Answer:
3.25
With 3.254, rule A applies and 3.254 rounded to the nearest hundredth is: 3.25
AND
107.7
Rounded to the nearest 0.1 or
the Tenths Place.
107.747
You rounded to the nearest tenths place. The 7 in the tenths place rounds down to 7, or stays the same, because the digit to the right in the hundredths place is 4.
107.7
When the digit to the right is less than 5 we round toward 0.
107.747 was rounded down toward zero to 107.7
The expression -8(2t2-3t) represents the height, in feet, of a soccer ball after it is kicked. Which value represents the initial
velocity of the soccer ball?
Answer: 5
Step-by-step explanation:
#1 Original price: $50
Discount: 30%
What is the sale price?
To find the sale price, we need to apply the discount to the original price:
Discount = 30% = 0.30
Original price = $50
Discount amount = Discount x Original price
= 0.30 x $50
= $15
Sale price = Original price - Discount amount
= $50 - $15
= $35
Therefore, the sale price after a 30% discount on an original price of $50 is $35.
Answer:
$35
Step-by-step explanation:
70% of 50 is 35
What is the product of (3a 2)(4a2 – 2a + 9)?a. 12a3 − 2a + 18b. 12a3 + 6a + 9c. 12a3 − 6a2 + 23a + 18d. 12a3 + 2a2 + 23a + 18
Answer:
12a^3 +2a^2+23a+18
Step-by-step explanation:
Ernesto takes a sheet of paper and makes a diagonal cut from one corner to the opposite corner, making two triangles. The cut he makes is 20 inches long and the width of the paper is 16 inches. What is the paper's length?
Answer:
3`1
Step-by-step explanation:
Answer: 12 inches
Step-by-step explanation:
The 20 inches is the hypotenuse whereas 16 inches is the height of the triangle respectively.
So,
let keep the paper's length as x inches.
20² = 16² + x²
x² = 20² - 16²
x = [tex]\sqrt{20^{2}-16^{2} }[/tex]
x = 12
I am not sure if this is right because i have never done a sum in inches before but I hope this helps sorry if it is not right.
help!?!??!?!!?!!?!!?!?!??
Answer:
C.)
Step-by-step explanation:
1.) First find the slope of the perpendicular line. The slope of the perpendicular line is the number that when multiplied by the slope of the initial line, you get -1. The slope of the line is 3/8, and -8/3 * 3/8 is -1, therefore -8/3 is the slope of the new line.
2.) Now, since we have a slope if you look at the options, there is only one option with the slope -8/3, which is option C.).
A movie theater wanted to determine the average rate that their diet soda is purchased. An employee gathered data
on the amount of diet soda remaining in the machine, y, for several hours after the machine is filled, x. The following
scatter plot and line of fit was created to display the data.
Soda Machine
Amount of Diet Soda
Find the y-intercept of the line of fit and explain its meaning in the context of the data.
The y-intercept is -4.8. The machine starts with 4.8 ounces of diet soda.
O The y-intercept is -4.8. The machine loses about 4.8 fluid ounces of diet soda each hour.
O The y-intercept is 40.98. The machine starts with 40.98 ounces of diet soda.
O The y-intercept is 40.98. The machine loses about 40.98 fluid ounces of diet soda each hour.
Option D is incorrect because it gives a value that is not consistent with the scatter plot and the line of fit. The correct answer is: The y-intercept is 40.98. The machine starts with 40.98 ounces of diet soda.
What is y-intercept?In the context of a graph, the y-intercept is the point where a straight line or curve intersects the y-axis. It is the value of the dependent variable (y) when the independent variable (x) is equal to zero. Mathematically, it is the value of y when x = 0 in the equation of the line or curve. The y-intercept is also known as the initial value or starting point of the graph. It is often denoted as (0, b), where b is the y-coordinate of the intercept point.
The y-intercept of the line of fit represents the starting point of the relationship between the two variables, in this case, the amount of diet soda in the machine and the time after the machine is filled. Therefore, the y-intercept value of 40.98 means that the machine starts with 40.98 ounces of diet soda when it is filled.
Option A is incorrect because the y-intercept is a starting value, and it does not represent a rate of change.
Option B is also incorrect because it suggests that the machine loses 4.8 fluid ounces of diet soda every hour, which is not what the y-intercept represents.
Option D is incorrect because it gives a value that is not consistent with the scatter plot and the line of fit.
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A certain Midwestern University claims that 90% of their football players complete the degree in five years. The NCAA investigates his claim by selecting a random sample of 20 football players, that have been part of the program within the past 5 years. 14 of these players received their degree, while 6 did not receive their degree. If you were the investigator, what would you conclude about the universities claim? Explain your reasoning with probability.
We can conclude that 10% of the football players complete their degree in five years.
How to investigate the university's claim?
To investigate the university's claim, we can set up a hypothesis test.
Null hypothesis: The proportion of football players who complete their degree in five years is 0.9.
Alternative hypothesis: The proportion of football players who complete their degree in five years is less than 0.9.
We can use a one-tailed binomial test to test this hypothesis. The test statistic is the number of football players who received their degree in our sample of 20. Under the null hypothesis, this follows a binomial distribution with n=20 and p=0.9.
We can calculate the probability of observing 14 or fewer football players receiving their degree if the true proportion is 0.9:
P(X ≤ 14) = Σ P(X = i) for i = 0 to 14
= binom.cdf(14, n=20, p=0.9)
≈ 0.001
This means that the probability of observing 14 or fewer football players receiving their degree in a random sample of 20, if the true proportion is 0.9, is only 0.001.
If we use a significance level of 0.05, this means we reject the null hypothesis if the p-value is less than 0.05. Since our p-value is much smaller than 0.05, we can reject the null hypothesis and conclude that the proportion of football players who complete their degree in five years is less than 0.9.
Therefore, based on the given sample, we cannot conclude that 90% of the football players complete their degree in five years.
So, we can conclude that 10% of the football players complete their degree in five years.
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We can conclude that 10% of the football players complete their degree in five years.
How to investigate the university's claim?To investigate the university's claim, we can set up a hypothesis test.
Null hypothesis: The proportion of football players who complete their degree in five years is 0.9.
Alternative hypothesis: The proportion of football players who complete their degree in five years is less than 0.9.
We can use a one-tailed binomial test to test this hypothesis. The test statistic is the number of football players who received their degree in our sample of 20. Under the null hypothesis, this follows a binomial distribution with n=20 and p=0.9.
We can calculate the probability of observing 14 or fewer football players receiving their degree if the true proportion is 0.9:
P(X ≤ 14) = Σ P(X = i) for i = 0 to 14
= binom.cdf(14, n=20, p=0.9)
≈ 0.001
This means that the probability of observing 14 or fewer football players receiving their degree in a random sample of 20, if the true proportion is 0.9, is only 0.001.
If we use a significance level of 0.05, this means we reject the null hypothesis if the p-value is less than 0.05. Since our p-value is much smaller than 0.05, we can reject the null hypothesis and conclude that the proportion of football players who complete their degree in five years is less than 0.9.
Therefore, based on the given sample, we cannot conclude that 90% of the football players complete their degree in five years.
So, we can conclude that 10% of the football players complete their degree in five years.
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write an expression to show the total number of hours erica works in a week that she babysits for 8 hours.
The total number of hours Erica works in a week when she babysits for 8 hours can be expressed using the following equation: Total Hours Worked = 8 x Number of Babysitting Days per Week In this case, since Erica babysits for 8 hours.
To calculate the total number of hours Erica works in a week, we need to consider the number of days she babysits and the number of hours she works per day. Erica babysits for 8 hours per day. Therefore, we can use the equation
Total Hours Worked = Number of Days x Hours per Day
To calculate the total number of hours she works in a week. We can substitute these values into the equation:
Total Hours Worked = 1 x 8 = 8 hours.
This means that Erica works a total of 8 hours per week when she babysits for 8 hours.
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This graph represents a proportional relationship.
The graph represents a proportional relationship is a false statement.
Based on the graph, it appears to represent a linear relationship, where there is a constant rate of change between the two variables. However, it is not possible to determine if it represents a proportional relationship without knowing the specific context of the data being plotted.
If the variables represented in the graph have a proportional relationship, then the graph should be a straight line that passes through the origin (0,0). This is because in a proportional relationship, as one variable increases, the other variable increases or decreases in proportion to the change in the first variable. Therefore, the ratio between the two variables should remain constant.
If the graph in this example does not pass through the origin, then it is likely that the relationship between the variables is not proportional.
As sufficient data is not provided so this statement is false.
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the appropriate test to compare one sample to another sample to see if one is greater than another in some way is called a(n) ?
The appropriate test to compare one sample to another sample to see if one is greater than another in some way is called a "two-sample hypothesis test."
In a two-sample hypothesis test, the null hypothesis states that there is no significant difference between the means of the two samples, while the alternative hypothesis states that there is a significant difference between the means.
The specific type of test used depends on several factors, including the type of data being compared, the sample sizes, and the level of significance desired. Commonly used tests include the t-test, z-test, and ANOVA.
It is important to choose the appropriate test for the data being analyzed to ensure accurate results and conclusions.
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suppose that the return for a particular large-cap domestic stock fund is normally distributed with a mean of 14.4% and standard deviation of 4.4%. (a) what is the probability that the large-cap stock fund has a return of at least 17%? (round your answer to four decimal places.)
The probability that the large-cap domestic stock fund has a return of at least 17% is approximately 0.2776 or 27.76%.
To calculate the probability that the large-cap domestic stock fund has a return of at least 17%, we need to find the z-score first.
The z-score formula is:
Z = (X - μ) / σ
Where X is the desired return (17%), μ is the mean of 14.4%, and σ is the standard deviation of 4.4%.
Z = (17 - 14.4) / 4.4 = 2.6 / 4.4 ≈ 0.591
Now, using a z-table or calculator, we can find the probability associated with this z-score. The area to the left of the z-score in a standard normal distribution is approximately 0.7224. Since we want the probability of a return of at least 17%, we need to find the area to the right of the z-score.
P(X ≥ 17%) = 1 - P(X < 17%) = 1 - 0.7224 = 0.2776
So, the probability that the large-cap domestic stock fund has a return of at least 17% is approximately 0.2776 or 27.76%.
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For each inequality, find two values for x that make the inequality true and two values that make it false
A. x+3>70
B. x+3<70
C. -5x<2
D. 5x<2
to find two values for x that make an inequality true, we can choose values that satisfy the inequality, while to find two values that make it false, we can choose values that do not satisfy the inequality.
How to solve inequality?
A. x+3>70
To make this inequality true, we can choose x = 67 or x = 100. For x = 67, we have 67 + 3 > 70, and for x = 100, we have 100 + 3 > 70. To make the inequality false, we can choose x = 68 or x = 69. For x = 68, we have 68 + 3 < 70, and for x = 69, we have 69 + 3 < 70.
B. x+3<70
To make this inequality true, we can choose x = 0 or x = 1. For x = 0, we have 0 + 3 < 70, and for x = 1, we have 1 + 3 < 70. To make the inequality false, we can choose x = 67 or x = 100. For x = 67, we have 67 + 3 > 70, and for x = 100, we have 100 + 3 > 70.
C. -5x<2
To make this inequality true, we can choose x = 0 or x = -1/5. For x = 0, we have -5(0) < 2, and for x = -1/5, we have -5(-1/5) < 2. To make the inequality false, we can choose x = -2 or x = 1/5. For x = -2, we have -5(-2) > 2, and for x = 1/5, we have -5(1/5) > 2.
D. 5x<2
To make this inequality true, we can choose x = 0 or x = 1/5. For x = 0, we have 5(0) < 2, and for x = 1/5, we have 5(1/5) < 2. To make the inequality false, we can choose x = -2 or x = -1/5. For x = -2, we have 5(-2) > 2, and for x = -1/5, we have 5(-1/5) > 2.
In summary, to find two values for x that make an inequality true, we can choose values that satisfy the inequality, while to find two values that make it false, we can choose values that do not satisfy the inequality.
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Graph the inequality h_< 46.
Open circle and move to the left.
which describes the relationship between a tangent line to a circle and the radius of the circle drawn to the point of tangency?
A. The tangent line and the radius are parallel
B. The tangent line and the radius are perpendicular
C. The tangent line is the perpendicular bisector of the radius
D. The radius is the perpendicular bisector of the tangent line.
Solve for x. Assume that lines which appear tangent are tangent.
A. 8
B. 10
C. 7
D. 9
The tangent line and the radius are perpendicular. and the value of x is 8
Describing the relationshipThe correct answer is B. The tangent line and the radius are perpendicular.
In a circle, the tangent line at a point on the circle is perpendicular to the radius drawn to that point. This is a fundamental property of circles.
Calculating the value of xThe value of x is calculated as
9 * x = 12 * 6
So, we have
9x = 72
Divide
x = 8
Hence, the value of x is 8
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Which expressions are equivalent to the one below? Check all that apply.
5x
A. 15^x/3
B. 5.5^x+1
C. (15/3)^x
D. 5^x
E. 5.5x-1
F.15^x/3^x
Answer:
C and D
Step-by-step explanation:
The expression 5x cannot be simplified further, but we can check which expressions are equivalent to it.The expressions that are equivalent to 5x are:
C. (15/3)^x = 5^xD. 5^xTherefore, the expressions that are equivalent to 5x are C. (15/3)^x and D. 5^x.None of the other expressions listed are equivalent to 5x.
A. 15^x/3 = (3*5)^x/3 = 3^x * 5^x/3, which is not equivalent to 5x.B. 5.5^(x+1) = 5.5^x * 5.5^1 = 5.5^x * 5.5, which is not equivalent to 5x.E. 5.5x-1 = 5x * 5^-1 = 5x/5, which is not equivalent to 5x.F. 15^x/3^x = (3*5)^x/3^x = 3^x * 5^x/3^x, which is not equivalent to 5x.The graph of the linear function passes through the points (4, 24) and (6,30).
What is the equation of the function?
Answer:
To find the equation of the linear function, we need to first determine its slope, which is the rate of change of the function with respect to its input. We can use the slope formula: slope = (change in y) / (change in x) Using the two given points, we have: slope = (30 - 24) / (6 - 4) = 3 Now that we know the slope, we can use the point-slope form of the equation of a line to find the equation of the function: y - y1 = m(x - x1) where m is the slope, and (x1, y1) is one of the given points. Let's use the point (4, 24): y - 24 = 3(x - 4) Expanding and simplifying, we get: y = 3x -
i need equivalent expressions
To fill up the blanks such that each rectangle has the same area, we could assign the following numbers:
1. First two upper boxes in the first rectangle 2 and 4. The side box, 3.
2. For the two upper boxes in the second rectangle, we could assign the numbers, 10 and 8.
How to find the area of a rectangleTo find the area of a rectangle, you should multiply the length of the rectangle by its width. So, for the first rectangle, the sum of the length will be 6 and this will be multiplied by the width, which is 3.
Also, for the second rectangle, we could sum up the length to be 10 + 8 which is 18. If multiplied by 1, the result of the area will be 18.
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Factorize quadratic expressions x-x+6
Thus, the two factors of quadratic expressions found after factorization are 2 and 3.
Explain about the factorization:Factorization is the technique of expressing a given statement as the product between two or more components. The sum of the greatest common factors of the integer coefficients and the same letters with the smallest powers yields the greatest common factor for two or more monomials.
Factorization in mathematics is the process of dividing a large number into smaller ones that, when multiplied together, gives you the original number. Factorization is the division of a number in with its factors or divisors. The factorization of the integer 12, for instance, might appear as 3 times 4.Given quadratic expressions:
= x² - 5x + 6
To factorize the expression, find the factors of 6 such that on addition it becomes -5.
= x² - 3x - 2x + 6
Taking x common from first two terms and -2 from last two terms.
= x(x -3) -2(x - 3)
Taking (x - 3) common.
= (x - 3)(x - 2)
x -3 = 0
x = 3
and, x - 2 = 0
x = 2
Thus, the two factors of quadratic expressions found after factorization are 2 and 3.
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Complete question:
Factorize quadratic expressions: x² - 5x + 6
Step-by-step explanation:
Perhaps you entered your question incorrectly
maybe you meant
x^2 -x + 6 THIS factors to
(x-3)(x+2) <=====if you multiply this out, you will see = x^2 -x+6
Are you familiar with the Quadratic Formula ?
for this Quadrtatic a = 1 b = -1 c = 6
if you plug these numbers into the Quadratic Formula you get the roots:
x = 3 and x = -2
then you know the quadratic is (x-3)(x+2)
What is the equation of the circle in standard form?
(x-4)² + (y− 1)² = 3
(x-4)² + (y - 1)² = 9
(x + 4)² + (y + 1)² = 3
(x+4)² + (y + 1)² – 9
The calculated value of the equation of the circle in standard form is: (x - 4)^2 + (y - 1)^2 = 9
What is the equation of the circle in standard form?The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center of the circle is (4, 1) and the radius is 3.
So we can plug in these values to get:
(x - 4)^2 + (y - 1)^2 = 3^2
Simplifying and expanding, we get:
(x - 4)^2 + (y - 1)^2 = 9
So the equation of the circle in standard form is: (x - 4)^2 + (y - 1)^2 = 9
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brown v board of education was passed in 1954. six years later in 1960, what was the percentage of black students attending integrated schools in the south? group of answer choices 2% 40% 20% 60%
In 1960, the percentage of black students attending integrated schools in the South was approximately 2%.
Brown v Board of Education was a landmark decision in 1954 that declared segregation in public schools unconstitutional. However, it took several years for this decision to be implemented in Southern states, and in 1960, only 2% of black students in the South attended integrated schools. This highlights the resistance to desegregation in Southern states and the slow pace of change. It also underscores the ongoing struggles for civil rights and the continued efforts to address systemic racism in education and other aspects of American society.
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After watching the Quest Kickoff video, which examines the work of an astronomer who searches for life on planets outside our solar system, think about the qualities that make for a skilled astronomer. What scientific attitudes are important to the work of an astronomer such as the one in the video? Record your thoughts.
NOTEBOOK
A deep appreciation for the natural world and a desire to contribute to advancing scientific knowledge is essential for pursuing a career in astronomy.
Astronomers require a combination of scientific attitudes to excel in their work. To begin with, they need to be curious and passionate about exploring the unknown universe. They should also have a strong sense of patience, persistence, and attention to detail, given that astronomical observations can take a long time to produce results. A capacity for critical thinking, creativity, and problem-solving is also crucial in analyzing and interpreting data, developing theories, and testing hypotheses. Additionally, astronomers must be skilled in communication, as they often collaborate with colleagues, present their findings to the scientific community and the public, and engage in public outreach and education efforts.
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an international calling plan charges 45 cents per minute or fraction of a minute for each call. what is the cost for making a 5 minute call?
The cost of making a 5-minute international call using the given international calling plan is $2.25.
To calculate the cost of making a 5-minute call using unitary method, we need to determine the cost per minute of the call, and then multiply it by the number of minutes.
Given that the international calling plan charges 45 cents per minute or fraction of a minute, we can say that the cost per minute is 45 cents.
Using the unitary method, we can determine the cost of 1 minute of the call as follows
1 minute = 45 cents
5 minutes = 5 x 45 cents = 225 cents
= $2.25
Therefore, the cost of making a 5-minute call is $2.25
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What is the formula for radius and circumference?
and how do you solve it?
What is the total area of the figure shown? Arrow-shaped composite shape formed by a rectangle and a triangle. The rectangle has a length of 12 meters and a width of 9 meters. The height of the triangle is 14 meters. The rectangle has two vertices 5 meters apart from the vertices of triangle.Type the correct answer in the box. Use numerals instead of words.
So the total area of the figure is 171 square meters.
What is area?Area is a measure of the size of a two-dimensional surface or region. It is usually expressed in square units, such as square meters (m²), square centimeters (cm²), or square feet (ft²). The concept of area is used in mathematics, geometry, and various fields such as architecture, engineering, and physics.
Here,
To find the total area of the figure, we need to find the areas of the rectangle and the triangle, and then add them together. The area of the rectangle is:
A = length x width
= 12 m x 9 m
= 108 m²
The area of the triangle is:
A = (1/2) x base x height
= (1/2) x 9 m x 14 m
= 63 m²
The base of the triangle is the same as the width of the rectangle, since they are adjacent and parallel. The other side of the triangle is the hypotenuse, which can be found using the Pythagorean theorem:
a² + b² = c²
where a and b are the legs of the right triangle, and c is the hypotenuse. In this case, we can use a = 9 m (the width of the rectangle) and b = 5 m (the distance between the rectangle and the triangle):
9² + 5² = c²
81 + 25 = c²
106 = c²
c ≈ 10.3 m
Therefore, the total area of the figure is:
total area = area of rectangle + area of triangle
total area = 108 m² + 63 m²
total area = 171 m²
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A rectangular prism is 4 meters long, 5 meters wide, and has a height of 7 meters. What is its surface area?
Answer:The surface area of rectangular prism is 166 m².
Step-by-step explanation:
Given that,
length l= 4 m
Width b= 5 m
Height h=7 m
Using the formula of surface area of rectangular prism
We substitute the value into formula
Hence, The surface area of rectangular prism is 166 m².
Kody and Josh rode bikes. Kody rode 35 miles in 80 minutes. Josh rode 102 minutes at a faster rate per mile than Kody. Find Kody's unit rate in minutes per mile. Then,
explain how you could use it to find a possible unit rate for Josh.
The unit rate for Kody is minutes per mile.
The possible unit rate for Josh is 2.91 minutes per mile. However, keep in mind that this is just one possible rate and there could be other values that satisfy the conditions given in the problem.
What is unit rate?Unit rate is a rate that is simplified to have a denominator of 1.
In other words, it is a ratio that compares two measurements with one of the measurements set to 1.
For example, if a car travels 100 miles in 2 hours, the unit rate for the car's speed would be 50 miles per hour.
Kody's unit rate can be found by dividing the time taken by the distance covered:
Unit rate = Time / Distance = 80 minutes / 35 miles = 2.29 minutes per mile (rounded to two decimal places).
To find a possible unit rate for Josh, we can use the fact that he rode at a faster rate per mile than Kody. Let's say that Josh's unit rate is x minutes per mile. Since he rode for 102 minutes, we can set up the following equation:
35 miles x (minutes per mile) = 102 minutes
Solving for x, we get:
x = 102 minutes / 35 miles = 2.91 minutes per mile
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?
What is the domain of the function f (x) = 2
O A.
O B.
O C.
O D.
all real number except 9
all real numbers except 3 and -3
all positive real numbers
all real numbers except 5 and 9
The domain of a function refers to the set of all values of the independent variable (usually denoted by x) for which the function is defined.
For the function f(x) = 2, the expression is defined for all real values of x. Therefore, the domain of f(x) is all real numbers.
If the function was f(x) = 2 / (x - 9), then the expression is undefined for x = 9. Therefore, the domain of f(x) would be all real numbers except 9.
If the function was f(x) = 2 / (x - 3)(x + 3), then the expression is undefined for x = 3 and x = -3. Therefore, the domain of f(x) would be all real numbers except 3 and -3.
If the function was f(x) = sqrt(2x - 5), then the expression is only defined for values of x that make the argument of the square root non-negative. Therefore, the domain of f(x) would be all real numbers such that 2x - 5 >= 0, which simplifies to x >= 5/2.
Therefore, without more information about the function, it is impossible to determine its domain.
A cube-shaped box has a volume of 64 cubic inches. If the box is packed full of cubes with edge lengths of 1 inch, how many cubes can fit along one side of the box?
A) 2 cubes
B) 4 cubes
C) 8 cubes
D)16 cubes
Answer:
B) 4 cubes
Step-by-step explanation:
V = s³
[tex] s = \sqrt[3]{V} [/tex]
[tex] s = \sqrt[3]{64} [/tex]
[tex] s = 4 [/tex]
Estimate the area of the rectangle.
Responses
A 8 square units8 square units
B 20 square units20 square units
C 14 square units14 square units
D 10 square units
The area of the rectangle is [tex]$\sqrt{20} \times \sqrt{5} = \sqrt{100} = 10$[/tex] square units. Thus, the answer is option D: 10 square units.
What is area of rectangle?
The area of a rectangle is the measure of the region enclosed by its sides, and is calculated by multiplying the length and the width of the rectangle. The unit of area is typically expressed in square units, such as square meters (m²) or square feet (ft²).
The area of rectangle can be calculated using the distance formula to find the length and width of the rectangle, and then multiplying them. Let A(2,1), B(1,3), C(5,5), and D(6,3) be the coordinates of the rectangle's vertices.
The distance between A and B is [tex]\sqrt{(1-2)^2 + (3-1)^2} = \sqrt{1+4} = \sqrt{5}$.[/tex]
The distance between B and C is [tex]\sqrt{(5-1)^2 + (5-3)^2} = \sqrt{16+4} = \sqrt{20}$.[/tex]
So, the length of the rectangle is [tex]\sqrt{20}$ units.[/tex]
The distance between A and D is [tex]\sqrt{(6-2)^2 + (3-1)^2} = \sqrt{16+4} = \sqrt{20}$.[/tex]
The distance between C and D is [tex]\sqrt{(6-5)^2 + (3-5)^2} = \sqrt{1+4} = \sqrt{5}$.[/tex]
Therefore, the width of the rectangle is [tex]\sqrt{5}$ units.[/tex]
Hence, the area of the rectangle is [tex]$\sqrt{20} \times \sqrt{5} = \sqrt{100} = 10$[/tex] square units. Thus, the answer is option D: 10 square units.
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For what values of x and y are the triangles to the right congruent by HL?
Based on the Hypotenuse-Length Theorem of Congruence (HL), the values of x and y that will make both right triangles congruent are:
x = 8 and y = 4
Two triangles are said to be congruent if the three sides and three angles are equal in all directions.
Two right triangles are congruent to each other by the Hypotenuse-Length Congruence Theorem (HL) if the hypotenuse length and one leg of a right triangle are congruent to the hypotenuse length and leg of the other right triangle.
Thus, for both right triangles to be congruent, it means that their hypotenuse and one of their corresponding legs must be equal to each other.
Use this to create two equations and solve them to get the value of x and y.
Equal Hypotenuse:
x + y = 3y
⇒ x +4 -4 = 3y -4
⇒ x = 3y -4 ------------------------ (1)
Equal legs:
x = y +4 _______________ (2)
Find the value of y, by substituting x = (y + 4) into eqn. 1.
y + 4 = 3y - 4
⇒ -2y = -8
⇒ y = 4
Find the value of x, by substituting y = 4 into eqn. 2.
x = y + 4
⇒ x = 4 +4
⇒ x = 8
Therefore, based on the Hypotenuse-Length Theorem of Congruence (HL), the values of x and y that will make both right triangles congruent are: x = 8 and y = 4.
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