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
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.
Marcus estimates that 230 people will attend the choir contrary there was an actual total of 300 people who attended the choir concert What is the answer to this?
Answer:
23% error since 70 more people attended then Marcus counted
Step-by-step explanation:
To calculate the percent error:
We find the difference between the predicted and the actual.We divide the absolute value of the difference by the actual recorded value.Convert to a percent by multiplying by 100 and adding a % sign.Marcus counted 230 and 300 actually came.
1. 230-300=-70 but the absolute value is 70
2. 70/300=0.23
3. 0.23(100)=23%
Marcus' estimated had a 23% error.
This box plot shows the heights ( in feet) of a sample of pine trees
Answer:
D
Step-by-step explanation:
The new one would have a long gap between the Q3 and the maximum of the box and whisker plot, therefore it should create a positive skew.
Answer:
D
Step-by-step explanation:
It will be positively skewed (skewed right), because that means that the longer segment would be on the right side of the median.
140 is a major outlier and a lot larger than the more central numbers. However, it's only 1 number, so the median (location of the "box") would remain similar or the same. The end on the right, however, would move far to the right, extending the "line" on that side.
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The money used in Saudi Arabia is the Riyal.The exchange rate is 4 Riyals to 1 dollar. How many Riyals would you receive if you exchange 5 dollars?
Answer:
20 Riyals you receive if you exchange 5 dollars .
Step-by-step explanation:
As given
The money used in Saudi Arabia is the Riyal.
The exchange rate is 4 Riyals to 1 dollar.
i.e
4 Riyals = 1 dollar.
Now calculate for 5 dollars.
5 × 4 Riyals = 5 × 1 dollars
20 Riyals = 5 dollars
Therefore 20 Riyals you receive if you exchange 5 dollars .
Hannah has 89 of a pound of birdseed. She put 23 of a pound of birdseed into her bird feeders. How much birdseed does Hannah have remaining?
Final answer:
Hannah initially has 8/9 of a pound of birdseed, and after using 2/3 of a pound, she has 2/9 of a pound remaining, which is calculated by finding a common denominator and subtracting the second fraction from the first.
Explanation:
Hannah initially has 8/9 of a pound of birdseed. After filling her bird feeders with 2/3 of a pound of birdseed, we need to subtract the amount she used from what she started with to find out how much birdseed she has left. The calculation we'll use is:
(initial amount) - (amount used) = (amount remaining)
In fractional form, this becomes:
8/9 - 2/3
To subtract these fractions, we need a common denominator. The smallest common denominator for 9 and 3 is 9. We can convert 2/3 to 6/9 by multiplying the numerator and the denominator by 3. After this step, the subtraction is straightforward:
8/9 - 6/9 = 2/9
Hence, Hannah has 2/9 of a pound of birdseed remaining.
100 points!
Only 29 and 31
Rewrite each expression eliminating fractions. Simplified answers may be left in terms of any of the six trigonometric functions.
Answer:
29. sec w csc w - sec^2 w
31. - tan ^2 k (sin k +1)
Step-by-step explanation:
29. (cot w -1)/ (1- sin^2 w)
We know that cot w = cos w/ sin w
and 1 - sin^2 w = cos ^2 w
Replace these identities into the expression
(cos w/ sin w -1)
-----------------------
cos ^ 2 w
Get a common denominator on top of sin w
(cos w/ sin w -sin w/sin w)
-----------------------
cos ^ 2 w
(cos w - sinw)/ sin w
-----------------------
cos ^ 2 w
(cos w - sinw)
-----------------------
cos ^ 2 w sin w
Split into 2 terms
cos w sin w
----------------------- - ---------------------------------
cos ^ 2 w sin w cos ^2 w sin w
Cancel cos w in the first term and sin w in the second term
1 1
----------------------- - ---------------------------------
cos w sin w cos ^2 w
We know that 1/cos = sec and 1/ sin = csc
sec w csc w - sec^2 w
31. sin k/ (1 - csc k)
We know that csc k is 1/ sin k
sin k
---------------
1 - 1/ sin k
Get a common denominator for the bottom
sin k
---------------
sin k/ sin k - 1/ sin k
sin k
---------------
(sin k-1)/ sin k
Multiply by sin k/sin k
sin k * sin k
---------------
(sin k-1)/ sin k * sin k
sin k * sin k
---------------
(sin k-1)
sin^2 k
---------------
(sin k-1)
Multiply by (sin k +1)/ (sin k +1) so we can get rid of the denominator
sin^2 k (sin k +1)
-------------------------
(sin k-1) (sin k +1)
Foiling out the denominator, we get (sin^2 k-1)
sin^2 k (sin k +1)
-------------------------
(sin^2 k-1)
Factoring out a -1 from the denominator
sin^2 k (sin k +1)
-------------------------
-1 (1 - sin^2 k)
1 - sin ^2 k = cos ^ k
sin^2 k (sin k +1)
-------------------------
-1 (cos ^2 k)
sin ^2k/ cos ^2 k = tan ^2 k
- tan ^2 k (sin k +1)
Consider 8x² - 48x = -104.
Write the equation so that
a = 1: x² + ___ x = ___
When a = 1, the equation becomes x² - 6x = -13.
To rewrite the equation 8x² - 48x = -104 with the coefficient of x² as 1,
you can divide the entire equation by 8,
which is the coefficient of x²:
(8x² - 48x) / 8 = (-104) / 8
Now, simplify:
x² - 6x = -13
So, when a = 1, the equation becomes x² - 6x = -13.
for such more question on coefficient
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A store makes shirts and jackets to sell each shirt costs $4 to make and each jacket costs $25 to make
he data set represents the number of miles Mary jogged each day for the past nine days.
6, 7, 5, 0, 6, 12, 8, 6, 9
The outlier of the data set is
Answer:
zero is your answer.
Step-by-step explanation:
Your answer for this question is 0
given f(x)=2^x, g(x) is found by translating f(x) three units right and two units down. Which function below shows g(x)?
[tex]\bf ~\hspace{10em}\textit{function transformations} \\\\\\ \begin{array}{llll} f(x)= A( Bx+ C)^2+ D \\\\ f(x)= A\sqrt{ Bx+ C}+ D \\\\ f(x)= A(\mathbb{R})^{ Bx+ C}+ D \end{array}\qquad \qquad \begin{array}{llll} f(x)=\cfrac{1}{A(Bx+C)}+D \\\\\\ f(x)= A sin\left( B x+ C \right)+ D \end{array} \\\\[-0.35em] ~\dotfill\\\\ \bullet \textit{ stretches or shrinks horizontally by } A\cdot B\\\\ \bullet \textit{ flips it upside-down if } A\textit{ is negative}\\ ~~~~~~\textit{reflection over the x-axis}[/tex]
[tex]\bf \bullet \textit{ flips it sideways if } B\textit{ is negative}\\ ~~~~~~\textit{reflection over the y-axis} \\\\ \bullet \textit{ horizontal shift by }\frac{ C}{ B}\\ ~~~~~~if\ \frac{ C}{ B}\textit{ is negative, to the right}\\\\ ~~~~~~if\ \frac{ C}{ B}\textit{ is positive, to the left}\\\\ \bullet \textit{ vertical shift by } D\\ ~~~~~~if\ D\textit{ is negative, downwards}\\\\ ~~~~~~if\ D\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{ B}[/tex]
with that template in mind, let's check
C = -3, three units to the right
D = -2, two units down.
[tex]\bf f(x)=2^x~\hspace{10em}\stackrel{\stackrel{C=-3\qquad D=-2}{\cfrac{}{}}}{g(x)=2^{x-3}-2}[/tex]
Answer:
g(x) = 2^(x - 3) - 2.
Step-by-step explanation:
Translating 2^x 3 units to the right . This is done by changing it to 2^(x - 3). Then 2 units down is given by 2^(x -3) - 2.
Answer is 2^(x - 3) - 2.
Lynn has 4 more books than Jose. If Lynn gives Jose 6 of her books, how many more will Jose have than Lynn?
A pipe that's 36 inches long needs to be cut into 2 1/4-inch long pieces. How many pieces can be cut from this length of pipe? Hint: Change 2 1/4 to an improper fraction before calculating.
A. 9
B. 11 1/4
C. 4 4/9
D. 16
Answer:
16 inches
Step-by-step explanation:
2 1/4 = 9/4 inches
Number of pieces = 36 / 9/4
= 36 + 4/9
= 16 inches
Answer:
D. 16
Step-by-step explanation:
I divided 36 from 2.25.
Please answer this question!! 20 points and brainliest!
Answer:
A
Step-by-step explanation:
We can find the surface area of the object by adding the surface areas of each part. We have many rectangle faces to count and two triangular faces. Each has a formula for the area. We will find the area of each and then add them all together.
Triangle - 0.5 *b*h
Rectangle - b*h
Triangles
There are two triangles on either side. The height is 1.5. The base is 1.8.
0.5(1.5)(1.8)=1.35 meters squared
Since there are two, we will add 1.35+1.35 in our final calculation.
Rectangles
We will start by calculating the largest rectangle on the side. It has height of 4 and a base of 2.5 (shown above left).
4(2.5)=10
Since there are two (one we can see and one we can't), we will add 10+10 in our final calculation.
Next we calculate the top and bottom. The height is 3 and the base is 2.5 on top. But the bottom sticks out more and adds 1.8 to its base.
Top - 3(2.5)=7.5
Bottom-3(2.5+1.8)=12.9
Finally, we will calculate the front side and back(not visible) as well as the slant up front. The back side has height 4 and base 3. The front side has base 3 and height 4-1.5=2.5. The slant has base 2.3 and height 3.
Back - 4(3)=12
Front- 3(2.5)=7.5
Slant - 3(2.3)=6.9
We add all together for the total surface area: 1.35+1.35+10+10+7.5+12.9+12+7.5+6.9=69.5 meters squared.
first number plus twice a second number is 8. Twice the first number plus the second totals 31. Find the number
Answer:
1st number = 18
2nd number = -5
Step-by-step explanation:
Let x be first number and y be the 2nd number.
We have been given that first number plus twice a second number is 8. We can represent this information as:
[tex]x+2y=8...(1)[/tex]
We are also given that twice the first number plus the second totals 31. We can represent this information as:
[tex]2x+y=31...(2)[/tex]
Now let us solve our system of equations using substitution method.
From equation (1) we will get,
[tex]x=8-2y[/tex]
Substituting [tex]x=8-2y[/tex] in equation (2) we will get,
[tex]2(8-2y)+y=31[/tex]
[tex]16-4y+y=31[/tex]
[tex]-4y+y=31-16[/tex]
[tex]-3y=15[/tex]
[tex]y=\frac{15}{-3}[/tex]
[tex]y=-5[/tex]
Therefore, the second number is -5.
Now let us substitute y=-5 in equation (1).
[tex]x+(2*-5)=8[/tex]
[tex]x-10=8[/tex]
[tex]x=8+10[/tex]
[tex]x=18[/tex]
Therefore, the first number is 18.
The numbers are x = 18 and y = -5.
The question involves a system of linear equations and requires solving for two unknown numbers. Using variables to represent these numbers, the system can be written as:
First equation: x + 2y = 8,
Second equation: 2x + y = 31.
To solve this system, follow these steps:
Multiply the first equation by 2, obtaining the equation 2x + 4y = 16.
Subtract the second equation from this result to eliminate x, leading to 3y = -15.
Divide by 3 to find y = -5.
Substitute y = -5 into any original equation to solve for x.
Using the first equation: x + 2(-5) = 8, therefore x = 18.
The solution to the system is x = 18 and y = -5.
Show and justify the steps for solving ax=bx+c where a≠b. then use the literal equation's solution to obtain the solution of 2x=x+7.
Answer: [tex]\bold{x=\dfrac{c}{a-b},\qquad x=7}[/tex]
Step-by-step explanation:
Move all of the x's to one side and everything else on the other.
ax = bx + c
ax - bx = c subtracted bx from both sides
x(a - b) = c factored out the like term of "x" on left side
[tex]x=\dfrac{c}{a-b}[/tex] divided both sides by (a- b)
2x = x + 7 → a = 2, b = 1, c = 7
[tex]x=\dfrac{7}{2-1}[/tex]
x = 7
Check:
2x = x + 7
-x -x
x = 7
Each year the soccer team, Peterson United, plays 25 games at their stadium. The owner of Peterson United claimed that last year the mean attendance per game at their home stadium was 24500.
Based on the owner´s claim, calculate the total attendance for the game at Peterson United´s home stadium last year.
High school question
3 parts
Answer:
Total attendance for the game at stadium last year was 612500.
Step-by-step explanation:
Each year the soccer team, Peterson United, plays 25 games at their stadium.
The owner of Peterson United claimed that last year the mean attendance per game was = 24500
Since mean of any set of data = [tex]\frac{sum of data}{Total number of data}[/tex]
When we apply this rule in this question formula becomes as
24500 = [tex]\frac{x}{25}[/tex]
x = 25 × 24500
x = 612500
suppose a normal distribution has a mean of 38 and a standard deviation of 2. What is the probability that a data value is between 36 and 43? Round your answer to the nearest 10th of a percent.
Answer:
83.5 %
Step-by-step explanation:
The mean is 38 and the standard deviation is 2
38 -2 is 36
36 is one standard deviation below the mean.
43 - 38 is 5
5/2 is 2 1/2 times the standard deviation above the mean
-1< z< 2.5
P ( Z<2.5 )−P (Z<−1 )
P ( Z<−1)=1−P ( Z<1 )
P ( Z<2.5 )-1+P ( Z<1 )
Using the standard normal table
0.9938 - 1 +0.8413
0.8351
83.51%
Rounding to the nearest tenth
83.5%
Yep the answer's 83.5%
godspeed in your adventures in cheating >:)
Which expression is equal to (In picture)
[tex]3x^2+12x-15=3(x^2+4x-5)=3(x^2+5x-x-5)\\\\=3[x(x+5)-1(x+5)]=3(x+5)(x-1)\\----------------------\\4x^2+4x-8=4(x^2+x-2)=4(x^2+2x-x-2)\\\\=4[x(x+2)-1(x+2)]=4(x+2)(x-1)\\----------------------\\2x^2-8=2(x^2-4)=2(x^2-2^2)=2(x-2)(x+2)\\----------------------\\x^2-5x=x(x-5)\\----------------------\\\\\dfrac{3x^2+12x-15}{4x^2+4x-8}\cdot\dfrac{2x^2-8}{x^2-5x}\\\\=\dfrac{3(x+5)(x-1)}{4\!\!\!\!\diagup_2(x+2)(x-1)}\cdot\dfrac{2\!\!\!\!\diagup^1(x-2)(x+2)}{x(x-5)}\\\\\text{Canceled:}\ (x-1)\ \text{and}\ (x+2)[/tex]
[tex]=\boxed{\dfrac{3(x-2)(x+5)}{2x(x-5)}}[/tex]
What is 2.45 x 10^4
A: 2450
B: 24,500
C: 245
D: 245,000
Answer:
I got B. 24,500 as my answer.
Step-by-step explanation:
2.45 x (10^4) = 24,500
What is the average rate of change of the function over the interval x = 0 to x = 5? f(x)=4^2x+1
Enter your answer, as a simplified fraction,
Answer: 838860
=========================================
Plug in x = 0 to find y = f(x)
f(x) = 4^(2x+1)
f(0) = 4^(2*0+1)
f(0) = 4
The point (0,4) is on the function curve
Plug in x = 5 and compute
f(x) = 4^(2x+1)
f(5) = 4^(2*5+1)
f(5) = 4194304
The point (5, 4194304) is on the function curve
Once you have these two points, you use the slope formula to compute the average rate of change
m = (y2-y1)/(x2-x1)
m = (4194304 - 4)/(5-0)
m = 838860
The slope of the line through the two points found earlier is m = 838860, so this is the average rate of change on f(x) for the interval from x = 0 to x = 5.
An angle measures 50° more than the measure of a supplementary angle. What is the measure of each angle?
Answer:
65° and 115°
Step-by-step explanation:
let one angle be x then the other is x + 50
The sum of 2 supplementary angles = 180°, hence
x + x + 50 = 180
2x + 50 = 180 ( subtract 50 from both sides )
2x = 130 ( divide both sides by 2 )
x = 65
Thus the 2 angles are x = 65° and x + 50 = 65 + 50 = 115°
is 3/7 and 8/28 proportional or not proportional??
Answer:
Yes they are proportional!
The factors of 28 that we can use in these fractions are 4 and 7. To find if these are proportional, just simplify 8/28. To do so divide the numerator (top) and the denominator (bottom) both by 4...
8/28 / 4 = 4/7
Now that you have both with the same denominator, you can evaluate if 3/7 and 4/7 are proportional. In this case they are.
Hope this helps! :)
An alloy composed of nickel, zinc, and copper in a 4:1:2 ratio. How many kilograms of each metal are needed to make 35 kg of this alloy?
Answer:
Step-by-step explanation:
The ration of weight of nickel, zinc, and copper in an alloy is 4:1:2
The weight of alloy can be written as
4x + x + 2x = 35 Kg
7 x = 35 Kg
X = 35 Kg/7
X = 5kg
The weight of nickel the alloy = 4 x 5 = 20 Kg
The weight of zinc in the alloy= 1 x 5 = 5kg
The weight of copper in the alloy = 2 x 5 = 10 kg
Answer:
Let X be the amount of each metal needed to make 35 Kg of this alloy.
We know that an alloy is composed of nickel, zinc and copper in a proportion
4:1:2. Therefore, we have:
[tex]4X+1X+2X = 35[/tex]
[tex]7X = 35[/tex]
[tex]X=\frac{35}{7}[/tex]
[tex]X=5[/tex] Kg
Therefore, there are [tex]5 \times 4 =20[/tex] kg of nickel, [tex]1 \times 5 =5[/tex] kg of zinc, and [tex]2 \times 5 = 10[/tex] kg of copper needed to make 35 kg of this alloy
Marco books a room at a hotel. He spends $600 in total for a 3-night stay. Fill in each statement. For every 1 night that Marco stays at the hotel, he spends . If he stays at the hotel for 5 nights, he would spend .
Mario is setting up a new tent during a camping trip. The tent came with 7 feet of rope. The instructions were to use 34.5 inches of the rope to tie a tarp on top of the tent. Then, the remaining rope should be cut into 8 1/4 inch sections to tie the tent to stakes in the ground. Mario will use al the rope as instructed. Write and solve an equation to determine the number of 8 1/4 inch sections of rope Mario can cut from the rope.
Answer:
[tex] The\ number\ of\ 8 \frac{1}{4}\ inches\ of\ section\ of\ rope\ be\ 6. [/tex]
Step-by-step explanation:
As given
Mario is setting up a new tent during a camping trip.
The tent came with 7 feet of rope.
As 1 foot = 12 inches
Now convert 7 feet into inches.
7 feet = 7 × 12 inches
= 84 inches
As given
The instructions were to use 34.5 inches of the rope to tie a tarp on top of the tent.
Than
Rope left after tie a tarp on top of the tent = 84 - 34.5
= 49.5 inches
As given
[tex]The\ remaining\ rope\ should\ be\ cut\ into\ 8 \frac{1}{4}\ inch\ sections\ to\ tie\ the\ tent\ to\ stakes\ in\ the\ ground.[/tex]
i.e
[tex]The\ remaining\ rope\ should\ be\ cut\ into\ \frac{33}{4}\ inch\ sections\ to\ tie\ the\ tent\ to\ stakes\ in\ the\ ground.[/tex]
[tex]Let\ us\ assume\ the\ number\ of\ \frac{33}{4}\ inches\ section\ of\ rope\ be\ x.[/tex]
Than the equation becomes
[tex]\frac{33\times x}{4} = 49.5[/tex]
[tex]x = \frac{49.5\times 4}{33}[/tex]
[tex]x = \frac{198}{33}[/tex]
x = 6
Answer:
6
Step-by-step explanation:
Converting 7 ft to inches:
[tex]7*12=84[/tex] inches
34.5 inches is used, and remaining is used for [tex]8\frac{1}{4}[/tex] in. sections. Let [tex]x[/tex] be the number of [tex]8\frac{1}{4}[/tex] in. sections. ([tex]8\frac{1}{4}[/tex] can be written as 8.25 inches)We set up the equation as:
[tex]34.5+8.25x=84[/tex]
Now solving for [tex]x[/tex] would give us the number of 8.25 inch sections:
[tex]8.25x=84-34.5\\8.25x=49.5\\x=\frac{49.5}{8.25}=6[/tex]
Hence, Mario can cut 6 (8.25 inch) sections from the rope
SOMEONE PLEASE HELP ALL THE QUESTIONS R ATTACHED AS WELL AS THE ANSWERS!!!!!!!!!!!!!!
The error is in Step-4.
A negative exponent does NOT mean that the number turns negative. A negative exponent means the number is in the denominator.
(4)⁻⁴ means (1/4⁴) . That's (1/256) . All positive numbers.
Help please answer this .... is this bettter?
Answer:
x = 53 degrees
Step-by-step explanation:
Alternate exterior angles are equal if two parallel lines are cut by a transversal.
If m and n are parallel, then
3x-28 = 2x+25
Subtract 2x from each side.
3x-28-2x = 2x+25-2x
x-28 =25
Add 28 to each side.
x-28+28 = 25+28
x = 53
25 POINTS + BRAINLIEST IF YOU EXPLAIN ANSWER
Answer:
4th degree
Step-by-step explanation:
We can use a technique called finite differences.
In column 1 put the differences between the y values (y2-y1)
Keep adding columns with the differences until they become constant.
Column 2 is column1 differences (-2- -30) etc
Column3 is column 2 differences
Column 4 is column3 differences
The differences in column4 are the same, so we can stop
This will be a 4th degree polynomial
x y column 1 column 2 column 3 column 4
-2 30
-1 0 -30
0 -2 -2 28
1 0 2 4 -24
2 30 30 28 24 48
3 160 130 100 72 48
4 510 350 220 120 48
Answer:
4th degree ...........
The length of the poster is 16 inches. What is the length of this poster in centimeters. (1 inch = 2.54 centimeters)
Answer:
90
Step-by-step explanation: