how many ordered pairs ( m , n ) of positive integers, such that m > n , have the property that their squares differ by 96 ?
Answer:
4 pairs
Step-by-step explanation:
You want the number of pairs of integers (m, n) such that m > n and m² -n² = 96.
Difference of squaresThe difference of squares is factored as ...
m² -n² = (m -n)(m +n)
In order for the difference to be 96, these factors must be factors of 96.
Factors of 96The prime factorization of 96 is ...
96 = 2^5 · 3
The number of divisors of 96 is the product of the exponents of these factors, each increased by 1:
(5+1)(1+1) = 12
That is, there are 12/2 = 6 factor pairs ab such that a<b. They are ...
96 = 1·96 = 2·48 = 3·32 = 4·24 = 6·16 = 8·12
SolutionThe values of m and n relate to the values of 'a' and 'b' by ...
m = (a+b)/2
n = (b-a)/2
In order for m and n to be integers, the values of 'a' and 'b' must both have the same parity. That is, they must both be even, or both be odd. This eliminates the factor pairs 1·96 and 3·32, leaving 4 (a, b) pairs, hence 4 (m, n) pairs. Possible ordered pairs are ...
(m, n) ∈ {(25, 23), (14, 10), (11, 5), (10, 2)} . . . . . 4 pairs
a tree t with 50 end-vertices has an equal number of vertices of degree 2, 3, 4 and 5 and no vertices of degree greater than 5. what is the order of t?
The order of T is 82.
Let n and e be the order and number of edges of T respectively.
Also, let a, b, c and d be the number of vertices of degree 2, 3, 4 and 5 respectively.
Then, 50+a+b+c+d = n
And by the Handshake Lemma, the sum of the degrees of all the vertices of T is twice the number of edges, so we have
50+2a+3b+4c+5d = 2e
Since a=b=c=d then we have the following two equations,
50+4a = n
50+14a = 2e
But, T is a tree so e=n=1, then the two equations become,
50+4a = n ...............(1)
50+14a = 2(n-1) .............(2)
Solving for a in (1) we get,
a = [tex]\frac{n-50}{4}[/tex]
Plugging a into (2) we get,
50+14([tex]\frac{n-50}{4}[/tex]) = 2(n-1)
50+3.5n-175 = 2n-2
3.5n-2n = 175-50-2
1.5n = 123
n = [tex]\frac{123}{1.5}[/tex] = 82
Therefore, the order of T is 82.
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Simplify and Explain This Equation
-8-3(9-2x)+9x
The force of attraction, F of a body varies directly as its mass (m) and inversely as the square of the distance (d) from the body. When m=8kg and d=5meters, F=100 Newtons. Find F when m=2 kilograms and d=15 meters.
The force of attraction is 2.78 Newton when m = 2 kilograms and d = 15 meters.
According to the question,
We have the following information:
The force of attraction, F of a body varies directly as its mass (m) and inversely as the square of the distance (d) from the body. And when m=8kg and d=5meters then F=100 Newtons.
Now, let's take the constant for this case to be k.
We have the following expression:
F = km/[tex]d^{2}[/tex]
Putting the values in this expression to find the value of k:
100 = 8k/(5*5)
Using the cross multiplication method:
2500 = 8k
k = 2500/8
k = 312.5
Now, we have the following value of F:
F = 312.5*2/15*15
F = 625/225
F = 2.78 Newton
Hence, the force of attraction is 2.78 Newton when m = 2 kilograms and d = 15 meters.
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Two sides of a triangle are 30in and 30in, what are the possible lengths of the third side
Answer:
[tex]0 < x < 60[/tex]
Step-by-step explanation:
By the Triangle Inequality Theorem:
30 + 30 > x, so 60 > x ----> x < 60
30 + x > 30, so x > 0
x + 30 > 30, so x > 0
Putting these inequalities together, we have 0 < x < 60. So the length of the third side can be greater than 0 inches but less than 60 inches.
Which is the graph of f(x) = x² + 4 and the function g, a reflection of f(x) across the x-axis?
The reflection of function f(x) across the x-axis is [tex]-x^{2} -4[/tex].
It is given to us that the function is -
[tex]f(x)=x^{2} +4[/tex] --- (1)
and a function g
We have to find out the reflection of f(x) across the x-axis.
A function can be defined as an expression that assigns the value of each element of a given set to the another specific element of another given set.
To find out the reflection of f(x) across the x-axis, we have to take -f(x) instead of f(x).
Now,
[tex]-f(x)=-(x^{2} +4)\\= > -f(x)=-x^{2} -4[/tex]
Thus, the reflection of function f(x) across the x-axis is [tex]-x^{2} -4[/tex].
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The amounts below show the change (measured in feet) in water
level over four months. What is the average monthly change in
water level?
-2.54, -3.16, -2.82, 1.6
Answer: -1.73 ft
Step-by-step explanation:
(-2.54 - 3.16 - 2.82 + 1.6) / 4 = -1.73
imagine that you are a researcher conducting a survey for a client. you are trying to decide what sample size you need. what would the main advantage be of a larger sample size?
Large sample size gives more accurate information about population.
The main advantages of Large sample size is :
1. Large sample size provide stronger and more reliable information about parent population.
2. It will help researcher to capture outliers and sweeping out in sample.
3. Measuring large sample size allows one to divide the sample into segments .
4. Large sample size helps in obtaining a quality and precise mean .
Some disadvantages of Large sample size is :
1. Large sample size will be time consuming and more expensive to work with
2. It require higher financial involvement to complete the survey.
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What is the sale price of the oil change after the discount
The sale price of the oil change after the discount can be found to be $25.50.
How to find the sale price?When given a discount, the selling price of an item is found by the formula:
= Price of item before discount - Discount
The discount on the oil change, given the price of the oil change and the coupon is:
= Price of oil change x Coupon rate
= 30 x 15%
= $4.50
The sale price after the oil change is:
= Price of oil change before discount - discount
= 30 - 4.50
= $25. 50
The first part of the question is:
An oil change at city auto is regularly $30. Mr Allen has a coupon for 15% off.
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Determine the distance between each set of ordered pairs
(6 7/8,2) and (-3,2)
Answer:
Determine the distance between each set of ordered pairs
(6 7/8,2) and (-3,2)
Step-by-step explanation:
Well if you do delta y over delta x you do -2 - 2 over 3 - 67/8 which will be -4 over -5.375 which will equal 32 over 43 which is you M you b is one point (your choice) then y = mx + b so 2 = 32/43(-3) + b 2 = -96/43 + b then add -96/43 to both sides and you'll get -10/43 = b so y = 32/43x + -10/43
equation - y = 32/43x + -10/43
Mo puts $290000 into savings bonds that pay a simple interest rate of 5.2%. How much money will the bonds be worth at the end of 6 years
Answer:
in six years, Mo will have roughly $393090.4 in savings bonds.
Step-by-step explanation:
(This might not be the most efficient way to do it.)
year one - 290000 x 1.052 = 305080
year two - 305080 x 1.052 = 320944.16
year three - 320944.16 x 1.052 = 337633.256
year four - 337633.256 x 1.052 = 355190.185
year five - 355190.185 x 1.052 = 373660.075
year six - 373660.075 x 1.052 = 393090.399
in six years, Mo will have roughly $393090.4 in savings bonds.
a pet boarder keeps a dog-to-cat ratio of 5:2.if the boarder has room for 98 animals,then how many of them can be dogs
Answer:
70 dogs
Step-by-step explanation:
total animal is 98
total ratio is 5+2 = 7
calculate how much each ratio has
98/7 = 14
5 ratio are dogs, so multiply 14 with 5 to get 70 dogs
you can also do:
5/7 × 98 = 70
URGENT !!! Answer the following questions using the graph.
Describe the y-intercept in this scenario.
Describe the x-intercept in this scenario.
Finish the ordered pairs (2.5, ____ ) and (____,300).
What are the domain and range?
Domain: {h I ___ < h < ____}
Range: {f(h) I ___ < f(h) < ____}
The x-intercept is (8.75,0) and
The y-intercept is (0,700)
The ordered pairs are (2.5,500) and (5,300).
Given that,
In the picture we have a graph with line.
We have to find the x-intercept and y-intercept of the scenario and the ordered pairs (2.5, ___) and (___ , 300)
We know that,
What is x-intercept and y-intercept?The x intercept is the location where the line touches the x axis. The y intercept is the location where the line touches the y axis.
So,
The x-intercept is (8.75,0) and
The y-intercept is (0,700)
The ordered pairs are (2.5,500) and (5,300).
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Find the missing side. Help please
Answer:
Step-by-step explanation:
add 23+14=37-90= 53
answer=53
john recently bought a used car for $\$5000$ for his pizza delivery job. he gets $\$10$ for each pizza he delivers, but he has to spend $\$3$ on gas for each pizza he delivers. what is the minimum whole number of pizzas john must deliver in order to earn back the money he spent on the car he bought?
The minimum number of pizzas that john must deliver to earn back the money he spent on the car he bought is 715 pizzas .
In the question,
it is given that ,
the amount for which John bought the used car for pizza delivery = $5000
the amount John gets for 1 pizza he delivers = $10
the amount John spend on gas for 1 pizza he delivers = $3
So , the amount that John earned for 1 pizza delivered is = 10 - 3 = $7 .
the number of pizza required to sell = (price of car)/(amount earned from 1 pizza)
Substituting the values , we get
number of pizza required to sell = 5000/7 = 714.2857
rounding the answer to whole number ,
we get
= 715 pizzas
Therefore , The minimum number of pizzas that John must deliver to earn back the money he spent on the car he bought is 715 pizzas .
The given question is incomplete , the complete question is
John recently bought a used car for $5000 for his pizza delivery job. he gets $10 for each pizza he delivers, but he has to spend $3 on gas for each pizza he delivers. what is the minimum whole number of pizzas john must deliver in order to earn back the money he spent on the car he bought ?
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. Examine this graph. What are the y-intercept and slope of this line?
*
10 points
Captionless Image
Hey, so because you didn't include the image this is impossible to get the number values. But, essentially you'd solve it something like this.
Remember, the Y axis goes up, and its located when you are at X = 0.
The y intercept is the value of Y, when is 0.
The slope of a line is referred to as "rise/run"
Basically, you take any given spot. For example, from x = 0 to x = 2. The run is 2, because thats how much x value the line covers. So, in that "run" how much do you go up or go down? Lets say in those 2 x's that you travelled, how much did the height change? If you changed by going up 4 units, it would be (rise/run) 4/2 = 2. The slope would be 2. That means that for every time x increases by 1, Y goes up 2.
the product of two consecutive nonnegative integers is 240 find the integers and separate them with a comma
The product of two consecutive non-negative integers is 240 and the integers are 15,16.
What are consecutive integers?
The numbers that come after one another are known as consecutive integers. They proceed sequentially or alphabetically. For instance, a group of natural numbers is a sequence of integers. In mathematics, the term "consecutive" refers to an uninterrupted succession or a continuous flow of numbers, so that successive integers follow a sequence in which each succeeding number is one more than the one before it. The mean and median in an array of successive integers (or in numbers) are both identical. In the event that x is an integer, then x + 1 and x + 2 are also integers.Let the numbers be x, x+1
Product is x(x+1)=240
x² + x = 240
x² + x − 240 = 0
x² + 16x − 15x − 240 = 0
x(x+16) − 15(x+16) = 0
(x+16)(x−15)=0
x = 15,−16
So, the product of two consecutive non-negative integers is 240,then numbers are 15, 16.
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What is the explicit formula for this sequence?
6, 2, -2, -6, ...
A. an = -6+ (n − 1)(-4)
B. a = -4+ (n-1)6
C. an= 6+ (n-1)(-4)
D. an = 6+ (n-1)4
SUBV
Explicit formula for this sequence is (C) an= 6+ (n-1)(-4)
Given sequence,
6, 2, -2, -6.....
First term = a = 6
Difference = d = 2 - 6
= -4
No. of terms = n
Therefore, an = a + (n-1)d
= 6 + (n-1)(-4)
Hence, an= 6+ (n-1)(-4).
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Jake completes two trails and hikes less than 10 km. Which two trails did Jake hike? Explain.
Step-by-step explanation:
Great Meadow Loop and Jordan Pond Path
If the two triangles below are similar and the area of the larger triangle is 36, what is the area of the smaller triangle?
The area of the smaller triangle is 18 squared units which is determined by the property of similar triangles.
The area of the larger triangle is 36.
The two triangles are similar which is given in the question
We know that the corresponding sides of similar triangles are being in the same ratio.
As per the property of similar triangles,
A₁ : S₁ = A₂ : S₂
Here A₁ is area of the larger triangle, A₂ is area of the smaller triangle
S₁ is side of the larger triangle, and S₂ is side of the smaller triangle,
⇒ A₁ / S₁ = A₂ / S₂
⇒ 36 / 12 = A₂ / 6
⇒ 3 = A₂ / 6
⇒ A₂ = 18
Therefore, the area of the smaller triangle is 18 squared units.
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How do I graph y=3x+2
what is the goal of a hypothesis test? select one: a. to give a strong statement about a statistic b. to determine if the data can refute an assumption about a parameter or population c. to create a larger sample d. to provide a range of values that are reasonable for the parameter of interest
The goal of a hypothesis test is to determine if the data can refuse an assumption a parameter or a population.
Statistical hypothesis test:
A statistical hypothesis test is a technique for determining whether the available data are sufficient to support a specific hypothesis. We can make probabilistic claims about population parameters through hypothesis testing.
The purpose of a hypothesis test is to ascertain whether the data can refute a parameter or population assumption.
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92.3- (3.2 ÷ 0.4) × 2³
Answer:
28.3
Step-by-step explanation:
To solve this problem, we must follow PEMDAS, which must be used in solving all mathematical equations and expressions.
PEMDAS is the order you solve the equation/expression in. It is an acronym for:
P: Parenthesis
E: Exponents
M: Multiplication
D: Division
A: Addition
S: Subtraction
Keep in mind that multiplication and division are too be done left and right. You do not do multiplication first and division last, read left to right. Same goes with addition and subtraction.
This is the less confusing notation of PEMDAS.
P: Parenthesis
E: Exponents
M: Multiplication D: Division
A: Addition S: Subtraction
Now, lets solve the problem.
92.3 - (3.2 ÷ 0.4) × 2³
What do we do first? Well since parenthesis is first according to PEMDAS, we must solve the division in side of the parenthesis first.
3.2/0.4 = 8
Now we have:
92.3 - (8) × 2³
Remove the parenthesis as there is only one term inside and therefore not needed.
92.3 - 8 × 2³
What do we do next?
We have:
Subtraction: 92.3 - 8
Multiplication: -8 × 2³
Exponent: 2³
Because PEMDAS states exponents is to be solved in front of multiplication/division and addition/subtraction, we must do that.
What is an exponent?
2³
2 is the base, 3 is the exponent.
2³
Means multiplying 2 by itself 3 times:
2 × 2 × 2
8
Now we have:
92.3 - 8 × 8
Because PEMDAS states solving multiplication/division is prioritized over addition/ subtraction, we must solve 8 × 8
8 × 8 = 64
Now we have: 92.3 - 64
Solve.
28.3
Which choice is the value of two over five raised to the second?
four over twenty-five
two over twenty-five
two over ten
four over seven
in a city of 246,000 people, there are 34 coffee shops. what is the number of coffee shops per capita? round to 6 decimal places.
Coffee shops per capita in city is 7235.294118
What is per capita?
The Latin phrase "per capita" means "by head." In statistics, the term "per capita" is sometimes used in place of "per person" to denote the average per person.
Main Body:
total population of city = 246000
Total coffee shops = 34
Per capita shops for population can be calculated by dividing total population by total shops.
calculatiing,
Per capita shops = total population/ total shops
= 246000/34
= 7235.294118
Hence number of coffee shops per capita is 7235.294118
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Two sides of a parallelogram are 190 feet and 740 feet. The measure of the angle
between these sides is 88°. Find the area of the parallelogram to the nearest square
foot.
Answer: About 140,514 ft²
Step-by-step explanation:
The area of a parallelogram is base × height. The base is already determined as 740 ft. You can find the height(h) with trigonometry:
[tex]sin(88)=\frac{h}{190}\\ \\h=190sin(88)=189.88425713[/tex]
Now, multiply the values together:
[tex]740*189.88425713=140514.350[/tex]
Given ∠11≅∠13. Which lines, if any, must be parallel based on the given information? Justify your conclusion.
Based on the information given: A. c║d; converse of alternate exterior angles theorem.
What is the Converse of Alternate Exterior Angles Theorem?Exterior alternate angles are congruent based on the alternate exterior angles theorem. Conversely, it can also be stated that if two angles, which are exterior alternate angles, are congruent to each other, therefore the lines they lie on that is crossed by a transversal are proven to be parallel to each other based on the converse of alternate exterior angles theorem.
Given that angles 11 and 13 are congruent angles, the diagram also shows that they are alternate exterior angles. They lie on each lines, c and d, that are crossed by transversal a.
Therefore, based on the converse of alternate exterior angles theorem, lines c and d are parallel.
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A piont is two dimensional ?
Answer:
False.
Step-by-step explanation:
A point is a zero-dimensional object as it has no length, width or height. It has no size.
And it would be great if you can spell "point" correctly.
Write the equation of the function in
slope-intercept form:
A line with a slope of 2 and passes through (-8, 3)
The equation of the function in slope-intercept form with slope of 2 is y = 2x+19.
According to the question,
We have the following information:
Slope = 2 (Note that slope of a line is denoted by m.)
Points through which function is passing = (-8,3)
So, we have x' = -8 and y' = 3.
We know that the following equation is used to find the equation of the function or line:
(y-y') = m(x-x')
y-3 = 2{x-(-8)}
y-3 = 2(x+8)
y-3 = 2x+16
y = 2x+16+3
y = 2x+19
Hence, the equation of the function in slope-intercept form with slope of 2 is y = 2x+19.
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Which is the better buy?
37-ounce bottle of barbeque sauce for
$35.15
2-pound bottle of barbeque sauce for
$23.36
Answer:
2 pound bottle
Step-by-step explanation:
convert ounce to pounds:
37/16=2.3125
find price of 1 pound of bbq sauce:
35.15/2.3125=$15.2
23.36/2=$11.68
$11.68 is more cheaper so 2 pound bottle is better to buy