the relation represented on the graph
The vertical line test consists in tracing various vertical lines throughout the function and checking wheter this lines will touch the function more than once.
As we can see none of the lines touch the graph more than once, therefore this graph is a function.
The radius of a circle is 8 miles. What is the area of a sector bounded by a 180° arc?Give the exact answer in simplest form. ____ square miles.
Radius r:
r = 8 miles
Area of a circle:
A = π * r²
The area of a 180° arc is the half of the area of the entire circle:
A_arc = (π * r²)/2
Solving:
A_arc = 32π square miles
A plane flew 1225 mi in 5 hr with the wind. It would take 7 hr to travel the same distance against the wind. What is the speed of the plane in still air and the speed of the wind?
Answer:
wind speed 35 miles per hour
plane speed 210 miles per hour
Explanation:
Let us call p the place speed and w the wind speed.
We know that
p + w = 1225/5 = 245
p - w = 1225/7 = 175
adding the two equations above gives
2p = 245 + 175
2p = 420
dividing both sides by 2 gives
p = 210 miles per hour.
putting in the value of p in p + w = 245 gives
210 + w = 245
w = 245 - 210
w = 35 miles per hour
PLEASE HELP!!!!!!!!
Choose the two graphs that preserve congruence.
The two graphs that reserve congruence are in the option A and option D
What is congruency?Congruent refers to something that is "absolutely equal" in terms of size and shape.
The shapes hold true regardless of how we rotate, flip, or turn them.
Draw two circles with the same radius, for instance, cut them out, and stack them on top of one another. We will see that they will superimpose, or be positioned entirely on top of, one another. This demonstrates the congruence of the two circles.
The images on the graphs in the options A and options D both maintains congruence, The sizes are identical and has a rigid transformation
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find al the solutions for x.9. 8x2+19 = 54 +3x
8x^2 + 19 = 54 + 3x^2
Solving for x:
8x^2 - 3x^2 = 54 - 19
5x^2 = 35
x^2 = 35/5 = 7
x^2 = 7
x = sqrt(7) = 2.6458
x = -sqrt(7) = -2.6458
Answer:
It has two solutions:
x = 2.6458 and x = -2.6458
In the diagram, BAE is a semicircle, and mZACE = 28° . Based on your explorations, which of the following statements must be true. Select all that apply.
Since the arc BAE is a semicircle, it measures 180°, and the angle that inscribes it, that is, angle ∠BCE, has half the measure, so ∠BCE = 90°.
The angle ∠ACE inscribes the arc AE, so the arc AE has double the measure of the angle ∠ACE, so AE = 56°.
Calculating the measure of the arc AB, we have:
[tex]\begin{gathered} AB+AE=BAE \\ AB+56=180 \\ AB=180-56 \\ AB=124\degree \end{gathered}[/tex]So the first option is correct.
For the second option, these two angles inscribes the same arc (arc AE), so they have the same measure of half the measure of the arc. Therefore, they are congruent, so the second option is correct.
For the third option, there is nothing that proves that these angles are congruent, so the third option is false.
For the fourth option, there is nothing that proves that AC is a diameter, so the fourth option is false.
For the fifth option, the angle ∠BDE inscribes an arc of 180° (semicircle), so it has half the measure of the arc, therefore ∠BDE = 90°. So the fifth option is correct.
How many gallons each of 25% alcohol and 10% alcohol should be mixed to obtain 15 gal of 21% alcohol?Gallons ofPure AlcoholGallons ofSolutionХy15Percent(as a decimal)25% = 0.2510% = 0.121% =How many gallons of 25% alcohol should be in the mixture? gal
Answer
11 gallons of the 25% alcohol is required for the mixture.
4 gallons of the 10% alcohol is required for the mixture.
Explanation
Let the number of gallons of 25% alcohol required be x
Let the number of gallons of 10% alcohol required be y
The total amount of gallons required is 15 gallons. In mathematical terms,
x + y = 15
The alcohol content of the 15 gallons is to be 21%.
21% of 15 gallons = 3.15 gallons
From the first statements,
Let the number of gallons of 25% alcohol required be x
Let the number of gallons of 10% alcohol required be y
25% of x gallons = 0.25x gallons
10% of y gallons = 0.10y gallons
0.25x + 0.10y = 3.15
We can then bring these two equations together to solve simultaneously
x + y = 15
0.25x + 0.10y = 3.15
Solving this simultaneously with the calculator, we obtain
x = 11 gallons
y = 4 gallons
Hope this Helps!!!
evaluate the expression:[tex] \frac{8}{5 - 1} \times (3 + 6) \times 3[/tex]A)102B)-12C)62D)54
To evaluate the expression:
[tex]\frac{8}{(5-1)}\cdot(3+6)\cdot3[/tex]We can use PEMDAS. We can start from the parenthesis, then, since we have divisions and multiplications, and they have the same precedence, we can start doing the evaluation from the left to the right. Then, we have:
[tex]\frac{8}{(4)}\cdot(9)\cdot3=2\cdot9\cdot3=54[/tex]The answer is 54 (option D).
What is the surface area of the prism below2.5 m2 m10 m1.5 m
The lateral surface area is the total surface area minus the area of the two triangular faces at the top and bottom of the prism. t
Three volunteers are chosen at random from a group of 20 people to help at a camp. How many unique groups of volunteers are possible?
In mathematics, a combination is a selection of items from a set that has distinct members
Formula
[tex]^n_{^{}}C_r=\frac{n\text{ !}}{(n-r)!r!}[/tex]Where
n = 20
r =3
[tex]\begin{gathered} ^{20}C_3=\frac{20\text{ !}}{(20-3)!3!} \\ \\ \\ ^{20}C_3=\frac{20\text{ !}}{17!3!} \\ \\ \\ ^{20}C_3=\frac{20\text{ }\times19\times18\times17!}{17!3!} \\ ^{20}C_3=\frac{20\text{ }\times19\times18}{3!} \\ \\ ^{20}C_3=\frac{20\text{ }\times19\times18}{3\times2\times1} \\ ^{20}C_3=\frac{20\text{ }\times19\times18}{6} \\ \\ ^{20}C_3=20\text{ }\times19\times3 \\ \\ \\ ^{20}C_3=1140 \end{gathered}[/tex]The final answer
1140 unique groups of volunteers are possible
The total cost function for a product is given by C(x)=3x3−9x2−243x+1229, where x is the number of units produced and C is the cost in hundreds of dollars. Use factoring by grouping and then find the number of units that will give a total cost of at least $50,000. Verify the conclusion with a graphing utility.
C(x) = 3x³ − 9x² − 243x + 1229
500 = 3x³ − 9x² − 243x + 1229
Subtract 500 on each side, as follows:
0 = 3x³ − 9x² − 243x + 729
(3x³ − 9x²) - (243x - 729) = 0
3x² (x - 3) - 243 (x - 3) = 0
(x - 3)(3x² - 243) = 0
x = 3, x = 9, x = -9; in the context of this problem, where x is the number
of units produced, negative values of x must be omitted, so x = 3 and x = 9
So we can say that if either 3 units or 9 units are produced the total cost
for the product will be at least $50000
[0, 3] U [9,∞)
(1.26 x 10^3) + (1.12 x 10^4)
Work needs to be shown!!!
Answer:
Step-by-step explanation:
1260+11200=12460
so its, 1.246 x 10^4
Jim and Carla are scuba diving. Jim started out 8 feet below the surface. He descended 18 feet, rose 5 feet,and descended 9 more feet. Then he rested. Carla started out at the surface. She descended 16 feet, rose 5feet, and descended another 18 feet. Then she rested. Which person rested at a greater depth? Completethe explanation.
To solve this question, we need to use integers to express different altitudes. We can sketch each situation as follows:
To find the depth at which each person rested, we need to algebraically sum all these altitudes.
Jim:
-8 -18 + 5 - 9 = -30. He rested at 30 feet below the surface.
Carla:
-16 + 5 - 18 = -29. She rested at 29 feet below the surface.
Therefore, Jim rested at a greater depth (30 feet below the surface).
A model airplane is built at a scale of 1 inch to 32 feet. If the model plane is 8 inches long, the actual length of the airplane is blank feet.
If the model plane is built at a scale of 1 inch to 32 feet
let x be the actual length of the airplane
1/32 = 8/x
cross-multiply
x = 32 times 8
x = 256 inches
A rectangular patio measures 20 ft by 30 ft. You want to double the patio area by adding the same distance x to the length and width. Write and solve an equation to find a value for x, then find the new dimensions for the patio.
area = side x side = 20 x 30 =600 ft
we want to double the area, this is 1200 ft, therefore:
[tex]\begin{gathered} \mleft(x+20\mright)\mleft(x+30\mright)=1200 \\ x^2+50x+600=1200 \\ x^2+50x-600=0 \\ \mleft(x+60\mright)\mleft(x-10\mright)=0 \\ x+60=0 \\ x=-60 \end{gathered}[/tex]but, We don't measure in negative numbers, so we disregard this answer. and the other
[tex]\begin{gathered} x-10=0 \\ x=10 \end{gathered}[/tex]therefore, the patio is now 30ft x 40ft
she has 78 inches of thread that she cut into 2 pieces. One piece is twice as long as the other piece. How long is each piece?
Answer:
One piece is 26 inches long while the other is 52 inches long.
We let:
x = one piece of the thread
2x = the other piece of the thread
Since the thread is 78 inches long,
2x + x = 78
Solve for x:
[tex]2x+x=78\Rightarrow3x=78[/tex][tex]\frac{3x}{3}=\frac{78}{3}\Rightarrow x=26[/tex]Since one piece of the thread is 26 inches long, the other piece would be:
[tex]2x=2(26)=52[/tex]The other piece would be 52 inches long.
a farmer wants to build a fence in the shape of a parallelogram for his animals. The perimeter of the fence will be 600 feet, and the North/South fences are half of the length of the West/East fences. If fences are sold in 5 foot segments, how many fence segments does the farmer need to buy ?
Let's use the variable L to represent the length and W to represent the width.
If the perimeter is 600 ft, we have:
[tex]\begin{gathered} P=2L+2W\\ \\ 2L+2W=600\\ \\ L+W=300 \end{gathered}[/tex]The width is half the length, so we have:
[tex]\begin{gathered} W=\frac{L}{2}\rightarrow L=2W\\ \\ 2W+W=300\\ \\ 3W=300\\ \\ W=\frac{300}{3}\\ \\ W=100\text{ ft}\\ \\ L=200\text{ ft} \end{gathered}[/tex]Now, if each fence segment is 5 ft, we number of segments needed is:
[tex]\begin{gathered} \text{ fences for W1: }\frac{100}{5}=20\\ \\ \text{ fences for W2: }\frac{100}{5}=20\\ \\ \text{ fences for L1:}\frac{200}{5}=40\\ \\ \text{ fences for L2:}\frac{200}{5}=40\\ \\ \\ \\ \text{ total:}20+20+40+40=120\text{ fence segments} \end{gathered}[/tex]Determine the value of k for which the inequality $0.5<-4x+k\le12-k$ has the solution set $\left\{x|1.25\le x<2\right\} Need the answer ASAP.
The value of k for which the inequality has the given solution set is:
k = 8.5.
How to obtain the value of k?The inequality is presented as follows:
0.5 < -4x + k ≤ 12 - k.
Two inequalities compounded, hence the and operation is applied, which means that the solution set is composed by the elements that respect both conditions.
The solution set of the inequality is:
1.25 ≤ x < 2.
The lower bound of the solution is of 1.25, hence:
-4x + k ≤ 12 - k.
-4x + 2k ≤ 12
4x ≥ 2k - 12
x ≥ 0.5k - 3
Hence:
0.5k - 3 = 1.25
0.5k = 4.25
k = 4.25/0.5
k = 8.5.
The upper bound of the solution is of 2, hence:
-4x + k > 0.5
-4x > 0.5 - k
x < -0.125 + 0.25k
Hence:
-0.125 + 0.25k = 2
k = 2.125/0.25
k = 8.5. -> Which confirms the solution.
Missing InformationThe problem is given by the image shown at the end of the answer.
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the table below shows the minimum volume of water needed to fight a typical fire in rooms of various sizes. Find the rate of change. Explain the meaning of rate of change. Include the units in your answer.
Since the volume of water depends on the floor area, the floor area is the independent variable while the minimum water volume is the dependent variable.
Let x represents the floor area
Let y represent the miminum volume of water
The rate of change is the ration of the change in y to the change in x
Let the rate of change be represented by dy/dx
[tex]\begin{gathered} \frac{dy}{dx}=\text{ }\frac{y_2-y_1}{x_2-x_1} \\ \end{gathered}[/tex]Considering the first two rows of the table, since the rate of change does not differ despite the rows picked
[tex]\begin{gathered} x_1=25,y_1=39,x_2=50,y_2=78 \\ \frac{dy}{dx}=\text{ }\frac{78-39}{50-25} \\ \frac{dy}{dx}=\frac{39}{25} \\ \frac{dy}{dx}=1.56 \end{gathered}[/tex]The patient recovery time from a particular surgical procedure is normally distributed with a mean of 3 days and a standard deviation of 1.7 days. Let X be the recovery time for a randomly selected patient. Round all answers to 4 decimal places where possible.a. What is the distribution of X? X ~ N(,)b. What is the median recovery time? daysc. What is the Z-score for a patient that took 4.1 days to recover? d. What is the probability of spending more than 2.4 days in recovery? e. What is the probability of spending between 2.7 and 3.4 days in recovery? f. The 80th percentile for recovery times is days.
Given
mean = 3 days
standard deviation = 1.7 days
Find
a. What is the distribution of X?
b. What is the median recovery time?
c. What is the Z-score for a patient that took 4.1 days to recover?
d. What is the probability of spending more than 2.4 days in recovery?
e. What is the probability of spending between 2.7 and 3.4 days in recovery?
f. The 80th percentile for recovery times
Explanation
a) Distribution of X is given by X ~ N( 3 , 1.7)
b) for the normal distibution ,the median is the same as the mean .
so , the median recovery time is 3 days
c) z - score for the patient that took 4.1 days to recover is
[tex]\begin{gathered} z=\frac{X-\mu}{\sigma} \\ \\ z=\frac{4.1-3}{1.7} \\ \\ z=0.64705882352\approx0.6471 \end{gathered}[/tex]d) probability of spending more than 2.4 days in recovery
[tex]\begin{gathered} P(X>2.4)=P(\frac{X-\mu}{\sigma}>\frac{2.4-3}{1.7}) \\ \\ P(X>2.4)=P(Z>-0.3529) \\ P(X>2.4)=P(Z<0.3529) \\ \\ P(X>2.4)=0.6379 \end{gathered}[/tex]e) probability of spending between 2.7 and 3.4 days in recovery
[tex]\begin{gathered} P(2.7f) 80th percentile for recovery times = [tex]\begin{gathered} P(XFinal AnswerHence , the above are the required answers.
Consider the equation. Determine whether the graph of the equation is wider or narrower than the graph of Y=1/2x^2+1
It's important to observe that the number that multiples the x is 1/2, that is, it's less than zero.
PLEASE HELP MEEEEMatch the properties with the steps to solving the following equation(Sz - 6) =2+9Distributive Property4x - 3 = x + 9Subtraction Property ofEquality3x - 3=93x = 12Addition Property ofEqualityDivision Property ofEqualityx = 4Previouse
apply the distributive property
[tex]4x-3=x+9[/tex]apply the substraction property
Billy used four colors to divide 4.20 by 4. Which model shows 4.20/4?
Given that Billy used four colours to divide 4.20 by 4
To Determine: The model that shows this division
Solution:
It can be observed that 1 represents 100 small boxes
Dividing 4.20 by 4 would give
[tex]\frac{4.20}{4}=1.05[/tex]1.05 would be represented by small boxes
[tex]1.05\times100=105[/tex]If we used colours, then each of the colours would have 105 boxes
From the above explanation, the model that shows 4.20/4 is OPTION D
In the diagram, M is the midpoint of BD and AC. Name two triangles that are congruent.BADABMCX ADMAA ABDACBAA CDM ABMADAB ABCD
From the question,
The two triangles that are congruent are :
Triangle CDM = Triangle ABM --------- Option C
The graph above: One to one function Function but not one to one Relation but not a function
The given graph represents a function, also notice that it is a one to one function because using the vertical line theorem we have that for all vertical lines it only intercepts the graph in just one point.
In ∆MNO , o = 790cm. < O=50° and
Using the law of sines:
[tex]\begin{gathered} \frac{o}{\sin(O)}=\frac{m}{\sin (M)} \\ \frac{790}{\sin(50)}=\frac{m}{\sin (25)} \\ m=\frac{790\cdot\sin (25)}{\sin (50)} \\ m=435.834278 \end{gathered}[/tex][tex]\begin{gathered} \frac{o}{\sin(O)}=\frac{n}{\sin (N)} \\ \frac{790}{\sin(50)}=\frac{n}{\sin (105)} \\ n=\frac{790\cdot\sin (105)}{\sin (50)} \\ n=515.6358793 \end{gathered}[/tex]Using the heron formula:
[tex]\begin{gathered} s=\frac{790+435.834278+515.6358793}{2} \\ s=870.7350787 \\ so\colon \\ A=\sqrt[]{870.7350787(870.7350787-790)(870.7350787-435.834278)(870.7350787-515.6358793)} \\ \end{gathered}[/tex][tex]\begin{gathered} A=104194.335 \\ A=104194cm^2 \end{gathered}[/tex]Pattern A follows the rule "add 2" and Pattern B follows the rule "subtract 2." Pattern A: 1, 3, 5, 7, 9 Pattern B: 10, 8, 6, 4, 2 Which ordered pairs are formed from combining a term in Pattern A with its corresponding term in Pattern B? Select all correct answers. A (1, 3) B (1, 10) C (3, 6) D (5, 4) E (5, 6) F (7, 4) I Need Help Please
To get the required combining pairs : B, C , D, E , F
B (1, 10) = Pattern A - 1 and pattern B = 10
C (3, 6) = pattern a = 3 and pattern b = 6
D (5, 4)= pattern A = 5 and pattern B = 4
E (5, 6) = Pattern A = 5 and pattern B = 6
F (7, 4) = Pattern A = 7 and pattern B = 4
Given ; Pattern A : 1, 3, 5, 7, 9
Pattern B: 10, 8, 6, 4, 2
To get ; A (1, 3) B (1, 10) C (3, 6) D (5, 4) E (5, 6) F (7, 4)
Thus to get the required pair we have to match it the following way
A ( 1, 3 ) = Pattern A - 1 and pattern B - ? thus not correct
B (1, 10) = Pattern A - 1 and pattern B = 10
C (3, 6) = pattern a = 3 and pattern b = 6
D (5, 4)= pattern A = 5 and pattern B = 4
E (5, 6) = Pattern A = 5 and pattern B = 6
F (7, 4) = Pattern A = 7 and pattern B = 4
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The sum of two integers is 463, and the larger number is 31 more than 5 times the smaller number. Findthe two integers.
SOLUTION
Let the smaller number be x
Let the larger number be y
Since the larger number is 31 more than 5 times the smaller number, it folllows:
[tex]y=5x+31[/tex]The sum of the two integers is 463, it follows
[tex]x+y=463[/tex]Substitute y=5x+31 into the equation
[tex]x+5x+31=463[/tex]Solve for x
[tex]\begin{gathered} 6x=463-31 \\ 6x=432 \\ x=\frac{432}{6} \\ x=72 \end{gathered}[/tex]Substitute x=72 into y=5x+31
[tex]\begin{gathered} y=5\left(72\right)+31 \\ y=360+31 \\ y=391 \end{gathered}[/tex]Therefore the two integers are 72 and 391
Construct a system of equations for the word problem. Do not solve. In the space provided, type the answer without any space between the letters, numbers, or symbols.
Given:
The sum of two numbers is 21, and their difference is 9.
Let x and y be the two numbers.
Equation 1: The sum of two numbers is 21.
[tex]\text{ x + y = 21}[/tex]
Equation 2: Their difference is 9.
[tex]\text{ x - y = 9}[/tex]