A student memorized the statement the product of any rational number and an irrational number is irrational holds true for non-zero rational number
Solution:
A rational number is a number that can be written as a fraction.
When a rational number fraction is divided to form a decimal value,
it becomes a terminating or repeating decimal.
An Irrational Number is a real number that cannot be written as a simple fraction. It cannot be expressed as a fraction with integer values in the numerator and denominator.
When an irrational number is expressed in decimal form, it goes on forever without repeating.
"The product of a rational number and an irrational number is SOMETIMES irrational."
If you multiply any irrational number by the rational number zero, the result will be zero, which is rational.
Any other situation, however, of a product of rational and irrational will be irrational.
A better statement would be:
"The product of a non-zero rational number and an irrational number is irrational."
Let us understand this with a example
[tex]\text { non - zero rational number } \times \text { irrational number }=\text { irrational number }[/tex]
Consider [tex]\frac{1}{2} \rightarrow rational number[/tex]
[tex]\frac{\sqrt{3}}{4} \rightarrow irrational number[/tex]
Now multiply both,
[tex]\frac{1}{2} \times \frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}=0.433012701[/tex]
We got 0.433012701 which is a irrational number because in decimal form, it goes on forever without repeating.
Thus we can frame two statements:
"The product of a rational number and an irrational number is SOMETIMES irrational." "The product of a non-zero rational number and an irrational number is irrational."Answer:
True.Step-by-step explanation:
"The product of any rational number and an irrational number is irrational"
To know if the statement is true or false, we can test an example. Lets use the rational number [tex]\frac{3}{2}[/tex] and the irrational number [tex]\pi[/tex]. Id we calculate the product of these two we would have:
[tex]\frac{3}{2}\pi= 4.71238898038469...[/tex]
As you can observe, the product is an irrational number, because it's infinite and not periodic, it doesn't have any pattern.
Therefore, the statement is true.
Can someone please help me?
GEORGINA WAS GIVEN THAT THE LENGTH OF THE RECTANGLE WAS 2.5 INCHES LONGER THEN ITS WIDTH AND THE PERIMETER OF THE RECTANGLE WAS 75.4 INCHES.
FIND THE LENGTH AND WIDTH OF THE RECTANGLE Alegebraically. SHOW ALL WORK
Answer:
Length of the rectangle is [tex]18.85\ inches[/tex],and the width of the rectangle is [tex]16.35\ inches[/tex]
Step-by-step explanation:
Let the width of the rectangle be [tex]x[/tex] inches
Then
The length of the rectangle will be [tex]x+2.5[/tex] inches
So perimeter [tex]=2(length + width)[/tex]
Plugging the values:
⇒ [tex]2(x+x+2.5)=75.4[/tex]
⇒ [tex]2(2x+2.5)=75.4[/tex]
⇒ [tex]4x+10=75.4[/tex]
Subtracting [tex]10[/tex] from both sides.
⇒ [tex]4x=65.4[/tex]
⇒ [tex]x=\frac{65.4}{4}=16.35\ inches[/tex]
So the width of the rectangle is [tex]16.35\ inches[/tex]
And the length of the rectangle is [tex]18.85\ inches[/tex]
Solve the system of linear equations by addition
x+y=-9
x+y=5
Answer:
No Solution.
Step-by-step explanation:
x+y=-9
x+y=5
------------
x+y=-9
-(x+y)=-5
---------------
x+y=-9
-x-y=-5
-------------
0=-14
The local Orchestra has been invited to play at a festival. There are 111 members of the orchestra and six are licensed to drive large multiple passenger vehicles. Busses 25 people but are much more expensive to rent passenger vans for 12 people. Create a system of equations to find the small list number of buses there orchestra can rent.
Answer:
Small list number of buses [tex]=3[/tex],the orchestra can rent.
Step-by-step explanation:
Form an equation representing the number of vehicles needed.
Let the number of buses be [tex]x[/tex] and the number of vans be [tex]y[/tex]
We have six drivers so,
[tex]x + y\leq 6[/tex] for both vehicles.
And it's an inequality,where we can equate with by removing the inequality to close in our answer.
[tex]x + y = 6[/tex]
Then form an equation representing the total number of seats [tex]K[/tex] in vehicles for the orchestra members.
[tex]25[/tex] people for bus and [tex]12[/tex] people for vans can be written in terms of seats.
So,[tex]K = 25x + 12y[/tex]
In terms of an inequality it can be written as,
[tex]25x + 12y\geq111[/tex],[tex]111[/tex] is the maximum numbers of the orchestra.
Solving the system of inequality.
We have
[tex]x + y = 6[/tex]
Re-arranging y.
[tex]y = 6 - x[/tex]
Plugging the values in [tex]25x + 12y\geq111[/tex] by removing the inequality and considering it as an equation.
[tex]111 = 25x + 12y = 25x + 12(6-x)[/tex]
[tex]111 = 25x + 72 - 12x[/tex]
[tex]111 - 72 = 13 x[/tex]
[tex]39 = 13 x[/tex]
[tex]x = 3[/tex]
So small list number of buses are [tex]3[/tex] there orchestra can rent.
The graph shows how many apples Erin can pick if she maintains a constant rate. What is the linear equation for this relationship? How many apples can she pick per hour?
An equation that represents the relationship between the x and y-values is y = 75x.
Erin can pick 75 apples per hour.
What is a proportional relationship?In Mathematics and Geometry, a proportional relationship is a type of relationship that produces equivalent ratios and it can be modeled or represented by the following mathematical equation:
y = kx
Where:
y represents the y-variable.x represents the x-variable.k is the constant of proportionality.Next, we would determine the constant of proportionality (k) by using various data points as follows:
Constant of proportionality, k = y/x
Constant of proportionality, k = 75/1 = 150/2
Constant of proportionality, k = 75.
Therefore, the required linear equation that models the graph can be written as follows;
y = kx
y = 75x
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Write the quadratic equation whose roots are 4 and 5, and whose leading coefficient is 4.
The quadratic equation whose roots are 4 and 5, and whose leading coefficient is 4, is: [tex]\[\bo{4x^2 - 36x + 80 = 0}\][/tex]
Step 1
To write a quadratic equation with given roots and a specified leading coefficient, we can use the fact that if p and q are the roots of a quadratic equation, the equation can be written as:
[tex]\[a(x - p)(x - q) = 0\][/tex]
where a is the leading coefficient.
Given roots: [tex]\( p = 4 \)[/tex] and [tex]\( q = 5 \)[/tex]
Leading coefficient: [tex]\( a = 4 \)[/tex]
Step 2
Let's substitute these values into the equation:
[tex]\[4(x - 4)(x - 5) = 0\][/tex]
Now, we need to expand this equation. First, expand the product [tex]\((x - 4)(x - 5)\):[/tex]
[tex]\[(x - 4)(x - 5) = x^2 - 5x - 4x + 20 = x^2 - 9x + 20\][/tex]
Step 3
Next, multiply this expression by the leading coefficient [tex]\( 4 \)[/tex]:
[tex]\[4(x^2 - 9x + 20) = 4x^2 - 36x + 80\][/tex]
Thus, the quadratic equation whose roots are 4 and 5, and whose leading coefficient is 4, is:
[tex]\[\bo{4x^2 - 36x + 80 = 0}\][/tex]
Mary covered her kitchen floor with
10 tiles. The floor measures 6 feet long
by 5 feet wide. The tiles are each 3 feet
You have to write an equation though not solve it.
long and w feet wide.
Final answer:
The question pertains to writing an equation for a mathematics-related scenario, emphasizing the importance of understanding equations as expressions of concepts rather than just tools for calculation.
Explanation:
The subject of this question appears to revolve around forming an equation that could pertain to a mathematics problem. The use of mathematical equations to convey concepts is crucial; they aren't just mechanisms for inserting and churning numbers, but they are sentences that express concepts precisely and purposefully. Writing out such equations in both words and symbols ensures comprehension of the underlying reasoning.
Moreover, when teaching mathematics, it's beneficial to allow students to create their own formulas. This practice not only solidifies the understanding of concepts but also emphasizes that the symbols used in equations are arbitrary and that what matters is the concept they represent.
Remembering the correct order of operations is essential, which can be recalled with the mnemonic 'Please Excuse My Dear Aunt Sally.' Also, an iterative relation often used in advanced mathematics is not an equation per se, but rather a set of instructions for approximating a solution. This iterative process requires a different approach compared to straightforward equation solving.
If you buy a home and make a relatively small down payment, what happens to the monthly payments and total payment for you loan?
Answer:
Both monthly payment and total payment will increase.
Step-by-step explanation:
If I buy a home and make a relatively small down payment, then the due payment will be more.
If I pay for my home through a bank and I fix the number of installments to pay back the due amount after down payment (say, monthly installment for 5 years), then the amount of installment payment will definitely increase if my down payment is less.
Again, if the down payment is less, the remaining amount to pay back will rise and the bank calculates interest on the remaining amount to pay. Therefore, the amount of interest will rise in value and finally the total payment after interest for my loan will increase. (Answer)
If you suspect that "hours studied" and "score on a math test" are positively correlated, what does that mean?
A. More hours of studying tend to go with higher scores on the test.
B. Fewer hours of studying tend to go with lower scores on the test.
C. More hours of studying tend to go with lower scores on the test.
D. Both A and B are true.
I think its D but im not sure completely
Answer:
D. Both A and B are true
Step-by-step explanation:
Why would you study to decrease scores? so C. Is automatically wrong. Now A. is true and B. is true. D says both A and B are true so D is correct.
Answer:
D
Step-by-step explanation:
D. Both A and B are true.
ASAP! A biologist is to set up an aquarium in his laboratory. He intends to buy 11 piranhas and 5 sharks. He has a budget of £200. If p represents one piranhas and s one shark, write down an equation for the price of one piranhas in terms of the price of sharks.
Answer:
p = (200 -5s)/11
Step-by-step explanation:
I think you meant p is the price of piranhas, and a is the price of one shark.
You have:
11p + 5s = 200
11p = 200 - 5s
p = (200-5s)/11
Does that help? ;)
PLEASE SHOW YOUR WORK. thanks
Mrs. Bailey had 12 pieces of candy. She gave away some candy and has 3 pieces left. What is the percent decrease of candy?
Answer:
75%
The drawings will help
Refer to the first one to help understand the second.
Greenfield village is spread across 240 acres of land. Only 37% of the land is used for the exhibits, while the rest of the land consists of forest, rivers and pastures. How many acres are used for exhibits?
Answer:
The land which are used for exhibits are 89 acres (approximately).
Step-by-step explanation:
Given:
Village is spread across 240 acres of land. 37% of land is used for the exhibits.
Now, we have to find acres of land which are used for exhibits.
So, here it is given that land used for exhibits is 37%, and we have to calculate the quantity from the percentage given:
According to question:
37% of 240 acres of land.
[tex]\frac{37}{100}\times 240[/tex]
[tex]0.37\times 240[/tex]
[tex]88.80[/tex]
So, the land used for exhibits are 88.80 acres.
Therefore, the land which are used for exhibits are 89 acres (approximately).
the diameter of the sweep of the main rotor of the eh101 helicopter is 18.6 meters. A toy model of it has a sweep of 31 centimeters. What is the scale of the model?
Answer:
The actual length of 60 cm is modeled as 1 cm length.
Step-by-step explanation:
The diameter of the sweep of the main rotor of the helicopter is 18.6 meters i.e. 1860 cm.
Now, the toy model of the helicopter has a sweep of 31 cm.
Therefore, the scale in which the model is made is 1860 : 31 i.e. 60 : 1 ratio.
So, the actual length of 60 cm is modeled as 1 cm length. (Answer)
Answer:
The scale of the model to the actual is 1:60
Step-by-step explanation:
The scale of the model may be computes as the ratio of the diameter of the original sweep to the diameter of the model. Given that the diameter of the sweep of the main rotor of the eh101 helicopter is 18.6 meters and that of the model is 31 cm, the scale of the model
= 31cm to 18.6m
since 100cm = 1m,
18.6m = 18.6 * 100cm
= 1860 m
31cm to 18.6m = 31cm to 1860cm
= 1 to 60
A rectangular auditorium seat 2842 people. The number of seats in each row exceeds the number of rows by 9. Find the number of seats in each row.
Answer:25,578
Step-by-step explanation:
Multiply 2,842 by 9
Give formula to calculate Selling price when profit is given in percentage.
Factor the polynomial x^4-16. How many real zeros does the function g(x)=x^4-16 have?
Answer:
(x-2)(x+2)(x^2 + 4)
Step-by-step explanation:
This can be done by difference of squares:
(x^2 - 4) (x^2 + 4)
And now (x^2 - 4) can be further factored:
(x-2)(x+2)(x^2 + 4) is your answer
Clearly, There are only 2 zeroes that we can get from this, 2 and -2
Hope this helps!
If you think I helped please mark brainliest! Would really appreciate!
Answer:
[tex]\large\boxed{x^4-16=(x^2+4)(x+2)(x-2)}[/tex]
[tex]\large\boxed{g(x)=x^4-16\ \text{has two real zeros}\ x=-2\ \text{and}\ x=2}[/tex]
Step-by-step explanation:
[tex]x^4-16=x^4-2^4=x^{(2)(2)}-2^{(2)(2)}\\\\\text{use}\ (a^n)^m=a^{(n)(m)}\\\\=(x^2)^2-(2^2)^2\\\\\text{use}\ a^2-b^2=(a+b)(a-b)\\\\=(x^2+2^2)(x^2-2^2)=(x^2+4)(x+2)(x-2)[/tex]
The zeros of g(x):
[tex]g(x)=x^2-16=(x^2+4)(x+2)(x-2)\\\\g(x)=0\iff(x^2+4)(x+2)(x-2)=0\iff x^2+4=0\ \vee\ x+2=0\ \vee\ x-2=0\\\\x^2+4=0\qquad\text{subtract 4 from both sides}\\x^2=-4\qquad\bold{FALSE}\\\\x+2=0\qquad\text{subtract 2 from both sides}\\\boxed{x=-2}\\\\x-2=0\qquad\text{add 2 to both sides}\\\boxed{x=2}[/tex]
Evaluate x + y when x = -94 and y = 24
A: -118
B: -70
C: 70
D: 118
The simplification of the equation when x = -94 and y = 24 will be B: -70
What is a simplification of an expression?Usually, simplification involves proceeding with the pending operations in the expression. Simplification usually involves making the expression simple and easy to use later.
We have been given an equation as x + y
We have to find the solution when x = -94 and y = 24
Therefore, substitute the values;
x + y = (-94 +24)
Now Evaluate;
(-94 +24) = -70
Hence, the correct answer is B.
Therefore, The simplification of the equation when x = -94 and y = 24 will be B: -70
Learn more about simplification;
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find the HCF of following !! a) 15,63,99 B) 66,33,78
Answer:
(a) 3, (b) 3
Step-by-step explanation:
HCF is highest common factor. Each number is split into multiples while the common factors are then multiplied together.
(a) HCF of 15, 63, 99
15 = 3 X 5 ----------------------------- (i)
63 = 3 X 3 X 7 ---------------------------(ii)
99 = 3 X 3 X 11 ---------------------------(iii)
From the three equations, 3 is the only common factor; 3 appears in equations (i), (ii) and (iii)
hence the HCF of 15, 63 and 99 is 3
(b) HCF of 66, 33, 78
66 = 2 X 3 X 11 ----------------------------- (iv)
33 = 3 X 11 ---------------------------(v)
78 = 2 X 3 X 13 ---------------------------(vi)
From the three equations, 3 is still the only common factor; 3 appears in equations (iv), (v) and (vi)
hence the HCF of 166, 33, 78 is 3
Give your answer in its simplest form.
Use [tex]\frac{a^n}{a^m}=a^{n-m}[/tex].
[tex]
\dfrac{2m^2t^6}{m^4t^2}=2m^{2-4}t^{6-2}=\boxed{2m^{-2}t^4}
[/tex]
Hope this helps.
Answer:
2m⁻²t⁴
Step-by-step explanation:
Rule:
If aᵇ / aˣ, then aᵇ ⁻ ˣ
[tex]\frac{2m^2t^6}{m^4t^2} = 2 m^{2-4} t^{6-2} = 2m^{-2} t^{4}[/tex]
Mohit pays rupees 9000 as an amount on the sum of rupees 7000 that he had borrowed for 2 years.
Find the rate of interest.
pls answer fast
I will mark u as the brainliest
Answer:
The required rate of interest = 14.285%
Step-by-step explanation:
Here, The principal amount = 7,000
The final paid amount = 9,000
Time = 2 years
Let us assume the rate of interest = R
Now, Simple Interest = Amount - Principal
= 9,000 - 7,000 = 2,000
So, the SI on the given principal for 2 years = 2,000
[tex]\textrm{SIMPLE INTEREST } = \frac{P \times R \times T}{100}\\\implies 2000 = \frac{7,000 \times R \times 2}{100}\\\implies R = \frac{2,000 \times 100}{7,000 \times 2} = 14.285[/tex]
or, R = 14.285%
Hence, the required rate of interest = 14.285%
There are 24% of boys in a school. If the number of girls are 456, find the total number of students in the school.
Answer:
600 students
Step-by-step explanation:
100% - 24% = 76%
76% = 456
100%= x
76x = 456x100
x = 45600÷76
x= 600 students (total)
Answer: 600
Step-by-step explanation:
Since there are 24% of boys in a school , this means that the number of girls will be :
100% - 24 %
= 76%
That means there are 76% of girls in the school.
Let the total number of students in the school be x, this means that
24% of x = number of boys
76% of x = number of girls
Since the total number of girls is given to be 456 , then
76% of x = 456
0.76x = 456
x = 456/0.76
x = 600
That means there are 600 students all together in the school
write 2 formulas you use for finding the circumference of a circle
Answer: the 2 equations are 2(pi)r and (pi)d.
Step-by-step explanation: the radius is from the center of the circle to the edge. And the diameter is double that. R is radius and d is diameter since the diameter is double the radius, you don’t need to multiply it by 2. Hope this helps!
Sheila has $47 to buy graphing pads for her math class. Each pad costs $8,
How many pads can she buy? Do not include units in your answer.
Answer:
5 tablets can be bought
Step-by-step explanation:
47/8= 5
my teacher keeps explaining what a "vertical line test" is and how it helps solve a linear equation plz explain and dumb it down for me hahahaha thx.(Im new here btw)
The vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x. If the vertical line you drew intersects the graph more than once for any value of x then the graph is not the graph of a function.
Please someone help with this and explain it as best as you can. Thank you ;)
Answer:
the answer is B
Step-by-step explanation:
a circle has a circumference of 320π cm. what is it radius
Answer:
The radius of the circle is 160cm
Step-by-step explanation:
circumference of circle = 2πr
circumference = 320π cm
2πr = 320π
2r = 320
r = 160 cm
[tex] \frac{1}{3}x - 5 + 171 = x[/tex]
Answer:
x=249
Step-by-step explanation:
1/3x-5+171=x
1/3x+166=x
166=x-1/3x
166=2/3x
x=166/(2/3)
x=(166/1)(3/2)
x=498/2
x=249
Which of the following lines is parallel to x = 4
O 1) 3y = 9
O 2) y = 4
O 3) x = 2
O4) x = y
Answer:
O4) x = y
Step-by-step explanation:
I took this before
Does anyone mind helping me with this? It seems simple enough.
The question is: Radical x-3 = radical x + 3 (the +3 is outside of the radical). We are solving for X in this question.
Answer:
x = 4
Step-by-step explanation:
It is given that [tex]\sqrt{(x - 3)} = \sqrt{x} + 3[/tex], and we have to solve the given equation for x.
Now, we have [tex]\sqrt{(x - 3)} = \sqrt{x} + 3[/tex]
Squaring both sides we get
[tex][\sqrt{(x - 3)}] ^{2} = [\sqrt{x} + 3]^{2}[/tex]
⇒ x - 3 = x + 6√x + 9 {Since we know the formula of (a + b)² = a² + 2ab + b²}
⇒ 6√x = - 12
⇒ √x = -2
Again squaring both sides we get
⇒x = 4 (Answer)
Find the equation of the linear function represented by the table below in slope-intercept form.
x
0
1
2
3
4
y
2
7
12
17
22
The equation of the function represented by the table is y = 5 x + 2
Step-by-step explanation:
The equation of the linear function in the slope-intercept form is
y = m x + b, where
m is the slope of the lineThe formula of the slope is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex] , where [tex](x_{1},y_{1})[/tex] and [tex](x_{1},y_{1})[/tex] are any two points lie on the lineb is the y-intercept, means the line intersects the y-axis at point (0 , b)The table:
→ x : 0 : 1 : 2 : 3 : 4
→ y : 2 : 7 : 12 : 17 : 22
Let us chose any two points from the table to calculate the slope
∵ Points (1 , 7) and (2 , 12) lie on the line
∴ [tex]x_{1}[/tex] = 1 and [tex]x_{2}[/tex] = 2
∴ [tex]y_{1}[/tex] = 7 and [tex]y_{2}[/tex] = 12
- Substitute these values in the formula of the slope
∵ [tex]m=\frac{12-7}{2-1}=\frac{5}{1}[/tex]
∴ m = 5
Substitute the value of m in the form of the equation
∵ y = m x + b
∴ y = 5 x + b
∵ The line intersects the y-axis at point (0 , b)
∵ From the table point (0 , 2) lies on the line
∴ b = 2
∴ y = 5 x + 2
The equation of the function represented by the table is y = 5 x + 2
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Help please I don’t get this
Answer:
y = -2x - 1
Step-by-step explanation:
Formula is y = mx + b
m = slope
slope = change in y/change in x = (-1 - 3) / (0 - -2) = -4/2 = -2
So equation is now y = -2x + b
To find b, substitute any value from the table into the equation. Let's use (-2,3)
y = -2x + b
3 = -2(-2) + b
3 = 4 + b
-1 = b
So equation is y = -2x - 1
Answer:
C. This can be found by looking at when x=0. If y=-1, then that means that the y intercept is -1, so plug in 1 to either one of the middle two equations and see if it works.
Step-by-step explanation: