The best unit of measure for the liquid in family-sized container of apple cider is the liter - True.
The metric unit of volume is known as the litre (international spelling) or liter (American English spelling) (SI symbols L and l,[1] with another symbol also used: ℓ). It is equivalent to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cubic metre (m3). A litre or cubic decimetre fills a space of 10 cm × 10 cm × 10 cm (see figure) and is therefore equal to one-thousandth of a cubic metre. One litre of liquid water has a weight of almost precisely one kilogram, because the kilogram was initially defined in 1795 as the weight of one cubic decimetre of water at the temperature of melting ice (0 °C). However, subsequent redefinitions of the metre and kilogram mean that this relationship is no longer exact.
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1. Use the following information and the map of downtown Seattle to answer questions one and two. Harrison St, Thomas St, and Denny Way are parallel. On Broad St, the distance between Mercer St and Denny Way is 0.7 miles. The distance between those same streets on Aurora Ave is 0.45 miles.
a) On Aurora Ave the distance between Thomas St to Denny Way is 0.2 miles. What is the distance
between these two streets on Broad St? Round your answer to the nearest tenth of a mile.
The distance between these two streets on Broad St is 0.3 miles.
We can use proportions to solve the problem.
Let's assume x is the distance between Thomas St and Denny Way on Broad St.
We know that on Aurora Ave, the distance between Thomas St and Denny Way is 0.2 miles, and on Broad St, the distance between Mercer St and Denny Way is 0.7 miles.
So, we can set up the following proportion:
0.2/0.45 = x/0.7
Simplifying, we get:
x = (0.2/0.45) * 0.7
x = 0.3111 miles
Rounded to the nearest tenth, the distance between Thomas St and Denny Way on Broad St is 0.3 miles.
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Solve For X Question 1
Answer:
x=52
Step-by-step explanation:
Set equal to 180
x-30+3x+2=180
solve for x
4x-28=180
4x=208
x=52
plug in to check
Lim (x,y)->(0,0) f (x,y) = L if f (x,y) gets close to l as we approach (0,0) along the x-axis (y = 0) and along the y-axis (x = 0)?
If f(x,y) gets close to L as we approach (0,0) along the x-axis (y = 0) and along the y-axis (x = 0), the limit of f(x,y) as (x,y) approaches (0,0) is L.
We can follow these steps:
Step 1: Approach (0,0) along the x-axis (y = 0)
Evaluate the limit of f(x,0) as x approaches 0. If the limit exists and is equal to L, proceed to step 2.
Step 2: Approach (0,0) along the y-axis (x = 0)
Evaluate the limit of f(0,y) as y approaches 0. If the limit exists and is equal to L, proceed to step 3.
Step 3: Conclude the limit
If both the limits in steps 1 and 2 exist and are equal to L, then the limit of f(x,y) as (x,y) approaches (0,0) is L. Otherwise, the limit does not exist.
So, if f(x,y) gets close to L as we approach (0,0) along the x-axis (y = 0) and along the y-axis (x = 0), the limit of f(x,y) as (x,y) approaches (0,0) is L.
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It’s asking me to write a quadratic equation in that form with the roots
Answer:
x^2-11x+28=0
Step-by-step explanation:
(x-7)(x-4)=0
= x^2-7x-4x+28=0
= x^2-11x+28=0
Hank pours himself a jar full of
milk to drink with his pie. The jar is 6"
high and 3" in diameter. What is
the volume of the jar?
Answer:
13.5pi inches^3
Step-by-step explanation:
the jar is a cylinder.
the volume of a cylinder is the height * base area, and the base area is a circle. the area of a circle is pi*r^2.
the radius is half the diameter, so it is 1.5.
the area of the circle is pi * 1.5^2 = 2.25pi.
multiply this by the height, and we get 13.5pi. the units is cubic inches, or inches^3.
if the original length and width of the paper is twice its current length and width, calculate the original area, in cm, of the paper in terms of a and b.
Answer:
[tex]4a^{2} + 4ab[/tex]
Step-by-step explanation:
as I can see in the image posted by you,
current length = a + b
current width = a
according to the condition, original paper length and width was twice of their current values,
original length = 2(a+b)
original width = 2a
therefore,
original area = 2(a+b) * 2a
original area = 4(a+b)a
original area = [tex]4a^{2} + 4ab[/tex]
Hopefully you found the answer helpful
Leeze is making labels in the shape of parallelograms. Each label has an area of 18 square centimeters and a base of 6 centimeters. What is the height of each label?
The height of the parallelogram is 3 cm.
What is area?
The region that an object's shape defines as its area. The area of a figure or any other two-dimensional geometric shape in a plane is how much space it occupies.
What is a parallelogram?A parallelogram is a quadrilateral with two pairs of parallel sides. This means that opposite sides of a parallelogram are parallel and have the same length. Additionally, opposite angles of a parallelogram are congruent (i.e., have the same measure). Some common properties of parallelograms include:
The opposite sides of a parallelogram are equal in length.
The opposite angles of a parallelogram are equal in measure.
The consecutive angles of a parallelogram are supplementary (i.e., they add up to 180 degrees).
The diagonals of a parallelogram bisect each other (i.e., they intersect at their midpoint).
Here the given ,
Area of the parallelogram = 18 square centimeters.
Base b = 6 cm
Area of parallelogram = bh square unit
=> 18=6*h
=> h= 18/6 = 3 cm
Hence the height of the parallelogram is 3 cm.
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Which expression is equivalent to
(x-1)^2 x^2+-6
———— * ————
x^2-x-12 x^2+6x-5
If no denominator equals 0?
The expression is equivalent to (x - 2)(x - 5)/(x - 4)(x - 1), as long as the denominators do not equal 0.
What is Algebraic expression ?
An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (such as addition, subtraction, multiplication, and division), as well as grouping symbols like parentheses.
We can begin by factoring the denominators of both fractions as follows:
[tex]x^{2}[/tex]- x - 12 = (x - 4)(x + 3)
[tex]x^{2}[/tex]- + 6x - 5 = (x - 1)(x + 5)
Substituting these expressions, the given expression becomes:
[(x - 1)(x-1) [tex]x^{2}[/tex]-+ (-6)]/[(x - 4)(x + 3)] * [(x + 5)/(x - 1)(x + 5x - 5)]
Simplifying, we can cancel out the (x - 1) and (x + 5) terms in the numerator and denominator:
[(x - 1) * [tex]x^{2}[/tex]- + (-6)]/[(x - 4)(x + 3)] * [1/(x - 5)]
Expanding the numerator:
( [tex]x^{3}[/tex]- [tex]x^{2}[/tex]- - 6)/( [tex]x^{2}[/tex]- - x - 12) * [1/(x - 5)]
Factoring the numerator:
( [tex]x^{2}[/tex]- + x - 6)(x - 5)/( [tex]x^{2}[/tex]- - x - 12)
Factoring again:
(x + 3)(x - 2)(x - 5)/(x - 4)(x + 3)(x - 1)
Canceling out the (x + 3) terms:
(x - 2)(x - 5)/(x - 4)(x - 1)
Therefore, the expression is equivalent to (x - 2)(x - 5)/(x - 4)(x - 1), as long as the denominators do not equal 0.
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The value of a book is $258 and decreases at a
rate of 8% per year. Find the value of the book
after 11 years.
Answer:
The value of the book after 11 years is $103.11
The equation should be 258*0.92, 11 times repeating.
The answer should be 103.106443494, but if you round it the answer is 103.11
[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &258\\ r=rate\to 8\%\to \frac{8}{100}\dotfill &0.08\\ t=years\dotfill &11\\ \end{cases} \\\\\\ A = 258(1 - 0.08)^{11} \implies A=258(0.92)^{11}\implies A \approx 103.11[/tex]
a manufacturer knows that their items have a normally distributed length, with a mean of 14.7 inches, and standard deviation of 4.8 inches. if 25 items are chosen at random, what is the probability that their mean length is less than 12.4 inches?
The probability that the mean length of the 25 items is less than 12.4 inches is approximately 0.0009
We can solve this problem by using the Central Limit Theorem (CLT), which states that the sample mean of a large enough sample size from any distribution with a finite mean and variance will follow a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the population mean is 14.7 inches, the population standard deviation is 4.8 inches, and the sample size is 25. Thus, the mean of the sample means is also 14.7 inches, and the standard deviation of the sample means is 4.8 inches / sqrt(25) = 0.96 inches.
To find the probability that the mean length of the 25 items is less than 12.4 inches, we can standardize the sample mean using the z-score formula
z = (x - mu) / (sigma / sqrt(n))
where x is the sample mean we want to find the probability of (in this case, 12.4 inches), mu is the population mean (14.7 inches), sigma is the population standard deviation (4.8 inches), and n is the sample size (25).
Substituting these values, we get
z = (12.4 - 14.7) / (4.8 / sqrt(25)) = -3.13
We can then use a standard normal distribution table or calculator to find the probability that a standard normal variable is less than -3.13, which is approximately 0.0009.
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z varies directly as x^2. If z= 8 when x= 2, find z when x= 6.
Thus, the value of z = 72, when x = 6 for the given condition that z varies directly with the square of x.
Define about the direct proportion:The connection among two variables is known as a direct proportion when their ratios are comparable to a fixed value.
To the contrary, a direct proportion occurs when a change in one quantity prompts a commensurate change in the other amount, or when a change in one quantity prompts a change in the other quantity.The proportional sign (∝) is used to represent a direct proportion. For instance, the sentence "x ∝) y" can be used to indicate the relationship between two variables x and y.Given data:
z varies directly as x².
z= 8 when x= 2.
So,
z ∝ x²
z = k x²
k = z/ x² (k is the constant of proportionality)
z= 8 when x= 2.
k = 8/(2)²
k = 2
Now, when x = 6
z = k x²
z = 2 * (6)²
z = 72
Thus, the value of z = 72, when x = 6 for the given condition that z varies directly with the square of x.
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The test statistic of z=−1.91 is obtained when testing the claim that p=1/2. a. Using a significance level of α=0.10, find the critical value(s). b. Should we reject H 0 or should we fail to reject H 0 ? Click here to view page 1 of the standard normal distribution table. Click here to view page 2 of the standard normal distribution table. a. The critical value(s) is/are z= (Round to two decimal places as needed. Use a comma to separate answers as needed.) b. Choose the correct conclusion below. A. Reject H 0 . There is sufficient evidence to warrant rejection of the claim that p=1/2. B. Fail to reject H 0 . There is sufficient evidence to warrant rejection of the claim that p=1/2. C. Fail to reject H 0 . There is not sufficient evidence to warrant rejection of the claim that p=1/2. D. Reject H n . There is not sufficient evidence to warrant rejection of the claim that p=1/2.
a. The critical value(s) is/are z = -1.645, 1.64
b. Fail to reject H0.
How to determine critical valueThere is not sufficient evidence to warrant rejection of the claim that p = 1/2. The test statistic of z = -1.91 is obtained when testing the claim that p = 1/2.
We want to know whether to reject H0 or fail to reject H0 using a significance level of α = 0.10.
Using the standard normal distribution table, the critical value(s) are z = ± 1.645 (rounded to two decimal places).
Since -1.91 is less than -1.645, we fail to reject H0.
In other words, there is not sufficient evidence to warrant rejection of the claim that p = 1/2.
Therefore, the correct conclusion is "Fail to reject H0.
There is not sufficient evidence to warrant rejection of the claim that p = 1/2."
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help asap will give brainliest!!!!
Are angles C and D supplementary angles: B. No.
What is a supplementary angle?In Mathematics, a supplementary angle simply refers to two (2) angles or arc whose sum is equal to 180 degrees. Mathematically, a supplementary angle can be calculated by using this mathematical equation:
C + D = 180
Where:
C and D are measure of the angles subtended.
Additionally, the sum of all of the angles on a straight line is always equal to 180 degrees. In this scenario, we can reasonably infer and logically deduce that the sum of the given angles are not supplementary angles:
58 + 102 = 160° ≠ 180°
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Rewrite each equation without absolute value for the given conditions y=|x-3|+|x+2|-|x-5| if 3
help asap will give brainliest!
Step-by-step explanation:
VOLUME of pyramid = 1/3 base area * height
Base area is a square = 3 x 3
= 1/3 ( 3x3) * 16 = 48 in^3
expedia would like to test if the average round-trip airfare between philadelphia and dublin is less than $1,200. the correct hypothesis statement would be
The correct hypothesis statement in this case would be "The average round-trip airfare between Philadelphia and Dublin is less than $1,200".
The hypothesis statement for the test of the average round-trip airfare between Philadelphia and Dublin being less than $1,200 would be “The average round-trip airfare between Philadelphia and Dublin is less than $1,200”.What is hypothesis?A hypothesis is a suggested explanation for a phenomenon or a theory that is tested using various experiments.
It is the first step in research that aids in forming a research question and establishes a framework for the study.The hypothesis statement:In hypothesis testing, the hypothesis statement is a declarative statement or an assertion that specifies the existence or non-existence of a phenomenon. In hypothesis testing, there are two hypothesis statements: null hypothesis and alternative hypothesis.
The null hypothesis would be that the average round-trip airfare between Philadelphia and Dublin is equal to or greater than $1,200. The alternative hypothesis would be that the average round-trip airfare between Philadelphia and Dublin is less than $1,200.
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simplify (x-y/√x-√y)-√x
Heidi swims 50 meters at an average speed of 1.25 meters per second
How many seconds did it take her?
Answer: 62.5 would be the answer.
Step-by-step explanation: You have to multiply how many meters per second by the whole total meters.
50x1.25=62.5.
*********************************************************************************************
Hope this helps! ^^
Examine the loan amortization table for a $210,000, 15-year mortgage with an APR of 3.8%. The borrower paid an extra $100 each month towards the principal.
Determine the missing amounts.
Using a loan amortization calculator, we can generate a table that shows the borrower's monthly payments, the interest paid, the principal paid, and the remaining balance after each payment.
What is equation?An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x + 3" equals the value "9."
Using a loan amortization calculator, we can generate a table that shows the borrower's monthly payments, the interest paid, the principal paid, and the remaining balance after each payment. Here is the loan amortization table for the $210,000, 15-year mortgage with an APR of 3.8%, assuming the borrower pays an extra $100 towards the principal each month:
Month Payment Interest Paid Principal Paid Extra Principal Paid Remaining Balance
1 $1,529 $662 $297 $100 $209,703
2 $1,529 $657 $301 $100 $209,402
3 $1,529 $652 $306 $100 $209,096
4 $1,529 $647 $311 $100 $208,783
5 $1,529 $642 $316 $100 $208,463
6 $1,529 $637 $321 $100 $208,137
7 $1,529 $632 $326 $100 $207,803
8 $1,529 $627 $331 $100 $207,463
9 $1,529 $622 $336 $100 $207,116
10 $1,529 $617 $341 $100 $206,762
11 $1,529 $611 $347 $100 $206,411
12 $1,529 $606 $352 $100 $206,052
13 $1,529 $601 $357 $100 $205,686
14 $1,529 $596 $362 $100 $205,322
15 $1,529 $590 $368 $100 $204,950
16 $1,529 $585 $373 $100 $204,570
17 $1,529 $580 $378 $100 $204,182
18 $1,529 $574 $384 $100 $203,787
19 $1,529 $569 $389 $100 $203,383
20 $1,529 $563 $395 $100 $202,972
21 $1,529 $558 $400 $100 $202,552
22 $1,529 $552 $406 $100 $202,125
23 $1,529 $547 $411
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Write a story problem to go with the multiplication problem 3 x 7/8. Then, solve the problem.
Answer:
Step-by-step explanation:
Story Problem:
Samantha is baking cookies and her recipe calls for 3 cups of flour. She only has a bag of flour that is 7/8 full. If each cup of flour weighs the same, how much flour does Samantha have left after taking out the 3 cups needed for the recipe?
Solution:
To solve the problem, we need to multiply the amount of flour in the bag by 3/1 (which is the same as multiplying by 3).
3 x 7/8 = (3 x 7)/8 = 21/8
So, Samantha has 21/8 cups of flour in the bag.
To find out how much flour she has left after taking out the 3 cups needed for the recipe, we need to subtract 3 from 21/8:
21/8 - 3 = 21/8 - 24/8 = -3/8
Samantha has -3/8 cups of flour left in the bag, which means she doesn't have enough flour to make the recipe. She needs to get more flour before she can continue baking.
Answer:
Look below
Step-by-step explanation:
Ken drinks 7/8 of a carton of milk each day. How much milk does
he drink in 3 days?
7/8*3
21/8
2 5/8 cartons of milk a day
Answer the question below in the screenshot, thanks! (please answer quickly, I'm on edge so I can't move on!)
The exact answer on Edge is
"Use the product of powers property to simplify the numerator by removing the parentheses. Follow the order of operations by removing the innermost parentheses first. Cube the quantity to get the product of 2 to the third power, r to the 6th power, and t to the third power, or 2^3r^6t^3 in the numerator. "
2.5 * 10^5 ____ 4.2 * 10^-7
i need to compare these
Answer: 250,000 > .00000042
Step-by-step explanation:
2.5 times 10^5 means move the decimal to the right 5 times.
2.5
1. 25.
2. 250.
3. 2,500.
4. 25,000.
5. 250,000.
250000 __ 4.2 times 10^-7
10^-7 would mean move the decimal to the left that many times.
4.2
1. .42
2. .042
3. .0042
4. .00042
5. .000042
6. .0000042
7. 00000042
250,000 > .00000042
a sample of 1200 computer chips revealed that 53% of the chips do not fail in the first 1000 hours of their use. the company's promotional literature claimed that more than 50% do not fail in the first 1000 hours of their use. is there sufficient evidence at the 0.10 level to support the company's claim? state the null and alternative hypotheses for the above scenario.
The null hypothesis is that the proportion of computer chips that do not fail in the first 1000 hours of use is equal to 50%. The alternative hypothesis is that the proportion is greater than 50%.
At the 0.10 level of significance, we will reject the null hypothesis if the test statistic is greater than 1.28. The test statistic can be calculated using the formula:
[tex]z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$[/tex]
where [tex]\hat{p}$[/tex] is the sample proportion, [tex]p_0$[/tex] is the hypothesized proportion under the null hypothesis, and $n$ is the sample size.
In this case, we have [tex]\hat{p} = 0.53$, $p_0 = 0.50$, and $n = 1200$.[/tex] Substituting these values into the formula, we get:
[tex]z = \frac{0.53 - 0.50}{\sqrt{\frac{0.50(1-0.50)}{1200}}} = 2.77$[/tex]
Since the test statistic is greater than 1.28, we reject the null hypothesis and conclude that there is sufficient evidence at the 0.10 level to support the company's claim that more than 50% of computer chips do not fail in the first 1000 hours of their use.
Therefore, the null hypothesis is equal to 50%, and the alternative hypothesis is 50%. The test statistic is 2.77, which is greater than the critical value of 1.28 at the 0.10 level of significance, so we reject the null hypothesis and conclude that there is sufficient evidence to support the company's claim.
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The value of x is__?
The measure of is angle 1 is__?
The measure of angle 2 is __?
The measure of angle 3 is __?
The measure of angle 4 is __?
The measure of angle 5 is __?
The measure of angle 6 is __?
The measure of angle 7 is __?
The measure of angle 8 angles 2 and 3 are____ angles.
The value of x is 24
The measure of is angle 1 is 101°
The measure of angle 2 is 79°
The measure of angle 3 is 101°
The measure of angle 4 is 101°
The measure of angle 5 is 79°
The measure of angle 6 is 79°
The measure of angle 7 is 101°
The measure of angle 8 is 79°
What are angles on a parallel line?Angles in parallel lines are angles that are created when two parallel lines are intersected by another line called a transversal.
3x +19 = 5x-29 ( opposite angle)
collect like terms
5x-3x = 29+19
2x = 48
x = 24
angle 1 = 180-(3×24+7)
= 180- 79
= 101°
angle 2 = 3x+7 = 79°( opposite angles)
angle 3 = 101° ( opposite angles)
angle 4 = angle 1 = 101° ( corresponding angles)
angle 5 = 79° ( corresponding angles)
angle 6 = angle 5 = 79°( opposite angle)
angle 7 = angle 4 = 101 ( corresponding angles)
angle 8 = angle 7 = 79 ( opposite angle )
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The simple intrest owed on a loan of 5,600after 4 years is 1,008 what percent reprsents the annual in intrest rate on the loan
The annual interest rate on the loan is 4.5% using simple interest formula and concepts.
To calculate the annual interest rate on the loan, we need to use the simple interest formula:
Simple Interest = (P × r × t) / 100
P stands for principal, r stands for annual interest rate, and t stands for time in years.
Given principal amount is $5,600, the simple interest earned after 4 years is $1,008. Therefore, we can plug in values into formula and solve for r:
$1,008 = (5,600 × r × 4) / 100
Multiply LHS and RHS by 100 and divide by 5,600 × 4, we will obtain:
r = $1,008 / $22400 = 0.045 or 4.5%
Hence, the annual interest rate on the loan is 4.5%.
It is significant to remember that simple interest does not compound and has a linear relationship to time. This indicates that the interest charged on the loan is the same each year and does not compound. Short-term loans and personal loans frequently employ simple interest. However, compound interest, which takes into account the cumulative interest over time, is utilised for long-term loans such as mortgages. To make wise financial decisions, it is crucial to comprehend the distinction between simple and compound interest.
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Obtain the derivative of the following function using the addition and subtraction rule:
F(x) = 5x10 - 2x2 + x - 1
Answer:
Step-by-step explanation:
To obtain the derivative of F(x) = 5x^10 - 2x^2 + x - 1 using the addition and subtraction rule, we can take the derivative of each term separately and then add or subtract the resulting derivatives.
The derivative of 5x^10 is:
(5x^10)' = 50x^9
The derivative of -2x^2 is:
(-2x^2)' = -4x
The derivative of x is:
(x)' = 1
The derivative of -1 is:
(-1)' = 0
Putting these derivatives together using the addition and subtraction rule, we get:
F'(x) = (50x^9) - (4x) + (1) + (0)
Simplifying this expression, we get:
F'(x) = 50x^9 - 4x + 1
Therefore, the derivative of F(x) = 5x^10 - 2x^2 + x - 1 using the addition and subtraction rule is F'(x) = 50x^9 - 4x + 1.
A bakery can make 3 cheesecakes for every 4 blocks of cream cheese. Which
table represents the relationship between the number of cheesecakes the
bakery makes and the number of blocks of cream cheese the bakery uses?
A
C
Cheesecakes
3
6
12
Cheesecakes
3
9
12
Cream Cheese
(blocks)
4
7
13
Cream Cheese
(blocks)
4
16
20
B
D
Cheesecakes
3
9
12
Cheesecakes
9
10
11
Cream Cheese
(blocks)
4
12
16
Cream Cheese
(blocks)
12
13
14
Since this bakery can make 3 cheesecakes for every 4 blocks of cream cheese, a table that represent the relationship between the number of cheesecakes the bakery makes and the number of blocks of cream cheese the bakery uses is: B. table B.
What is a proportional relationship?In Mathematics, a proportional relationship produces equivalent ratios and it can be modeled or represented by the following mathematical equation:
y = kx
Where:
y represent the number of blocks of cream cheese.x represent the number of cheesecakes.k is the constant of proportionality.In order to have a proportional relationship, the variables representing the number of blocks of cream cheese and the number of cheesecakes must have the same constant of proportionality:
Constant of proportionality, k = y/x
Constant of proportionality, k = 4/3 = 8/6 = 12/9 = 16/12
Constant of proportionality, k = 4/3.
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14. Ella is making a corner bench
seat. She plans to outline the seat
cushion in a decorative ribbon.
How much ribbon will she need?
3.6 ft
6.25 ft
4.8 ft
Answer:
14.65
Step-by-step explanation:
3.6 ft + 6.25+4.8=14.65
What are the solutions to the system of conics?
y^2/4−x^2/2=1
y^2=8(x+1)
Drag the ordered pairs into the box to correctly complete the table.
The cοmplete table with the οrdered pairs is:
(2-√6,2√(6-2√6)) (2+√6,2√(6-2√6)) (2+√6,-2√(6-2√6))
(2-√6,2√(6+2√6)) (2+√6,-2√(6+2√6)) (2+√6,2√(6+2√6))
What is quadratic equatiοn?A quadratic equatiοn is a secοnd-degree pοlynοmial equatiοn in οne variable οf the fοrm: [tex]ax^2 + bx + c = 0.[/tex]
Tο find the sοlutiοns tο the system οf cοnics, we can use substitutiοn tο eliminate οne variable and οbtain a quadratic equatiοn in the οther variable. Then, we can sοlve this quadratic equatiοn tο find the pοssible values οf the remaining variable.
Frοm the given system οf cοnics:
[tex]y^2/4 - x^2/2 = 1 ...(1)[/tex]
[tex]y^2 = 8(x + 1) ...(2)[/tex]
We can eliminate [tex]y^2[/tex] frοm equatiοn (1) by multiplying bοth sides by 4:
[tex]y^2 - 2x^2 = 4[/tex]
Substituting the value οf [tex]y^2[/tex] frοm equatiοn (2), we get:
[tex]8(x + 1) - 2x^2 = 4[/tex]
Simplifying and rearranging, we get:
[tex]x^2 + 2x - 2 = 0[/tex]
We can sοlve this quadratic equatiοn using the quadratic fοrmula:
[tex]x = (-2 \± \sqrt{(2^2 - 4(1)(-2))}) / (2(1))[/tex]
x = (-2 ± √12) / 2
x = -1 ± √3
Substituting these values οf x in equatiοn (2), we can find the cοrrespοnding values οf y:
When x = -1 + √3, we get:
[tex]y^2[/tex] = 8((-1 + √3) + 1) = 8√3
y = ±2√(2√3)
Therefοre, the sοlutiοns are:
(-1 + √3, 2√(2√3)) and (-1 + √3, -2√(2√3))
When x = -1 - √3, we get:
[tex]y^2[/tex] = 8((-1 - √3) + 1) = -8√3
This equatiοn has nο real sοlutiοns fοr y.
Therefοre, the cοmplete table with the οrdered pairs is:
(2-√6,2√(6-2√6)) (2+√6,2√(6-2√6)) (2+√6,-2√(6-2√6))
(2-√6,2√(6+2√6)) (2+√6,-2√(6+2√6)) (2+√6,2√(6+2√6))
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So, the bounce heights form a geometric sequence: 27, 18, 12. Two-thirds of 12 is 8, so on the fourth bounce, the ball will reach a height of 8 feet.
In this case, the bounce heights form a geometric sequence with a common ratio of 2/3. , the ball reaches a height of [tex]8[/tex] feet on the third bounce,
What is the geometric sequence?The definition of a geometric sequence is a set of numbers where each term is created by multiplying the preceding term by a fixed factor.
If the bounce heights form a geometric sequence with common ratio "r", we can write:
27 = initial height
[tex]18 = 27r[/tex]
[tex]12 = 18r = 27 \times r^2[/tex]
Solving for "r", we get:
[tex]r = 2/3[/tex]
So, the fourth bounce height would be:
[tex](12) \times (2/3) = 8[/tex]
Therefore, the ball reaches a height of 8 feet on the third bounce, not the fourth bounce as stated in the original statement.
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