We must conclude that there are infinitely many primes of the form 3k + 2, where k is a non-negative integer.
To adapt the proof:
We can use a similar contradiction argument.
Suppose that there are only finitely many primes of the form 3k + 2, say p1, p2, ..., pn.
Let N = 3p1p2...pn + 2. Note that N is of the form 3k + 2 for some non-negative integer k.
Now, let's consider the prime factorization of N. Either N is prime and of the form 3k + 2, in which case we have found a new prime of the desired form, contradicting our assumption that there are only finitely many such primes. Or, N is composite and has a prime factorization consisting only of primes of the form 3k + 1 (since any prime of the form 3k + 2 would divide N). But this implies that N itself is of the form 3k + 1.
Now, let's consider the number M = 3N + 2. M is also of the form 3k + 2, and so must have a prime factorization consisting only of primes of the form 3k + 1. But since N is of the form 3k + 1, we have
M = 3(3p1p2...pn + 1) + 2 = 9p1p2...pn + 5. This means that M has a prime factorization consisting of primes of the form 3k + 2, which contradicts our assumption that there are only finitely many such primes.
Therefore, we must conclude that there are infinitely many primes of the form 3k + 2, where k is a non-negative integer.
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Suppose we have converted the augmented matrix of a system of equations into reduced row-echelon form. How do we then identify the dependent and independent (free) variables?
We can easily identify the dependent and independent (free) variables by looking at the matrix after converting augmented matrix.
When we convert the augmented matrix of a system of equations into reduced row-echelon form, we can easily identify the dependent and independent (free) variables by looking at the matrix. The columns that contain leading 1's correspond to the variables that are dependent on the others. The columns that do not contain leading 1's correspond to the variables that are independent or free.
For example, if we have a matrix in reduced row-echelon form that looks like this:
1 0 2 | 3
0 1 -1 | 2
0 0 0 | 0
We can see that the first and second columns have leading 1's, so the variables corresponding to those columns (in this case, x1 and x2) are dependent on each other. The third column does not have a leading 1, so the variable corresponding to that column (in this case, x3) is independent or free.
In general, we can say that the number of independent variables is equal to the number of columns without leading 1's, and the number of dependent variables is equal to the number of columns with leading 1's. By identifying the dependent and independent variables, we can then use this information to solve the system of equations using substitution or elimination.
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Are these correct or not?
There are 19 ounces
There are 107 visitors per day
He needs 63 boxes
What is the meaning of division in mathematics?1) We have to convert the money to cents thus we have;
$10.45 = 1045 cents
Then Number of ounces = 1045/ 55
= 19 ounces
2) The total number of days the museum was open is 365 - 4 = 361 days
Thus we have that the number of visitors in a day is; 38627/361
= 107 visitors per day
3) If 1 box holds 45 books
x boxes will hold 2835 books
x = 2835/45
x = 63 boxes as shown
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In a population of N = 10 scores obtained for a discrete variable for which is not a possible score, the smallest score is X = 8 and the largest is X = 20. What is the range for this population? 8 10 12 20
The range for this population is 12, which is calculated by subtracting the smallest score from the largest score (20-8 = 12). Since the variable is discrete, only whole numbers are possible scores, and the score that is not possible is not relevant to the range calculation.
To find the range for this population, you'll need to use the largest and smallest scores.
Given:
- Population (N) = 10
- Discrete variable
- Smallest score (X) = 8
- Largest score (X) = 20
To find the range, simply subtract the smallest score from the largest score:
Range = Largest score - Smallest score
Range = 20 - 8
Range = 12
So, the range for this population is 12.
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what is the surface area of 2in 5.2in 4in
The surface area of a rectangular prism of dimensions 2 in, 5.2 in and 4 in is given as follows:
S = 78.4 in².
What is the surface area of a rectangular prism?The surface area of a rectangular prism of height h, width w and length l is given by:
S = 2(hw + lw + lh).
This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.
The dimensions for this problem are given as follows:
2 in, 5.2 in and 4 in.
Hence the surface area of the prism is given as follows:
S = 2 x (2 x 5.2 + 2 x 4 + 5.2 x 4)
S = 78.4 in².
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Unit 7 right triangles and trigonometry homework 2 special right triangles
The values of x, y and z from the triangle are z = 12√2, x = 6 and y = 12
What is trigonometrical ratio of angles?Trigonometrical ratio of angles is used to find the sine, cosine, and tangent of angles in right triangles
The ratios are denoted by SOHCAHTOA
Tan A = Oppo/Adj
Tan 60 = 6√3/x
√3/ 1 = 6√3/x
Cross and multiply to have
x√3/3 = 6√3/√3
x = 6
To find y
First from pyth. rule
6² + (6√3)² = m²
36 + 36*3 = m²
144 = m²
m =√144 = 12
Using tan 45 = 1/1
Tan 45 = y/12
1/1 = y /12
y = 12
To find x w use sine
Sin 45 = 1/√2
1/√2 = 12/x
x = 12√2
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I need help please I will give my 10 points I have left.
Answer:
divide and 16 per pound...
Answer:
divide, 16
Step-by-step explanation:
To find the weight, we have to divide the total number of ounces, 96, by the unit rate, which is the baseline for the conversion, 16 ounces.
Hope this helps! :)
match the list on the left with the possible number of times the binary search algorithm splits the list when searching for a term in the list.
The possible number of times the binary search algorithm splits the list when searching for a term in the list varies depending on the length of the list and the position of the term within the list.
However, in general, the number of times the list is split during the binary search algorithm is proportional to the logarithm of the length of the list. So, for example, if the list has 8 items, the binary search algorithm may split the list up to 3 times (log base 2 of 8 is 3), while if the list has 1024 items, the binary search algorithm may split the list up to 10 times (log base 2 of 1024 is 10).
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A charity organization is having a fundraiser. P represents the fundraiser's profit (in dollars) if n tickets are sold. A negative profit means the expenses exceeded the income from tickets. P=70n-1500. What are the expenses of the fundraiser?
In the year 2000, the average car had a fuel economy of 21.17 MPG. You are curious as to whether this average is greater than today. The hypotheses for this scenario are as follows: Null Hypothesis: μ ≤ 21.17, Alternative Hypothesis: μ > 21.17. You perform a one sample mean hypothesis test on a random sample of data and observe a p-value of 0.0427. What is the appropriate conclusion? Conclude at the 5% level of significance.
This suggests that there is enough evidence to support the alternative hypothesis, indicating that the average fuel economy today is greater than 21.17 MPG.
Based on the given information, the null hypothesis (H₀) states that the average fuel economy is less than or equal to 21.17 MPG (μ ≤ 21.17), while the alternative hypothesis (H₁) states that the average fuel economy is greater than 21.17 MPG (μ > 21.17). You conducted a one-sample mean hypothesis test and observed a p-value of 0.0427.
To conclude at the 5% level of significance (α = 0.05), you compare the p-value to the significance level. Since the p-value (0.0427) is less than the significance level (0.05), you reject the null hypothesis.
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Let X1, X2, ... , X30 be a random sample of size 30 from a Poisson distribution with a mean of 2/3. Approximate (a) P 15 < 30 i=1 Xi 22 . (b) P 21 30 i=1 Xi < 27 .
(a) To approximate P(15 < ΣXi < 22), we can use the Central Limit Theorem (CLT) since we have a large enough sample size (n = 30) and the mean and variance of the Poisson distribution are both finite.
First, we need to find the mean and variance of the sample mean, which is also the mean and variance of the Poisson distribution:
μ = λ = 2/3
σ^2 = λ = 2/3
Next, we can standardize the random variable Z = (ΣXi - nμ) / sqrt(nσ^2) to have a standard normal distribution:
Z = (ΣXi - 30(2/3)) / sqrt(30(2/3)) = (ΣXi - 20) / sqrt(20)
Then, we can use a standard normal table or calculator to find the probability:
P(15 < ΣXi < 22) ≈ P(-2.74 < Z < -1.77) ≈ 0.038
Therefore, the approximate probability is 0.038.
(b) To approximate P(21 < ΣXi < 27), we can use the same method with the CLT.
First, we need to find the mean and variance of the sample mean:
μ = λ = 2/3
σ^2 = λ = 2/3
Next, we can standardize the random variable Z = (ΣXi - nμ) / sqrt(nσ^2) to have a standard normal distribution:
Z = (ΣXi - 30(2/3)) / sqrt(30(2/3)) = (ΣXi - 20) / sqrt(20)
Then, we can use a standard normal table or calculator to find the probability:
P(21 < ΣXi < 27) ≈ P(-0.68 < Z < 0.68) ≈ 0.495
Therefore, the approximate probability is 0.495.
Graph the absolute value function f(x) = |x – 2| on the coordinate plane.
The graph of the absolute value function f(x) = |x - 2| is given by the image presented at the end of the answer.
What is a translation?A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.
The four translation rules for functions are defined as follows:
Translation left a units: f(x + a).Translation right a units: f(x - a).Translation up a units: f(x) + a.Translation down a units: f(x) - a.The parent absolute value function is given as follows:
f(x) = |x|.
Which has the format of the V-graph, with vertex at the origin.
f(x) = |x - 2| is a translation right two units of f(x) = |x|, hence the vertex will be at the point (2,0).
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A Supermarket Offers a discount OF 5 Cent per EuroE How much will a Custo- mer pay for an article which is priced at €8500?
5. What is the Boolean duality principle?
The Boolean duality principle is a fundamental concept in Boolean algebra which states that any statement or expression in the algebra can be transformed into an equivalent form by interchanging the roles of AND and OR operators, as well as 0's and 1's.
The Boolean duality principle, also known as De Morgan's laws, is a fundamental concept in Boolean algebra that states that every Boolean expression remains valid if you perform the following steps:
1. Swap AND (⋅) operators with OR (+) operators and vice versa.
2. Replace all 1's with 0's and all 0's with 1's.
3. Complement (invert) all variables.
In other words, the Boolean duality principle allows us to derive a dual expression from an original Boolean expression by applying these transformations. This principle is useful in simplifying Boolean expressions and designing digital circuits.
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In the expression 30+40+70, Jillian added 30 and 40 and then 70, while Samuel added 30 and 70 and then 40. Who is correct? Explain your reasoning
As per the mathematical operation both Jillian and Samuel are correct.
Given expression = 30+40+70,
The methodology by Jillian = added 30 and 40 and then 70
The methodology by Samuel = added 30 and 70 and then 40.
Determining the result of the given equation:
30 + 40 + 70
= 140
Reviewing the operations of both Jillian and Samuel
Jillian
He added 30 and 40 and then 70:
30 + 40 = 70
70 + 70 = 140
Samuel
He added 30 and 70 and then 40
30 + 70 = 100
100 + 40 = 140
The result of both mathematical operations is 140. Thus, it can be stated that both are correct.
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Kenny is making creamy rice pudding his rice requires 4 cups of milk he has a quart of milk in his fridge does he have enough milk explain
Answer:
Yes
Step-by-step explanation:
Kenny will have enough milk to make his dessert because 1 quart contains 4 cups. Since he has 1 quart he has the equivalent to 4 cups.
*(Cauchy- Euler Equation): Find the general solution of the equation for the differential equation x^2y" + 4xy' -4y = 0 is (Select the correct answer) a. y = c_1x^-1 + c_2x^-1 ln x b. y = c_1x^-1 + c_2x^-2 c. y = c_1x^2 + c_2x d. y = c_1x + c_2x^-4 e. y = c_1x + c_2x^2
Given the following 2nd-order differential equation (Cauchy-Euler equation), [tex]x^2y''+4xy'-4y=0[/tex], find the general solution.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A differential equation in the form [tex]ax^2y''+bxy'+cy=0[/tex] , which is a Cauchy-Euler equation, can be solved in the following manner.
Where the characteristic equation => [tex]am^2+(b-a)m+c=0[/tex]
After solving for "m" the possible solutions are...
[tex]\bold{Real, \ Distinct \ Roots} \Rightarrow y=c_1x^{m_1}+c_2x^{m_2}\\\bold{ Repea ted \ Roots} \Rightarrow y=c_1x^{m}+c_2x^{m}ln(x)\\\bold{Complex \ Roots} \Rightarrow y=c_1x^{\alpha}cos(ln(x^\beta ))+c_2x^{\alpha}sin(ln(x^\beta )); \ m= \alpha \pm \beta i[/tex]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[tex]\Longrightarrow x^2y''+4xy'-4y=0 \ where \ a=1, \ b=4, \ and \ c=-4[/tex]
The characteristic equation:
[tex]\Longrightarrow (1)m^2+(4-1)m+(-4)=0 \Longrightarrow \boxed{m^2+3m-4=0}[/tex]
Solve for "m."
[tex]\Longrightarrow m^2+3m-4=0 \Longrightarrow (m-1)(m+4)=0 \Longrightarrow \boxed{m=1,-4}[/tex]
Note that m's are distinct and real.
[tex]Thus, \ \boxed{y=c_1x+c_2x^{-4}} \therefore Sol.[/tex]
Where the arbitrary constants c_1 and c_2 can be solved for given an initial condition.
The correct answer is option (e) [tex]$y = c_1 x + c_2 x^2$[/tex].
We can use the Cauchy-Euler equation to find the general solution of the given differential equation. The Cauchy-Euler equation is given by:
[tex]$a_n x^n y^{(n)}+a_{n-1} x^{n-1} y^{(n-1)}+\cdots+a_1 x y^{\prime}+a_0 y=0$[/tex]
where [tex]$a_n, a_{n-1}, \ldots, a_1, a_0$[/tex] are constants and [tex]$y^{(n)}, y^{(n-1)}, \ldots, y'$[/tex] denote the [tex]$n$[/tex]th, [tex]$(n-1)$[/tex]th, [tex]$\ldots$[/tex], first derivatives of y with respect to x.
In the given differential equation, we have [tex]$a_2 = 1$[/tex], [tex]$a_1 = 4$[/tex], and [tex]$a_0 = -4$[/tex]. Therefore, the Cauchy-Euler equation becomes:
[tex]$x^2 y^{\prime \prime}+4 x y^{\prime}-4 y=0$[/tex]
We assume a solution of the form [tex]$y=x^r$[/tex]. Substituting this into the differential equation, we get:
[tex]$$x^2 r(r-1) x^{r-2}+4 x r^1 x^{r-1}-4 x^r=0$$[/tex]
Simplifying, we get:
[tex]$$\begin{aligned}& r(r-1)+4 r-4=0 \\& r^2+3 r-4=0 \\& (r+4)(r-1)=0\end{aligned}$$[/tex]
So, we have two roots[tex]$r_1 = -4$[/tex] and [tex]$r_2 = 1$[/tex]. Therefore, the general solution of the differential equation is:
[tex]$y=c_1 x^{-4}+c_2 x^1$[/tex]
Thus, the correct answer is option (e) [tex]$y = c_1 x + c_2 x^2$[/tex].
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I need help asap what’s the answer
Option B. Theresa's work is correct its shows that Matrix multiplication is not commutative.
Is matrix multiplication commutative?
Clearly, it is true that AB and BA are distinct matrices, so matrix multiplication is not generally commutable.
Nevertheless, there are special conditions where the product of matrices could be commutable, such as when one of them is an id matrix or diagonal matrix with equivalent diagonal entries.
Hence we can say that Theresa's work is correct its shows that Matrix multiplication is not commutative. since the results are different
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Find the sector area
The area of the sector as shown in the diagram is 13.1 cm².
What is sector?A sector is said to be a part of a circle made of the arc of the circle along with its two radii.
To calculate the area of the sector, we use the formula below
Formula:
A = ∅πr²/360°............................ Equation 1Where:
A = Area of the sector∅ = Angle formed by the sector at the center of the circler = Radius of the circleFrom the question,
Given:
r = 5 cm∅ = π/3Note that 2π = 360°π = 3.14Substitute these values into equation 1
A = (π/3)×3.14×5²/2πA = 13.1 cm²Hence, the area of the sector is 13.1 cm².
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A student mows lawns on the weekends. It takes him 140 minutes to mow 4 lawns. What prediction can you make about the time he will spend this weekend if he has 12 lawns to mow?
It will take him 12 hours to mow 12 lawns.
It will take him 10 hours to mow 12 lawns.
It will take him 7 hours to mow 12 lawns.
It will take him 3 hours to mow 12 lawns.
Answer:
C. It will take him 7 hours to mow 12 lawns
Step-by-step explanation:
140 mins = 4 lawns
12 lawns = 4 x 3
140 x 3 = 420 minutes (12 lawns)
420 ÷ 60 = 7 hours
solve for y=
8y-9=14
y= ?
Step-by-step explanation:
8y-9=14
bring the 9 to the other side because of which the operator has to change to a plus sign.
8y=14+9=23
23÷8=
[tex] \frac{23}{8} [/tex]
or
[tex]2.875[/tex]
karma earns $36 in 3 hours. at this rate, how many dollars will he earn in 20 hours
Answer:
$240
Step-by-step explanation:
We Know
Karma earns $36 in 3 hours.
$36 in 3 hours = $12 in 1 hour
At this rate, how many dollars will he earn in 20 hours?
We Take
12 x 20 = $240
So, he earns $240 in 20 hours.
Answer: 240 dollars
Step-by-step explanation:
divide 36 dollars by 3 (=12)to know how much he earns per hour
then multiply the quotient by 20 which is 240
You select a marble without looking and then put it back. If you do this 72 times, what is the best prediction possible for the number of times you will pick a marble that is not blue?
Answer:
Step-by-step explanation: I might not be exact, cause im still a beginner, but, maybe halve of 72?
T/F; A line, ray, or segment that divides a line segment into two equal parts. It also makes a right angle with the line segment.
A line, ray, or segment that divides a line segment into two equal parts. It also makes a right angle with the line segment. The given statement is false.
A perpendicular bisector is a line, ray, or segment that intersects a line segment at its midpoint and creates two equal parts. It is called "perpendicular" because it always intersects the line segment at a 90-degree angle, but it does not necessarily make a right angle with the line segment. A line, ray, or segment that divides a line segment into two equal parts is called the perpendicular bisector.
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graph 4x-y=3 and 3x+y=4
Answer:
Step-by-step explanation:
Many fingerprint analysts claim that they report an average of 16 matching characteristics between crime scene fingerprints and the suspect's fingerprints. In her last 30 cases, a defense attorney calculated that fingerprint analysts only reported an average of M = 9.8 matching characteristics, with a standard deviation of s = 1.2. What statistical test would this attorney use to determine whether the average number of matching characteristics she observed differs significantly from that reported by fingerprint analysts?
a. Independent-samples t test
b. Z test
c. Single-sample t test
d. Dependent-samples t test
Independent-samples t test. This test is used to compare the means of two independent groups. The correct answer is a.
In this case, the average number of matching characteristics reported by the fingerprint analysts and the average number of matching characteristics calculated by the defense attorney. The standard deviation of s = 1.2 would be used in the calculation of the t-statistic.
To determine whether the average number of matching characteristics the defense attorney observed (M = 9.8) differs significantly from that reported by fingerprint analysts (16 matching characteristics), she should use a "Single-sample t test".
Here's why:
- The defense attorney is comparing a sample (her 30 cases) to a known population mean (16 matching characteristics).
- The standard deviation of the sample is provided (s = 1.2), but the population standard deviation is not known.
- An Independent-samples t test would be used if comparing two independent groups, which is not the case here.
So, the correct answer is c. Single-sample t test.
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What is the domain and range y=x^2+10x+26
The domain will be all real numbers and the range will be all real numbers greater than or equal to 1.
The domain of a function is the set of all possible input values that can be used as arguments for the function. In this case, there are no restrictions on the input values of x, so the domain is all real numbers.
Domain: All real numbers
Range:
To find the range of the function, we can complete the square to rewrite the function in vertex form:
f(x) = x^2 + 10x + 26
f(x) = (x + 5)^2 + 1
Since the square of a real number is always nonnegative, the minimum value of the function is 1, which occurs when x = -5. Therefore, the range of the function is all real numbers greater than or equal to 1.
Range: All real numbers greater than or equal to 1.
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Farmer Fred was a quizzical man who loved to talk in circles. One day when a group of beginning farmers came to his farm, they asked him real nicely "farmer fred, how many chickens and rabbits do you have on your farm"? Being the quizzical guy that he was, he didn't dare give them a straight answer, but he reported "you know Im farmer Fren and what i`m all about, in this cage there are 24 heads and 36 feet use a system of equation to figure it out give 2 equations.
The number of chicken and the rabbits in the farm are 8 and 16 respectively.
Given that Fred has 24 heads and 80 feet in his farm we need to find the number of chicken and the rabbits in the farm,
Let the number of chicken and the rabbits in the farm be c and r respectively.
Establishing the system of equations,
c + r = 24
c = 24 - r.........(i)
2c + 4r = 80
c + 2r = 40
c = 40 - 2r..........(ii)
Equating the equation, since their LHS are equal,
24 - r = 40 - 2r
r = 16
Put r = 16 in eq(i)
c = 40 - 32
c = 8
Hence, the number of chicken and the rabbits in the farm are 8 and 16 respectively.
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What if the measure of angle B?
A. 39.8
B. 48.59
C. 33.56
D. .015
Applying the tangent ratio, the measure of angle B in the image given is calculated as: A. 39.8°
How to Find the Measure of the Angle Using the Tangent Ratio?In order to find the measure of angle B, we will need to apply the tangent ratio which is expressed as the formula below:
tan ∅ = length of opposite side / length of adjacent side
From the image given, we have:
∅ = angle B
Length of opposite side = 10 cm
Length of adjacent side = 12 cm
tan B = 10/12
B = tan^(-1)(10/12)
Measure of angle B = 39.8°
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a piece of metal 24 inches by 20 inches is made into a box by cutting out sqaures of side x from each corner let V(x) be the volume of the box
write a function in standard form to represent the volume V(x) of the open box
staye an approximate domain for V(x)
calculate V(x)
To determine the volume of the box, we need to find the length, width, and height of the box. We can do this by subtracting twice the length of the square cutouts from the original length, twice the width of the square cutouts from the original width, and the height will be the length of the square cutouts.
Let's assume that the side length of the square cutouts is x. Then the length of the box is (24 - 2x), the width of the box is (20 - 2x), and the height of the box is x. Thus, the volume of the box can be expressed as:
V(x) = (24 - 2x) * (20 - 2x) * x
To simplify this expression, we can expand it using the distributive property and then combine like terms:
V(x) = 4x^3 - 88x^2 + 480x
The approximate domain for V(x) would be [0, 10] since the side length of the square cutouts cannot be greater than half the length or width of the original piece of metal, which is 12 and 10 respectively.
To calculate V(x), we simply substitute x into the formula for V(x):
V(x) = 4x^3 - 88x^2 + 480x
Let's say we want to calculate V(2):
V(2) = 4(2)^3 - 88(2)^2 + 480(2)
= 16 - 352 + 960
= 624
Therefore, the volume of the box when the side length of the square cutouts is 2 inches is 624 cubic inches.
Higher Order Thinking Suppose
you have the three cups shown at the
right. List two different ways you can
measure exactly 1 liter.
Plsss help
Two different ways we can measure exactly 1 Liter are:
Use of the 500 mL cup two times
Use the 300 mL cup 3 times in total and the 100 mL cup once
What is the volume of the cups?This question we want to know the different combination of the given cups that can lead to us getting a volume of 1 liter.
Now, we know that
1 L = 1000 mL
This means that we can make use of the 500 mL cup two times in total to get:
500 mL + 500 mL= 1000 mL = 1 L
Another way we can do the 1 liter is that we can use the 300 mL cup 3 times in total and the 100 mL cup once to get:
300 mL + 300 mL + 300 mL + 100 mL = 1000 mL = 1 L
Other ways are:
- use of the 500 mL cup two timese the 100 mL cup
- Take 10 times the 100-mL cup
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Complete question is:
Suppose you have a 100-mL cup,a 300-mL cup, and a 500-mL cup. list two different ways you can measure exactly 1 L