Answer:
D
Step-by-step explanation:
Answer: We can say that (a/b)^2 is a rational number
================================================
Proof:
Rational means the number is a ratio of two whole numbers
If 'a' is rational, then a = p/q for some whole numbers p,q (eg: p = 2, q = 3 so p/q = 2/3)
The same applies for b, which I'll let b = r/s, with r & s being whole numbers.
To avoid division by zero, I'm going to make p,q,r,s nonzero which leads to 'a' and 'b' being nonzero.
Now let's divide 'a' over b to get
a/b = (p/q) divided by (r/s)
a/b = (p/q) * (s/r) .... flip the second fraction; multiply
a/b = (ps)/(qr)
The new numerator is p*s which is a whole number (because the product of two whole numbers is also a whole number). The same goes for q*r in the denominator. So a/b is rational if both 'a' and 'b' are rational together.
We can extend this to (a/b)^2 as well
(a/b)^2 = [ (ps)/(qr) ] ^2
(a/b)^2 = [ (ps)^2 ]/[ (qr)^2 ]
(a/b)^2 = (p^2*s^2)/(q^2*r^2)
p is a whole number, and so is p^2. Same for s and s^2. Overall, the final numerator p^2*s^2 is also a whole number. Similarly, q^2*r^2 is a whole number.
So we have yet another ratio of two whole numbers proving that (a/b)^2 is a rational number.
Two times the sum of three consecutive odd integers is fifteen more than three times the largest of the integers. Find the integers?
Answer: 5, 7, 9
Step-by-step explanation:
1st#: 2k+1
2nd#: 2k+3
3rd#: 2k+5
2(2k+1 + 2k+3 + 2k+5) = 3(2k+5) + 15
2(6k + 9) = 6k + 15 + 15
12k + 18 = 6k + 30
6k = 12
k = 2
1st#: 2k+1 = 2(2) + 1 = 5
2nd#: 2k+3 = 2(2) + 3 = 7
3rd#: 2k+5 = 2(2) + 5 = 9
The school that Carlos goes to is selling tickets to the annual talent show. On the first day of ticket sales the school sold 4 senior citizen tickets and 1 child ticket for a total of $45. The school took in $190 on the second day by selling 12 citizen tickets and 14 child tickets. Find the price of a senior citizen ticket and the price of a child ticket.
Answer:
The price for both senior and child tickets are $9 each
Step-by-step explanation:
Since your just trying to find the price for senior citizen and child tickets, we'll use the equation for the first day:
4x + x = $45
5x = $45
x = 9
now plug it for 4x and x
4(9) = $36 <<divide $36 by 4 == $9
$9 for each child ticket
The amount Bryce earns babysitting in a month is represented by the equation a=10.5h. In the equation, a represents the total amount Bryce earns. The number of hours he babysits is represented by h. What is the constant of proportionality (unit rate) of a to h? A. 10.50/hr B.10.00/hr C.9.50/hr D.5.10/hr
Answer:
A. 10.5h
Step-by-step explanation:
a new car is sold for its sticker value of $19,400. three years later the customer returns to the car dealership to trade the car in. she is told that her car now has a value of $12,105. what is the rate of decline in the value of the car? In your final answer, include all of your calculations.
Answer:
100/19400*12105=62.3969072165 100-62.3969072165 = 37.60% (rounded)
Step-by-step explanation:
100% = 19'400 divide and multiply with the current value, this will give you the % deductions. Simply deduct the result from 100.
Answer:
copy paste version
Step-by-step explanation:
we know that the formula to calculate the depreciated value is equal to:
D = P(1-r)^t
where
D is the depreciated value
P is the original value
r is the rate of depreciation in decimal
t is Number of Time Periods
In this problem we have:
P = $19,400
D = $12,105
t = 3 years
substitute in the formula above and solve for r:
$12,105 = $19,400(1-r)^3
Simplify:
(12,105/19,400) = (1-r)^3
(1-r) = 3√(12,105/19,400)
r = 1 - 3√(12,105/19,400)
r = 0.1455
Convert to percentage:
r = 14.55%
ms jefferson spent $15 to buy 12.8 ounces of smoked trout what was the cost per pound
Answer:
Step-by-step explanation:
Pounds to Ounces: 1 ounce = 0.625 pounds
12.8 ounces = 0.8 pounds
15/0.8 = 18.75 dollars
18.75 dollars
A 2×2 square is centered at the origin. It is dilated by factor of 3. What are coordinates of the vertices of the square?
The 2x2 square centered at the origin has vertices at (-1,1), (1,1), (-1,-1), and (1,-1). When it is dilated by a factor of 3, the coordinates of the vertices are multiplied by 3, resulting in new vertices at (-3, 3), (3, 3), (-3, -3), and (3, -3).
Explanation:The subject here is Geometry, specifically dealing with transformations such as dilations on a plane. The starting square is 2x2 and centered at the origin means its vertices are at (-1,1), (1,1), (-1,-1), and (1,-1). A dilation by a factor of 3 from the origin means we are going to multiply every coordinate by 3. Doing this, we get the new vertices as (3*-1, 3*1) = (-3, 3), (3*1, 3*1) = (3, 3), (3*-1, 3*-1) = (-3, -3), and (3*1, 3*-1) = (3, -3). Therefore after the dilation, the vertices of the square will be at positions (-3, 3), (3, 3), (-3, -3), and (3, -3).
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You are paid $15.60/hr. Your deductions are FICA (7.65%), federal tax withholding (11.15%), and state tax withholding (6.5%). You work 15 hr/wk and save 10% each week. How much do you save each month?
a) $76.61
b) $79.29
c) $69.92
d) $71.04
Option C is the answer.
Explanation:
Amount paid per hour = $ 15.60
Total working hours in a week = 15
Hence, total earnings in a week are = 15.60 x 15 = $234
Deductions are =
7.65 % of 234 = [tex]\frac{7.65}{100}*234= 17.90[/tex]
11.15 % of 234 = $26.09
6.5% of 234 = $15.21
So totaling all deductions we get $59.20
Hence, earnings per week after deductions become = 234-59.20 = $174.80
As each week, 10% is saved, so amount saved each week is = 174.80 x 0.10 = $ 17.48
So, savings per month (4 weeks in a month) becomes = 17.48 x 4 = $69.92
Hence, option c) $69.92 is the answer.
If u(x)=-2x²+3 and v(x)=1/x, what is the range of (u ° v)(x)?
[tex]u(x)=-2x^2+3\\\\v(x)=\dfrac{1}{x}\\\\(u\ \circ\ v)(x)=-2\left(\dfrac{1}{x}\right)^2+3=-2\left(\dfrac{1}{x^2}\right)+3=-\dfrac{1}{x^2}+3\\\\\text{The range of}\ y=\dfrac{1}{x^2}\ \text{is all positive real numbers.}\\\\\text{The range of}\ y=-\dfrac{1}{x^2}\ \text{is all negative real numbers.}\\\\\text{The range of }\ (u\ \circ\ v)(x)=-\dfrac{1}{x^2}+3\ is\ (-\infty,\ 3)[/tex]
Answer:
(-∞,3)
Step-by-step explanation:
Imagine functions as little machines that turn one number into another number, the set of numbers that can enter the machine are called domain and the set of numbers that can exit the machine are called range. In this excercise we have to do function composition, which would be like having two machines and take the numbers that exit machine number 2 ([tex]v(x)[/tex]) and enter them into the machine number 1 ([tex]u(x)[/tex]). In math this is done by replacing the [tex]x[/tex] in [tex]u(x)[/tex] with the function [tex]v(x)[/tex] like this:
[tex]u(x)=-2x^{2} +3\\v(x)=\frac{1}{x}\\(u°v)=-2(\frac{2}{x})^{2} +3[/tex]
Now we need the range of the composed function, this will be the numbers that can come out of the machine number 1 when the numbers from the machine number 2 are entered to it.
So first which numbers will come out of machine number 2([tex]v(x)[/tex])? All but 0 because we there is no number that we can divide 1 by it that will give us the value 0 (not even zero itself because it is and indeterminate form).
We have now which numbers will enter machine number 1 (we dont have any restrictions in [tex]u(x)[/tex] to enter numbers)
The range of the composed function will be then the range of [tex]u(x)[/tex] less the value that we woud obtain by replacing [tex]x[/tex] with 0.
The range of [tex]u(x)[/tex] is (-∞,3] according to the attached graph and the value that we woud obtain by replacing [tex]x[/tex] with 0 is 3 so we would have (-∞,3).
A rectangle has an area of 40 square units. The length is 6 units greater than the width
Answer:
The width is 4 units, and the length is 10 units.
Step-by-step explanation:
area of rectangle = length * width
Let L = length; let W = width.
"The length is 6 units greater than the width.": L = W + 6
area = LW = 40
Since L = W + 6, we substitute L with W + 6.
(W + 6)W = 40
W^2 + 6W = 40
W^2 + 6W - 40 = 0
(W - 4)(W + 10) = 0
W - 4 = 0 or W + 10 = 0
W = 4 or W = -10
A width cannot be a negative number, so we discard the solution W = -10.
W = 4
L = W + 6 = 4 + 6 = 10
The width is 4 units, and the length is 10 units.
Answer:
10 by 4 or B
Step-by-step explanation:
took test on edge
Simplify the square root of 5 times 3 square root of 5
Answer:
15
Step-by-step explanation:
The square root of 5 times 3 square root of 5 is 5√3
How to simplify?[tex] \sqrt{5 \times 3} \sqrt{5} [/tex]
[tex] = \sqrt{15} \times \sqrt{5} [/tex]
[tex] = \sqrt{75} [/tex]
[tex] = \sqrt{25 \times 3} [/tex]
[tex] = 5 \sqrt{3} [/tex]
The simplification of the square root of 5 times 3 square root of 5 is 5√3
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what is the value of x in the diagram below ? A. 9 B. 6 C. 7 D. 8
Answer:
9
Step-by-step explanation:
In the larger triangle you divided 12 by 2 which gave you the length of its corresponding side. So if you do 12/6 to get 2, you divide 54 by 6 and get 9.
Carrie is looking for a job cutting hair. One option is self-employment at The Belmont Salon, where she would pay $438 per month to rent a station and keep all her earnings. Another option is to work at a franchise, where she just have to pay $2 for every haircut. if she performed a certain number of haircuts every month, the amount paid to either salon would be the same. how much would Carrie pay? how many haircuts would that be? ( Can you also show the steps please?)
Answer:
She would have to do 219 haircuts in a month for the costs to be the same, which would be $428.
Step-by-step explanation:
To find this we take the costs of the first place.
$438
Next we can look at the costs of the second place using x as the number of haircuts she'd do.
2x
Now we can set the two equal to each other to find the number of cuts.
438 = 2x
219 = x
Can someone help me on number 24?
Answer:
x must be less than or equal to 5 hours
Step-by-step explanation:
24. C= 2(x-2) +3
C must be less than 10
10> 2(x-2) +3
Distribute the 2
10> 2x-4+3
Combine like terms
10> 2x-1
Add 1 to each side
10+1> 2x-1+1
11> 2x
Divide each side by 2
11/2 >2x/2
5.5 >x
We can only have whole hours
x must be less than or equal to 5 hours
Subtract: (2x2 - 6x + 7) - (5x2 + 2x - 8).
2x^2 - 6x + 7 - (5x^2 + 2x - 8)
Distributive a -1 to each term in the parentheses.
2x^2 - 6x + 7 - 5x^2 - 2x + 8
Combine like terms.
-3x^2 - 8x + 15 is the expression after being subtracted.The subtraction of the given polynomials (2x² - 6x + 7) - (5x² + 2x - 8) results in the polynomial -3x² - 8x + 15.
To subtract the given polynomials, we rewrite the subtraction as an addition of the opposite. The original expression is (2x² - 6x + 7) - (5x² + 2x - 8). Changing subtraction to addition, we have (2x² - 6x + 7) + (-5x² - 2x + 8).
We combine like terms:
For x²: 2x² - 5x² = -3x²For x: -6x - 2x = -8xFor the constant: 7 + 8 = 15The result of the subtraction is therefore -3x² - 8x + 15.
On Sunday, Sheldon bought 3 and 1/2kg of plant food. He used 1 and 2/3kg on his strawberry plants and used 1/4 kg for his tomato plants .
How many kilogramsof plant food did Sheldon have left? Write one or more equations to show how you reached your answer.
Answer:
Step-by-step explanation:
3 1/2-1 2/3-1/4
How can you determine the domain and range of function.
Answer:
The domain of a function is the complete set of possible values of the independent variable.
In plain English, this definition means:
The domain is the set of all possible x-values which will make the function "work", and will output real y-values.
When finding the domain, remember:
The denominator (bottom) of a fraction cannot be zero
The number under a square root sign must be positive in this section
Step-by-step explanation:
Answer and explanation:
The set of all possible values of x that will make the function work, giving an output of real y values is said to be the domain of a function.
The domain of a function is determined by looking for the values of the independent variable (which is usually x) that are allowed to use.
While the complete set of all possible output values of the dependent variable (usually y), when we substitute the x values is called the range.
Range can be determined by the spread of the possible y-values (minimum y-value to maximum y-value).
Tom had 322 dollars to spend on 9 books. after buying them he had 16 dollars. How much did each book cost?
Answer:
He spent 34 dollars on each book
Step-by-step explanation:
first you subtract 16 from 322 and it would be 306 and then you divide 306 by 9 and that would be the total of each book!! <3
Answer:
The answer is $35
Step-by-step explanation:
First you divide 322 my 9 so you can get 35, that was the price of each book.
hope it helps:)
The length of a rectangle is 3yd longer than its width. If the perimeter of the rectangle is 50yd, find it’s length and width
Answer:
3(15)
Step-by-step explanation:
We found that the width is 11 yards and the length is 14 yards.
We are given that the length of a rectangle is 3 yards longer than its width and that the perimeter of the rectangle is 50 yards. To find the length and width, we can set up a system of equations based on the information provided.
Let the width of the rectangle be w yards. Then the length would be w + 3 yards. Because the perimeter of a rectangle is the sum of all its sides, we can express the perimeter P as:
P = 2 * ( L + B)
Substituting the given values:
50 = 2(w + 3) + 2w
Now, let's solve the equation:
50 = 2w + 6 + 2w
50 = 4w + 6
44 = 4w
11 = w
Thus, the width is 11 yards. The length is 14 yards (because it is 3 yards longer than the width).
find the area of a circle with a circumference of 11pi feet
Answer:
Step-by-step explanation:
did this on my hw got a 100
The area of the circle has a circumference of 11pie is 94.98 ft sq.
What is the area of the circle?The area of the circle is the region enclosed by a circle of radius r.
The area of the circle = [tex]\pi r^{2}[/tex]
It is given that
The circumference of a circle = [tex]2\pi r[/tex] = 11π
[tex]2\pi r[/tex] = 11π
r = 11/2
The area of the circle = [tex]\pi r^{2}[/tex]
= [tex]\pi \times (11/2)^{2}[/tex]
= [tex]3.14 \times 121/4\\[/tex]
= 94.98 ft sq
Thus, The area of the circle has a circumference of 11pie is 94.98 ft sq.
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Please help me ASAP:
Due to inflation there were two demand increases. After the second, the price for the certain item became 6 times the original. What was the percent increase of the second if the first increase was 50%?
Final answer:
To find the percent increase for the second demand increase when the original price is sextupled and the first increase was 50%, we calculate the ratio of the final price to the price after the first increase. The second price increase was determined to be 300% based on this calculation.
Explanation:
The question asks us to determine the percent increase of the second price change when the price of an item becomes six times its original price, with the first price increase being 50%. To solve this, let's assume the original price is $1 for ease of calculation. After a 50% increase, the new price would be $1.50. Now, we know that after the second increase the new price is 6 times the original, which means the price is now $6.00.
To calculate the percent increase of the second price change, we divide the final price by the price after the first increase: $6.00 / $1.50 equals 4. This means the price was quadrupled. To find the percentage increase, we subtract the initial value (1, since it was multiplied by 4) and then multiply by 100. Therefore, the percent increase for the second time is (4 - 1) × 100% which equals 300%. The second price increase was 300%.
Given the two expressions shown below: square root of 3 plus square root of 2 square root of 5 plus square root of 6 Which statement best describes the two expressions? Both are rational. Both are irrational. A is rational, but B is irrational. A is irrational, but B is rational.
Answer:
they are both irrational
Step-by-step explanation:
if you add them, then they turn out to be decimals that go on and have no pattern
Answer:
Both will be irrational
Los lados de un triangulo rectangulo miden 6cm, 8cm, y 10cm ¿cuanto mediran los catetos de un triangulo semejante al primero si tiene 5cm de largo en su hipotenusa?
Answer:
Los catetos miden 3 cm y 4 cm.
Step-by-step explanation:
5 cm/10 cm = 1/2
6 cm * 1/2 = 3 cm
8 cm * 1/2 = 4 cm
Mark and Melissa went to the garden center together to buy plants. Mark bought six ferns and one azalea for $80. Melissa bought five ferns and three azaleas for $84. how much was a single fern
Answer:
Price of single fern is $12
Price of single azalea is $8
Step-by-step explanation:
Let's assume
price of each fern is x
price of each azalea is y
Mark bought six ferns and one azalea for $80
so, we got first equation as
[tex]6x+y=80[/tex]
Melissa bought five ferns and three azaleas for $84
so, we got second equation as
[tex]5x+3y=84[/tex]
now, we can solve it
Firstly, we can solve for y from first equation and then plug that in second equation
[tex]6x+y=80[/tex]
[tex]y=80-6x[/tex]
now, we can plug that into second equation
[tex]5x+3(80-6x)=84[/tex]
[tex]-13x+240=84[/tex]
[tex]-13x=-156[/tex]
[tex]x=12[/tex]
now, we can solve for y
[tex]y=80-6\times 12[/tex]
[tex]y=8[/tex]
HW #35: similar polygons
Answer:
Q 9:
Because the two polygons are similar so AB ≈ PQ = SR
Scale factor is 25/20 = 1.25
Perimeter of ABCD = 14+20+14+20 = 68
We can find the length of SP by multiplying the scale factor with the AD so
SP = 14 * 1.24= 17.5
Perimeter of PQRS = 17.5+25+17.5+25 = 85
-----------------------------------------------------------------------------------
Q 10:
Because the two polygons are similar so AD ≈. EH
Scale factor is 7/14 = 0.5
Perimeter of ABCD = 12+14+13+26 = 65
Because the two polygons are similar so DC ≈ HG
and our scale factor is 0.5 so HG = 13/2 = 6.5
Perimeter of EFHG = 6 + 7 + 6.5 + 13 = 32.5
----------------------------------------------------------------------------------
Q 11:
The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers). And the formula for finding the Geometric mean of two numbers a and b is
[tex]\sqrt{a*b}[/tex]
So geometric mean of the 8 and 10 can be found as
[tex]\sqrt{8*2} = \sqrt{16} = 4[/tex]
-----------------------------------------------------------------------------------
Q 12:
Similarly we can using the above formula of finding the geometric mean of 5 and 45 as
[tex]\sqrt{5*45} = \sqrt{225} = 15[/tex]
-------------------------------------------------------------------------------
Q 13:
and we can find the geometric mean of 6 and 30 by using the same formula
[tex]\sqrt{6*30} = \sqrt{180} = 13.41[/tex]
write the equation of the line in slope intercept form with the given conditions (slope = 3 and passes through (1,-3))
Answer:
Because we have a point and slope, we can use at the beginning the point-slope form: y-y1=m(x-x1)
Step-by-step explanation:
m=3 , x1=1 , y=-3
y-(-3)=3(x-1)
y+3=3x-3 subtract 3 from both sides
y=3x-6 the answer in the slope-intercept form y=mx+b
which is more 4feet or 48inches
Answer:
4 feet = 48 inches
Step-by-step explanation:
In one foot there are 12 inches and since we have 4 of them we multiply to find how many total inces there are:
4x12=48
If you wanted to check the other side of it you would divide 48 inches by 12 inches to find out how many feet you have:
48÷12=4
Answer:
They are equal.
Step-by-step explanation:
A foot is made up of 12 inches
4 feet= 12*4 inches
4 feet= 48 inches
There you go!
The two are equal to each other
Which type of triangle, if any, can be formed with sides measuring 8 inches, 8 inches, and 3 inches
explain whether 8t-3y-4t is equivalent to 7t +(-3t)-3y
First you must simplify the equation. 7t+(-3t) is equal to 4t. Then we subract 3y from 4t. This brings us to 4t-3y. We also can simplify the first equation. We can move 4t to 8t and we then get 8t-4t-3y which is equal to 4t-3y. We know that 4t-3y=4t-3y.
Hopw this helps. <3
The solution is A = 4t - 3y
The value of the equation A is A = 4t - 3y
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
A = 8t - 3y - 4t be equation (1)
Let the equation B = 7t + ( -3t ) - 3y be equation (2)
Now , on simplifying the equation (1) , we get
A = 8t - 4t - 3y
A = 4t - 3y
So , the value of A is 4t - 3y
Now , on simplifying the equation (2) , we get
B = 7t + ( -3t ) - 3y
B = 7t - 3t - 3y
The value of B = 4t - 3y
Therefore , the value of A is equal to value of B
Hence , the equation A is equivalent to equation B
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A triangle with vertices at A(0, 0), B(0, 4), and C(6, 0) is dilated to yield a triangle with vertices at A′(0, 0), B′(0, 10), and C′(15, 0). The origin is the center of dilation. What is the scale factor of the dilation? A. 1.5 B. 2 C. 2.5 D. 3
Answer:
2.5
Step-by-step explanation:
just got it right
A $50 coat is put on sale for $35 find the percent of change
Answer:
70%
Step-by-step explanation:
To solve the equation you would do 35/50= ?/100
So step one would to be to multiply 35 by one hundred.
Then you divide by 50
then you should get 70.
so 70% is the answer
To calculate the percent change for a coat reduced from $50 to $35, you subtract the sale price from the original price, divide by the original price, and multiply by 100, resulting in a 30% decrease.
You first determine the amount of change in price, which is the original price subtracted from the sale price. In this case, the price decreased by $15 ($50 - $35). To find the percent change, divide the amount of change by the original price and then multiply by 100 to convert it to a percentage.
Percent Change = ($15 / $50) × 100
= 0.3 × 100
= 30%
Therefore, the coat was put on sale with a 30% decrease in price.