A rectangle has side lengths of a and b, which are non-zero integers. Determine possible values of a and b, which will make the diagonal of the rectangle a rational number.
A) a=1, b=1
B) a=4, b=5
C) a=3, b=4
D) a=1, b=2
Answers
Look at the picture.
Use the Pythagorean theorem:
[tex]a^2+b^2=d^2[/tex]
A) a = 1, b = 1
[tex]d^2=1^2+1^2\\\\d^2=1+1\\\\d^2=2\to d=\sqrt2\ \text{it's not rational number}[/tex]
B) a = 4, b = 5
[tex]d^2=4^2+5^2\\\\d^2=16+25\\\\d^2=41\to d=\sqrt{41}\ \text{it's not rational number}[/tex]
C) a = 3, b = 4
[tex]d^2=3^2+4^2\\\\d^2=9+16\\\\d^2=25\to d=\sqrt{25}\to d=5\ \boxed{:)}[/tex]
D) a = 1, b = 2
[tex]d^2=1^2+2^2\\\\d^2=1+4\\\\d^2=5\to d=\sqrt5\ \text{it's not rational number}}[/tex]
Answer: C) a = 3, b = 4.
Options A, B, and D do not yield a perfect square, while option C does, with a=3 and b=4 resulting in a rational diagonal length of 5.
The diagonal of a rectangle can be found using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this is represented as [tex]c^2 = a^2 + b^2,[/tex] where a and b are the sides of the rectangle and c is the diagonal.
Looking at the given options:
A) a=1, b=1: Here, [tex]a^2 + b^2 = 1^2 + 1^2 = 2,[/tex] it is not a perfect square, so the diagonal will not be rational.B) a=4, b=5: Here, [tex]a^2 + b^2 = 4^2 + 5^2 = 16 + 25 = 41,[/tex] it is not a perfect square, so the diagonal will not be rational.C) a=3, b=4: Here, [tex]a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25[/tex], it is a perfect square (5^2), so the diagonal is rational.D) a=1, b=2: Here, [tex]a^2 + b^2 = 1^2 + 2^2 = 1 + 4 = 5,[/tex] it is not a perfect square, so the diagonal will not be rational.Therefore, the possible values of a and b which will make the diagonal of the rectangle a rational number are given in option C: a=3 and b=4.
Related Questions
What expression can be used to determine the total cost of buying g pounds of granola for $3.25 per pound and f pounds of flour for $0.74 per pound?
Answers
The expression for determining the total cost of buying granola and flour is (3.25/g) * g + (0.74/f) * f. The values of g and f represent the number of pounds of each item. To find the total cost, substitute the values into the expression.
Explanation:The expression that can be used to determine the total cost is:
Total cost = ($3.25/g) * g + ($0.74/f) * f
where 'g' represents the number of pounds of granola and 'f' represents the number of pounds of flour.
For example, if you wanted to find the total cost of buying 5 pounds of granola and 2 pounds of flour, you would substitute g = 5 and f = 2 into the expression to get:
Total cost = ($3.25/5) * 5 + ($0.74/2) * 2 = $16.25 + $1.48 = $17.73
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The total cost of buying g pounds of granola at $3.25 per pound and f pounds of flour at $0.74 per pound can be determined using the expression 3.25g + 0.74f.
To determine the total cost of buying g pounds of granola for $3.25 per pound and f pounds of flour for $0.74 per pound, you can use the following expression:
3.25g + 0.74f
Here's a step-by-step explanation:
For granola, the cost per pound is $3.25. Therefore, for g pounds, the total cost is 3.25g.For flour, the cost per pound is $0.74. Therefore, for f pounds, the total cost is 0.74f.Add the costs together to get the total cost: 3.25g + 0.74f.This expression gives you the total cost based on the quantities of granola and flour you wish to purchase.
You invest $4000 in an account at 6.5% per year simple interest. How much will you have in the account at the beginning of the 9th year.
Answers
Answer:
$6340
Step-by-step explanation:
First figure out what 6.5% of 4000 is by multiplying not dividing because dividing will give you a way bigger number.
4000 x 6.5%=260
260 x 9=2340
2340+4000=6340
Lily swims a 100-meter race. She swims the first half in 27.8 seconds. She swims the second half in 30.12 seconds. About how much longer does it take her to swim the second half of the race than the first half? Choose the best estimate.
Answers
Lily swam a 100-meter race, completing the first half in 27.8 seconds and the second half in 30.12 seconds. The difference between these times reveals that it took Lily about 2.32 seconds longer to swim the second half of the race compared to the first half.
Explanation:The student, Lily, swam a 100-meter race with the first and second halves timed separately. She completed the first half in 27.8 seconds and the second half in 30.12 seconds. To find out how much longer it took her to swim the second half than the first, we need to subtract the first time from the second.
So, the calculation would be: 30.12 seconds (second half) - 27.8 seconds (first half) = 2.32 seconds. Therefore, it took Lily about 2.32 seconds longer to swim the second half of the race compared to the first half.
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Mallory has 535 e-mail messages in her inbox. She sorts them evenly among folders labeld family,work, school and recipes.How many messages does mallory put in each folder
Answers
Anwser could be 107 because 535/5=107
Eva and Emily can clean the entire house in 4 hours. Eva can do it by herself in 6 hours. How long would it take Emily to do it by herself?
Answers
Answer:
12 hours would take emily
Step-by-step explanation:
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The hospital shop buys cereal bars in packs of 4. In one day they sell 58 cereal bars. How many packets did they open tht day?
Answers
Joe ate 7 pies on wednesday. If that is only 1/4 of the world record for pie eating what is the pie record
Answers
Answer:
7 pie is the world record. That is 1/4 under the record and your Answer.
Step-by-step explanation:
Who come up with these dump questions??
A polynomial function has a root of zero with multiplicity one, and root of two with multiplicity four. If the function has a negative leading coefficient, and is of odd degree, which of the following is true.
- the function is positive on (-infinity, 0)
-the function is negative on (0,2)
-the function is negative on (2,infinity)
-the function is positive on (0, infinity)
Answers
Answer: A, B, and C
Step-by-step explanation:
root of zero (x = 0) with multiplicity of 1 (x = 0)¹ means that the graph CROSSES the axis at x = 0
root of two (x = 2) with multiplicity of 4 (x = 2)⁴ means that the graph TOUCHES the axis at x = 2
WHY?
odd numbered multiplicity crosses the x-axiseven numbered multiplicity touches the x-axis (it is a turning point)Leading Coefficient (end behavior of right side):
If the leading coefficient is positive, then right side goes to +∞If the leading coefficient is negative, then right side goes to -∞Degree of polynomial (left side behavior):
If degree is even, then left side is the same as the right sideIf the degree is odd, then left side is opposite of right sideThe given polynomial has a negative leading coefficent and odd degree so the left side goes to +∞ and the right side goes to -∞
OBSERVE INTERVALS:
(-∞, 0) is above the x-axis so POSITIVE(0, 2) is below the x-axis so NEGATIVE(2, +∞) is also below the x-axis x=2 was a turning point so NEGATIVESee graph to confirm statements above
Answer:
A, B, and C
Step-by-step explanation:
Had the same question on Edge :)
Each group member is permitted to consume one-third of a pizza. If there are twenty-four members in the group, how many pizzas should be ordered?
Answers
Answer:
8
Step-by-step explanation:
so they are one third of a pizza so thats 1/3 and 24 memebers so basically you just have to multiply 1/3 x 24 = 24/3 which is 8.
Answer:
8 pizzas should be ordered.
Step-by-step explanation:
To get the number of pizzas to be ordered, all we simple need to do is to multiply the numbers of the group members by the amount to be consume by each group member. That is;
Number of pizzas to be ordered = amount to be consume by each group member × number of group members.
amount to be consume by each group member = [tex]\frac{1}{3}[/tex]
Number of group members = 24
We can now proceed to insert our values into the formula
Number of pizzas to be ordered = amount to be consume by each group member × number of group members.
Number of pizzas to be ordered = [tex]\frac{1}{3}[/tex] × 24
= [tex]\frac{24}{3}[/tex]
= 8
Number of pizzas to be ordered = 8
Therefore, 8 pizzas should be ordered.
Which value must be added to the expression x^2+16 to make it a perfect square trinomial
Answers
Adding the term 8x we will get:
x² + 8x + 4² = (x + 4)²
How to make a perfect square trinomial?
The perfect square trinomial is:
(a + b)² = a² + 2ab + b²
Here we start with:
x² + 16
We can rewrite this as:
x² + 4²
We can see that the term missing is the cross one, which will be:
2*(x)*4 = 8x
Then we will get:
x² + 8x + 4² = (x + 4)²
In the diagram below, what needs to be labeled in order to prove triangle AHL and triangle EML are congruent by the SSS postulate
Answers
Answer:
HL = LM
Step-by-step explanation:
We have been given a diagram in which sides AH= EM and AL=EL.
Now we need to find which statement is missing to prove that triangles AHL and EML are congruent using SSS.
SSS means (side - side - side)
Two pair of equal sides are already given so we have completed SS of SSS.
Last "S" means we need one more pair of equal sides so that must be using remaining sides HL and LM
Hence final answer is HL = LM.
graft the solution to the inequality on the number line.
m>2.8
Answers
Answer:
See the graph of the given inequality in the attachment as shown below:
Step-by-step explanation:
Given the inequality: [tex]m > 2.8[/tex]
To graph this inequality on the number line.
Remember:
If the symbol is [tex](\geq or \leq)[/tex] then you fill in the dotIf the symbol is [tex](> or <)[/tex] then you do not fill in the dot.Since, in the given inequality the symbol is '>'
Now, sketch a number line and draw an open circle around the number 2.8.
You can see the number line as shown below in the attachment.
Find the least common multiple. 30, 25, and 4 a. 60 b. 3,000 c. 300 d. 150
Answers
Answer:
The least common multiple is 4.
Step-by-step explanation:
Which functions are continuous at x= -2? Select all that apply.
Answers
Answer:
(i) and (iii)
Step-by-step explanation:
for i)
f(x) = [tex]\frac{x^{2}-4}{x+2}[/tex]
=[tex]\frac{(x+2)(x-2)}{x+2}[/tex]
=(x-2)
As x =-2
f(x)=-2-2
f(x)=-4
so it is a continuous function
for ii)
from the result of (i) we can deduce the result of second and we see that it is not equal to zero so it is not a continuous function
iii)
f(x)=[tex]\frac{x^{2}+4x+4}{x+2}[/tex]
f(x)=[tex]\frac{(x+2)^{2}}{x+2}[/tex]
f(x)=[tex]\frac{(x+2)(x+2)}{x+2}[/tex]
f(x)=x+2
as x= -2
f(x)=-2+2
f(x)=0
so it is a continuous function
iv) From the result of iii we can deduct that
f(x)=0 because it have the same equation so it is not a continuous function
Answer:
The first and third functions are continuous at x=-2
Step-by-step explanation:
Please, see the attached files.
Thanks.
how can we write 128 in exponent form?
Answers
[tex] log_{n }(128) [/tex]
Answer: 2⁷
Step-by-step explanation:
128
∧ 2 64 ∧ 2 32 ∧ 2 16 ∧ 2 8 ∧ 2 4 ∧ 2 2128 = 2 * 2 * 2 * 2 * 2 * 2 * 2
= 2⁷
In 2010, the United States national debt was estimated to be nearly 13,000,000,000,000 dollars. Which of the following represents the national debt in scientific notation?
Answers
Answer: [tex]1.3 \times 10^{13}[/tex]
Explanation:
We are given, that in 2010, the United States national debt was estimated to be nearly 13,000,000,000,000 dollars.
We are required to find the national debt in scientific notation. The scientific notation is used to represent either very small number or very large number.
The scientific form of the given number is:
[tex]1.3 \times 10^{13}[/tex]
A rectangular garden is 17 feet wide. Its length is 3 feet more than twice the width. What is the area of the garden?
Answers
Answer:
a = 629
Step-by-step explanation:
a = w * L
w = 17
L = 2w + 3
a = (17) * (2w + 3)
a = (17) * (2(17) + 3)
a = 17 * (34 + 3)
a = 17 * (37)
a = 629
Lm is the midsegment of trapezoid ABCD ab=46 dc=125 what is lm
A. 103.5
B. 63
C. 79
D. 85.5
Answers
Answer: Choice D) 85.5
Work Shown:
To get the midsegment, we add up the sides parallel to this midsegment, then we divide by 2.
LM = (AB + DC)/2
LM = (46 + 125)/2
LM = 171/2
LM = 85.5
Answer: D is the answerb
Stu takes a job with a starting salary of $65,000 for the first year. He earns a 2% increase each year. How much does Stu make over five years?
$325,000
$331,500
$338,263
$358,826
Answers
Answer:
Stu makes $338,263 over five years.
Step-by-step explanation:
In order to find the total amount Stu would make after five years, you need to find out how much he will make each year based on the 2% increase. Starting with year 1 at $65,000, we need to take his total salary and multiply by 1.02 (or the whole salary plus an increase of 2%). So, in year 2, he would make $66,300. Again, we would take his year 2 total salary and multiply by 1.02 to get $67,626 for year 3. We apply these same steps to years four ($68,978.52) and five ($70,358.10). To find his income over the five years, we need to add his salary from all five years together to get a total of $338,263.
Answer:
Option C which is $ 338,263
Step-by-step explanation:
The salary for the first year = $65,000
Increase per year= 2 %
To find total money made in five years = ?
total earned in first year = $65,000
Total increase in money after first year = 2 % of 65,000
= [tex]\frac{2*65000}{100}[/tex]
=$ 1300
Total earned in second year = 65,000 + 1300
= $ 66,300
Total increase in money after second year = 2 % of 66,300
= [tex]\frac{2*66300}{100}[/tex]
=$ 1326
Total earned in third year = 66300 + 1326
= $ 67,626
Total increase in money after third year = 2 % of 67,626
= [tex]\frac{2*67626}{100}[/tex]
=$ 1352.52
Total earned in fourth year = 67626 + 1352.52
= $ 68978.52
Total increase in money after fourth year = 2 % of 68978.52
= [tex]\frac{2*68978.52}{100}[/tex]
=$ 1379.58
Total earned in fifth year = 68978.52 + 1379.58
= $ 70358.1
Total Made in all five years = 65000+66300+67626+68978.52+70358.1
= $ 338262.62
≅$ 338,263
Find the coordinates of the other endpoint of the segment, given its midpoint and one endpoint. (Hint: Let (x,y) be the unknown endpoint. Apply the midpoint formula, and solve the two equations for x and y.)
midpoint (-18,-12) endpoint (-9,-11)
Answers
[tex]\text{The formula of a midpoint:}\\\\\left(\dfrac{x_1+x_2}{2};\ \dfrac{y_1+y_2}{2}\right)[/tex]
[tex]\text{We have}\\\\\text{endpoint}\ (-9,\ -11)\to x_1=-9,\ y_1=-11\\\\\text{midpoint}\ (-18,\ -12)\\\\\text{Therefore we have the equations:}\\\\\dfrac{-9+x_2}{2}=-18\qquad\text{multiply both sides by 2}\\\\-9+x_2=-36\qquad\text{add 9 to both sides}\\\\\boxed{x_2=-27}\\\\\dfrac{-11+y_2}{2}=-12\qquad\text{multiply both sides by 2}\\\\-11+y_2=-24\qquad\text{add 11 to both sides}\\\\\boxed{y_2=-13}\\\\Answer:\ \boxed{(-27,\ -13)}[/tex]
The main show tank has a radius of 60 feet and forms a quarter sphere where the bottom of the pool is spherical and the top of the pool is flat. (Imagine cutting a sphere in half vertically and then cutting it in half horizontally.) What is the volume of the quarter-sphere shaped tank? Round your answer to the nearest whole number. You must explain your answer using words, and you must show all work and calculations to receive credit.
the equation given to solve this is v=4/3 pi radius by the power of 3
Answers
The given radius is = 60 feet
The volume of the sphere is given by = [tex]\frac{4}{3} \pi r^{3}[/tex]
So we will find the volume of the sphere using radius as 60 and pi as 3.14
Volume = [tex]\frac{4}{3}*3.14*60*60*60[/tex] = 902059.20 cubic feet
As quarter of anything is defined as its one-fourth part, so here the volume of the quarter of sphere can be calculated by finding one fourth of the calculated volume.
So volume of quarter sphere = [tex]\frac{902059.20}{4}[/tex]
= 225514.80 cubic feet
And rounding this to nearest whole number we get 225515 cubic feet as the answer.
Anumeha is mowing the lawns for a summer job. For every mowing job, she charges and initial fee of $10 plus a constant fee for each hour of work. Her fee for a 5-hour job, for instance is $35. Let F(t) denote Anumeha's for a single job F (measured in dollars) as a function of the number of hours t it took her to complete it. Write the functions formula
Answers
Answer:
F(t) = 10 +5t
Step-by-step explanation:
Income equals the initial fee plus the hours worked times the hourly rate
We know the initial fee = 10
hours = t
hourly rate =r
F(t) = 10 + r*t
We know a 5 hour job is 35 dollars
35 = 10 + r*5
Subtract 10 from each side
35-10 = 10-10 +5r
25 = 5r
Divide by 5
25/5 = 5r/5
5 =r
The hourly rate is 5
F(t) = 10 +5t
A line contains the points S (4, -10) and B (10, -16). Find the slope of ̅̅̅̅SB. Must show your work to receive full credit.
Answers
Answer:
The slope is -1
Step-by-step explanation:
To find the slope when we have 2 points, we can use the formula
m = (y2-y1)/(x2-x1)
= (-16--10)/(10-4)
= (-16+10)/(10-4)
= -6/6
= -1
The slope is -1
Which binomial expressions are factors of 5x^3+8x^2-7x-6
Answers
Answer:
x - 1 and x + 2
Step-by-step explanation:
By the factor theorem if (x - a) is a factor then f(a) = 0.
Try x - 1:_
f(1) = 5(1)^3 + 8(1)^2 - 7 - 6
= 5 + 8 - 7 - 7
= 0 so x - 1 is a factor
We can find the other factors by long division:-
5x^2 + 13x + 6
-----------------------------------
x - 1 ) 5x^3 + 8x^2 - 7x - 6
5x^3 - 5x^2
--------------------
13x^2 - 7x
13x^2 - 13x
------------------
6x - 6
6x - 6
-----------------
5x^2 + 13x + 6 = (5x + 3)(x + 2)
You conduct an experiment where you roll a die and record the results. You roll the die 1,230 times, and the results are shown in the table below:
Result Frequency
1 245
2 172
3 219
4 201
5 137
6 256
What is the experimental probability of rolling a 1 on your next roll?
A.
0.1667
B.
0.1992
C.
0.245
D.
0.3275
Part B. What is the theoretical probability of rolling a 3 on your next roll?
A.
0.167
B.
0.178
C.
0.245
D.
0.333
The probability of an event is
2/7
What are the odds of the same event?
A. 7/9
B. 5/7
C. 2/9
D. 2/5
If I could get some help That'd Be Greattt
Answers
Answer: Other guy is right for 1 & 2 but is wrong for the last question.
I got a 100 on connections.
(1) 0.1992
(2)0.167
(3) 2/5
Step-by-step explanation:
The experimental probability of rolling a 1 on the next roll is 0.1992 (B). The theoretical probability of rolling a 3 on a six-sided die is 0.1667 (A). The odds of an event with a probability of 2/7 are 2 to 5 (D).
Explanation:The experimental probability of rolling a 1 in this experiment is calculated by dividing the frequency of rolling a 1 by the total number of rolls. There were 245 occurrences of rolling a 1 out of 1230 total rolls. To find the experimental probability, you divide 245 by 1230, which gives you 0.1992.
Part B: The theoretical probability of rolling any particular number on a fair six-sided die is 1/6, because there are 6 possible outcomes and each outcome is equally likely if the die is fair. Therefore, the theoretical probability of rolling a 3 on the next roll is 0.1667 (which is equivalent to 1/6 when rounded to four decimal places).
When the probability of an event is 2/7, the odds of the event are 2 to 5 (not 2/5). This is because odds are expressed as the ratio of the number of successful outcomes to the number of unsuccessful outcomes. Since there are 2 chances of success and 5 chances of failure (7-2=5), the odds are 2 to 5.
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Montel sold 13 popcorn buckets and sold 13 fruit baskets for a fundraiser. The fruit baskets coast $20.75 each . If Montel raised a total of $469, how much did each popcorn buckets coast
Answers
Answer:
The popcorn buckets each cost around $15.34.
Step-by-step explanation:
This table shows the input and output values for an exponential function f(x) .
What is the ratio of outputs for any two inputs that are three values apart?
x −3 −2 −1 0 1 2 3
f(x) 8/27 8/9 8/3 8 24 72 216
Enter your answer, as a simplified fraction
Answers
Answer:
Given the table as shown below:
x y
-3 8/27
-2 8/9
-1 8/3
0 8
1 24
2 72
3 216
Taking any input variable x that are three apart:
i,e x = 0 and x = 3
The output values at x =0 and x = 3 are:
f(0) = 8 and f(3) = 216
Then;
The ratio of outputs is:
[tex]\frac{f(3)}{f(0)} = \frac{216}{8}= \frac{27}{1}[/tex] or 27 : 1
Therefore, the ratio outputs for any two inputs that are three values apart is: [tex]27: 1[/tex]
Answer:
0.5
Step-by-step explanation:
What is the value of a
A. 27.5
B. 45
C. 50
D. 90
Answers
Answer:
90 degrees
Step-by-step explanation:
PLEASE HELP 20pts
what is the number ? fifty
-three more then 3 times that number
-35 less than seven times a certain number is
what is the number? is twenty
-nine more than 12 times that number 21 less than fourteen times a certain number
Answers
Answer:
53+3=56
56 times -35
-1960-7
-1967
i think this is how you do it.
Step-by-step explanation:
Identify a pattern and sequence and fine the next number in the pattern. 54,52,26,24,_,_,_
Answers
Answer:
54, 52, 26, 24, 22, -6, -4
Step-by-step explanation: