Let assume that [tex]\sqrt7[/tex] is a rational number. Therefore it can be expressed as a fraction [tex]\dfrac{a}{b}[/tex] where[tex]a,b\in\mathbb{Z}[/tex] and [tex]\text{gcd}(a,b)=1[/tex].
[tex]\sqrt7=\dfrac{a}{b}\\\\7=\dfrac{a^2}{b^2}\\\\a^2=7b^2[/tex]
This means that [tex]a^2[/tex] is divisible by 7, and therefore also [tex]a[/tex] is divisible by 7.
So, [tex]a=7k[/tex] where [tex]k\in\mathbb{Z}[/tex]
[tex](7k)^2=7b^2\\\\49k^2=7b^2\\\\7k^2=b^2[/tex]
Analogically to [tex]a^2=7b^2[/tex] ------- [tex]b^2[/tex] is divisible by 7 and therefore so is [tex]b[/tex].
But if both numbers [tex]a[/tex] and [tex]b[/tex] are divisible by 7, then [tex]\text{gcd}(a,b)=7[/tex] which contradicts our earlier assumption that [tex]\text{gcd}(a,b)=1[/tex].
Therefore [tex]\sqrt7[/tex] is an irrational number.
Final answer:
To prove that √7 is irrational, we assume the opposite and show that it leads to a contradiction. We start by assuming that √7 is rational and can be expressed as a fraction. By squaring both sides of the equation and simplifying, we get an equation that leads to p and q being divisible by 7, which contradicts our initial assumption. Therefore, √7 must be irrational.
Explanation:
To prove that √7 is irrational using the method of contradiction, we assume the opposite, which is that √7 is rational. This means that it can be expressed as a fraction p/q, where p and q are integers with no common factors other than 1. We can then square both sides of the equation (√7)² = (p/q)² and simplify to get the equation 7 = p²/q². Rearranging, we have p²= 7q².
From this equation, we can deduce that p² is divisible by 7, which means p must also be divisible by 7. Let's represent p as 7r, where r is another integer. Substituting back into the equation, we get (7r)² = 7q², which simplifies to 49r² = 7q². Dividing both sides by 7, we have 7r² = q². This implies that q² is also divisible by 7, so q must also be divisible by 7.
However, if both p and q are divisible by 7, this contradicts our initial assumption that p/q has no common factors other than 1. Therefore, our assumption that √7 is rational must be false, and hence √7 is irrational.
Given cos A = 0.25 find angle A in degrees. Round your answer to the nearest hundredth.
34.12°
123.98°
56.78°
75.52°
Compute the requested average. Tim Worker receives the following income monthly from a second job. Jan. Feb. Mar. Apr. May June $200 $150 $250 $260 $285 $380 What is his average pay? (To the nearest hundredth.)
254.17255.37257.83259.85
A customer buys 3 bottles of water for ninety-nine cents each seven percent tax on each bottle what is the total bill
What are the approximate solutions of 4x2 + 3 = −12x to the nearest hundredth?
x ≈ −3.23 and x ≈ 0.23
x ≈ −2.72 and x ≈ −0.28
x ≈ 0.28 and x ≈ 2.72
x ≈ −0.23 and x ≈ 3.23
Use the chart above to determine the premium for 10-year term insurance. $10,000 policy for a 35-year-old male. Premium = _____?
6.11
61.10
611.00
6,110.00
Answer:
Option B 61.10
Step-by-step explanation:
Given is a chart giving the premium to be paid for 10 year term insurance for different age groups for an amount of 1000 policy.
We have to calculate the premium for a 35 year old man for 10 year term insurance for 10000 policy.
From the chart we find that for a 35 year old man premium = 6.11
This premium is for 1000 dollars.
Since here sum assured is 10000 dollars, we get the premium
=6.11(10) = 61.10 dollars
Option B is correct
Premium = 61.10 dollars
A rectangular prism has a volume of 360 in3. If the height measures 4 in., which base measurements could the prism have? 4 in. by 20 in. 45 in. by 45 in. 6 in. by 15 in. 5 in. by 19 in.
What is the smallest integer greater than -1/2 .
Group of kids rode horses to canyon. 20 eyes 30 legs between them. How many horses
Final answer:
To solve this problem, we set up equations based on the total number of eyes and legs. By solving the system, we find that the group consists of 5 horses.
Explanation:
The question asks us to solve a problem involving a group of kids and horses by using the given information about the total number of eyes and legs. Since humans have two eyes and two legs, and horses have two eyes and four legs, we can set up a system of equations to find the solution.
Let's denote the number of kids as 'k' and the number of horses as 'h'. Since each human and horse has two eyes, the total number of eyes is given by the equation: 2k + 2h = 20. Similarly, the total number of legs is given by the equation: 2k + 4h = 30. Solving these equations will give us the number of kids and horses in the group.
Solving for 'k' from the first equation gives us k = 10 - h. Substituting this into the second equation gives us 2(10 - h) + 4h = 30. Simplifying this equation results in 20 - 2h + 4h = 30, which further simplifies to 2h = 10, and thus h = 5. Therefore, there are 5 horses in the group.
The formula represents the surface area S of a cube with side lengths x.
S=6x^2
Solve for x.
The solution of x is √(S/6).
The formula to find the surface area of a cube with side lengths x is:
S = 6x2
To solve for x:
Divide both sides by 6: S/6 = x2
Take the square root of both sides to isolate x: x = √(S/6)
An angle measures 72° less than the measure of a complementary angle. What is the measure of each angle?
Answer: [tex]81^{\circ}\ \text{ and }\ 9^{\circ}[/tex]
Step-by-step explanation:
We know that the sum of two complementary angles is [tex]90^{\circ}[/tex].
Given : An angle measures 72° less than the measure of a complementary angle.
Let x be the angle , then the measure of other angle will be x-72.
Since they area complementary , then we have
[tex]x+x-72=90\\\\\Rightarrow\ 2x=90+72\\\\\Rightarrow\ 2x=162\\\\\Rightarrow\ x=\dfrac{162}{2}=81[/tex]
Therefore, the measure of angles are :
[tex]81^{\circ}\ \text{ and }\ (81-72)^{\circ}=9^{\circ}[/tex]
the inverse of an exponential function is the quadratic function? true or false?
Answers anybody?????
A circle has an arc of length 24pi that is intercepted by a central angle of 80 degrees. what is the radius of the circle?
Johnny is five years older than his brother. the sum of their ages is 37. how old is Johnny
How would paying off the lowest outstanding balance credit card first be beneficial over paying off the one with the highest interest rate?
Paying off the lowest outstanding balance credit card first can be beneficial for motivation and momentum, even if it's not the most mathematically efficient approach.
Explanation:When paying off credit card debt, it can be beneficial to start by paying off the credit card with the lowest outstanding balance. This approach is called the Snowball Method and is popularized by financial expert Dave Ramsey. The Snowball Method focuses on tackling the smallest debt first, regardless of the interest rate, as it provides a psychological boost and motivates individuals to continue paying off their debts.
By paying off the smallest debt first, you can quickly eliminate one of your credit card balances. This can give you a sense of accomplishment and momentum to continue paying off your other debts. It's important to note that while the Snowball Method may not be the most mathematically efficient approach in terms of saving on interest payments, it can be effective in helping individuals stay motivated and committed to paying off their debts.
look at the pictures below
This figure shows circle Z with chords AB and RS . mAR=55° mRB=66° AB=8 m RS=8 m What is mRS?
Answer:
The correct answer is, mRs = 121°
Step-by-step explanation:
From figure we get ,
mAR=55° mRB=66° AB=8 m RS=8 m
To find mRS
Using given information we can write,
mAB = mAR + mRB = 55 + 66 = 121°
Since AB=8 m RS=8 m,
AB = RS
⇒ mAB = mRS
mAB = 121°
Therefore mRS = 121°
The correct answer is given by,
mRS = 121°
The price of 6 pretzels is $5.10. Simon and Sofia bought 8 pretzels an shared the cost equally. How much did each person pay?
Answer:
$3.40
Step-by-step explanation:
To find out the amount each person paid, let's find the total cost of the pretzels first.
6 pretzels ----- $5.10
1 pretzel ----- $5.10 ÷6= $0.85
8 pretzels ----- 8($0.85)= $6.80
The total cost of 8 pretzels is thus $6.80.
Since the total cost was shared equally amongst two people, each person paid half of the total cost.
Amount each person paid
= $6.80 ÷2
= $3.40
By taking a quotient between the total cost and the number of people, we can see that each one pays $3.40
How much did each person pay?We know that the price of 6 pretzels is $5.10, then the price for each pretzel is given by the quotient:
$5.10/6 = $0.85
We know that Simon and Sofia bought 8 pretzels, so they pay:
P = 8*$0.85 = $6.80
And they divide that evenly between the two, so we can take the quotient:
p = $6.80/2 = $3.40
That is the amount that each person pays.
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describe the vertical asymptotes) and holes) for the graph of y=(x+2)(x+4)/(x+4)(x+1)
A. asymtotes: x=-1 and hole: x=4
B. asymtotes: x=2 and hole: x=-1
C. asymtotes: x= -1 and hole: x=-4
D. asymtotes: x=1 and hole: x=4
Can someone plzzzz help me!??
The measure of dispersion which is not measured in the same units as the original data is the
a. variance
b. standard deviation
c. median
d. coefficient determination
Variance is not expressed in the same units as the original data.
Variance is measured by:
Finding the difference between the mean of a data set and the variables in the data set Squaring this difference Summing up the squaresBecause variance is squared, it is no longer the same units as the rest of the data set. For instance, if the data set is in centimeters, the variance will be in centimeters².
In conclusion, the variance of a data set is not measured in the same unit as the data set.
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Final answer:
The measure of dispersion which is not measured in the same units as the original data is the variance.
Explanation:
The measure of dispersion which is not measured in the same units as the original data is the variance. The variance is a squared measure and does not have the same units as the data. On the other hand, the standard deviation is derived from the variance and it measures the spread in the same units as the data.
Then again, the standard deviation, which is the square foundation of the fluctuation, is estimated in similar units as the first information. It gives a proportion of the spread of the information that is all the more effectively interpretable. The middle is a proportion of focal inclination and doesn't give data about the scattering or spread of the information.
The coefficient of assurance, otherwise called R-squared, is a proportion of how well the relapse line fits the information and isn't straightforwardly connected with scattering.
Add the reciprocal of 1 1/5 to the sum of the numbers (+1.25) and (−1 3/4 ).
The answer would be 1/3 after adding the reciprocal of 1 1/5 to the sum of the numbers (+1.25) and (−1 3/4 ).
What is a fraction?Fraction number consists of two parts, one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.
It is given that:
The fraction is 1 1/5 = 6/5
The reciprocal of the above fraction = 5/6
= 1.25 - 1 3/4 + 5/6
= 1.25 - 7/4 + 5/6
= 1.25 - 11/12
= 4/12
= 1/3
Thus, the answer would be 1/3 after adding the reciprocal of 1 1/5 to the sum of the numbers (+1.25) and (−1 3/4 ).
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Which values are solutions to the inequality below?
Check all that apply.
√x ≤ 11
a) 121
b) 120
c) 111
d) -10
e) 122
f) no solutions
The values that are solutions to the inequality √x ≤ 11 are 121, 120, and 111.
Explanation:To find the values that are solutions to the inequality √x ≤ 11, we first need to square both sides of the inequality. This gives us x ≤ 121. So any value of x that is less than or equal to 121 will be a solution.
Now let's check each option:
a) 121: 121 is equal to 121, so it is a solution.b) 120: 120 is less than 121, so it is a solution.c) 111: 111 is less than 121, so it is a solution.d) -10: Negative numbers do not have real square roots, so -10 is not a solution.e) 122: 122 is greater than 121, so it is not a solution.f) no solutions: This option is not a valid solution.In summary, the values that are solutions to the inequality √x ≤ 11 are: 121, 120, and 111.
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Explain why the function f(x)=x^2 is not invertible
Calculus HELP PLEASE? A highway runs due north. A car traveling at 50 mph exits the highway at a 30° angle. After driving another 1.5 hours at the same speed and direction, how far away is the car from the highway?
Answer:
37.5 miles is the answer.
Step-by-step explanation:
Since car is travelling with a speed = 50 mph
Angle between the highway and the track on which the car is travelling = 30°
Now we know sin∅ = distance between highway and the car/Distance traveled by the car on the new track
Distance traveled by the car in 1.5 hours = speed × time
= 50 × 1.5 = 75 mi.
Therefore sin 30° = x/75
x = 75×sin30° = 75 × 1/2 = 37.5 mi.
Therefore the answer is 37.5 mi away.
Complete the statement If x/7 = 5/3, then x+7/7 =?
The volume, V, of a soap bubble is related to its surface area, A. In cubic centimeters, this relationship is shown by the formula V = 0.094A 3/2. What is the surface area of a bubble with a volume of 5 cm3 ?
The surface area of a soap bubble with a volume of 5 cm³, given the relationship V = 0.094 [tex]A^{3/2[/tex], can be calculated by rearranging and solving the formula, resulting in an approximate surface area of 36.3 cm².
To find the surface area of a soap bubble with a volume of 5 cm³, using the given relationship V = 0.094 [tex]A^{3/2[/tex], we need to rearrange the formula to solve for A (the surface area). We will take the following steps to find A:
First, divide both sides of the equation by 0.094 to isolate [tex]A^{3/2[/tex] on one side:
[tex]A^{3/2[/tex] = V / 0.094
Next, substitute V = 5 cm³ into the equation:
[tex]A^{3/2[/tex] = 5 / 0.094
Calculate the right-hand side:
[tex]A^{3/2[/tex] ≈ 53.19
Finally, raise both sides of the equation to the power of 2/3 to solve for A:
A = [tex](53.19)^{2/3[/tex]
Calculating the right-hand side gives us the surface area A. After performing these calculations, we find that the surface area of a bubble with a volume of 5 cm³ is approximately 36.3 cm².
Use the quadratic formula to determine the exact solutions to the equation.
x^2−2x−4=0
Enter your answers in the boxes.
x=__ or x=___
All side lengths of ΔJKL equal 2 units. A transformation maps ΔJKL to ΔJ'K'L'. The length of side J'K' is 5 units. Is this a rigid transformation?
No, there is no one-to-one mapping of all the points of the pre-image to the image.
No, at least one segment length is not preserved, making this a nonrigid transformation.
Yes, the vertices of the pre-image map to the vertices of the image.
Yes, all of the side lengths of the pre-image are proportionate to the image.
No, at least one segment length is not preserved, making this a nonrigid transformation. Option B is correct.
What is geometric transformation?It is defined as the change in coordinates and the shape of the geometrical body. It is also referred to as a two-dimensional transformation. In the geometric transformation, changes in the geometry can be possible by rotation, translation, reflection, and glide translation.
It is given that, all side lengths of ΔJKL equal 2 units. A transformation maps ΔJKL to ΔJ'K'L'. The length of side J'K' is 5 units.
A transformation that leaves a geometric figure's size or shapes unchanged is known as a rigid transformation. It is a unique type of metamorphosis in which a figure's size or shape is left unchanged.
From the definition, it is obtained that there is no rigid transformation because at least one segment length is not preserved.
Thus, no at least one segment length is not preserved, making this a nonrigid transformation. Option B is correct.
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what is the approximate distance between the points (-5 1) and (-2 3) on a coordinate grid
What is the domain of the relation graphed below? GET 50 POINTS!
Answer:
Domain is the set of x values
so domain is {-5 , -4 , -3 , 1 , 2 , 5}