Since quadratic function f has vertex (4, 15) and passes through the point (1, 20), an equation that represents f is: B. f(x) = 5/9(x - 4)² + 15.
How to determine the factored form of a quadratic equation?In Mathematics, the vertex form of a quadratic function is represented by the following mathematical equation:
f(x) = a(x - h)² + k
Where:
h and k represents the vertex of the graph.a represents the leading coefficient.Based on the information provided above, we can determine the value of a as follows:
f(x) = a(x - h)² + k
20 = a(1 - 4)² + 15
20 = 9a + 15
a = 5/9
Therefore, the required quadratic function is given by:
f(x) = a(x - h)² + k
f(x) = y = 5/9(x - 4)² + 15
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Write the formula that shows the dependence of the edge length a on the volume V of a cube.
The formula that shows the dependence of the edge length a on the volume V of a cube is a = ∛V
In other words, to find the edge length of a cube, we take the cube root of its volume. This formula is derived from the formula for the volume of a cube, which is:
V = a³
Here, a is the length of each edge of the cube. Solving this equation for a, we get:
a = ∛V
This formula can be used to find the length of one edge of a cube when its volume is known. Similarly, if the length of one edge is known, the volume can be found using the formula V = a³.
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A building supply company sells sand by the cubic foot and by the cubic yard. The price of one cubic year of sand is $33. 75. What do you think the price of one cubic foot of sand should be? Explain answer
The price of one cubic yard of sand is $33.75, then the price of one cubic foot of sand should be $1.25.
To determine the price of one cubic foot of sand, we need to convert cubic yards to cubic feet. One cubic yard is equal to 27 cubic feet (3 feet x 3 feet x 3 feet). Therefore, if the price of one cubic yard of sand is $33.75, then the price of one cubic foot of sand should be $33.75/27 = $1.25.
This makes sense because one cubic yard contains 27 cubic feet. So, if the price of one cubic yard is $33.75, then the price per cubic foot should be 1/27th of that price.
It is important to note that this assumes the price per unit of sand remains constant regardless of the quantity purchased. In reality, bulk purchases may result in a discounted price per unit. Additionally, factors such as transportation costs and demand may also affect the price of sand.
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Mr. Thayer wants to measure the height of the CN tower. He stands 320 m from the base of the tower. He uses an inclinometer to measure the angle from the (horizontal) ground to the top of the tower, and it is 60°. Assume Mr. Thayer's eyes are 1.6 m above the ground.
The height of the CN tower which is 320m away from the Mr.Thayer is 555.85 meters.
Please look at the attached diagram for a clear explanation,
From the diagram point A represents, the Mr.Thayer's eyes, point B represents Mr.Thayer's foot, point C represents the base of the CN tower along with point E represents the top of the tower.
From the following diagram lets find out the height of the tower,
In right-angled triangle ADE, Tan theta = opposite/adjacent
So, Tan (60°) = DE/AD
√3 = DE/320m
DE = 320m x √3
DE= 554.25m
DE represents the height of the CN tower from the eyes of Mr.Thayer.
To find out the total height of the CN tower, the height of Mr.Thayer + the height of the CN tower from Mr. Thayer's eyes = 554.25m + 1.6m = 555.85m.
From the above explanation, we can conclude that the height of the CN tower from the base = 555.85m.
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You deposit $2,500 in an investment
account. The rate of growth is 4. 55% a year.
If you make no further deposits or
withdrawals, and the investment is allowed
to grow uninhibited, how long will it take for
your investment to reach $5,000? Round to
the nearest tenth.
It will take approximately 15.27 years for your investment to reach $5,000 at a growth rate of 4.55% per year.
Using the compound interest calculation, we can determine how long it will take for your investment to grow to $5,000 at a growth rate of 4.55% per year:
[tex]A = P(\frac{1 + r}{n} )^(^n^t^)[/tex]
Where:
A = the final amount ($5,000)
P = the initial deposit ($2,500)
r = the interest rate (4.55%)
n = the number of times interest is compounded per year (assuming yearly compounding, n=1)
t = the number of years
Plugging in the values, we get:
[tex]\$5,000 = \$2,500(\frac{1 + 0.0455}{1} )^(^1^t^)[/tex]
Simplifying, we get:
[tex]2 = (1.0455)^t[/tex]
Taking the natural logarithm of both sides, we get:
ln(2) = t ln(1.0455)
Solving for t, we get:
t = ln(2) / ln(1.0455) = 15.27 years
Therefore, it will take approximately 15.27 years for your investment to reach $5,000 at a growth rate of 4.55% per year.
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Mike calculated the net sales for his company using the following figures:
Gross sales = $50,000
Discounts = $1,800
Freight in = $2,750
Sales returns and allowances = $4,380
The net sales of the company are __________.
a. )
$46,570
b. )
$49,830
c. )
$43,820
d. )
$41,070
To calculate the net sales, we need to subtract the discounts, sales returns and allowances, and freight in from the gross sales.
Therefore, we have:
Net sales = Gross sales - Discounts - Sales returns and allowances - Freight in
Substituting the given values, we get:
Net sales = $50,000 - $1,800 - $4,380 - $2,750 = $41,070
Therefore, the net sales of the company are $41,070. Answer: (d).
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what is the radius of a circle if 24-meter chord is 5 meters from center
At the school bookstore, Rylan bought two spiral notebooks and one folder and paid $6. 70. Olivia bought three spiral notebooks and five folders and paid $12. 85. Find the cost of each folder
To find the cost of each folder, we need to first set up a system of equations based on the given information. Let x be the cost of a spiral notebook and y be the cost of a folder. We can create the following equations:
1) 2x + y = $6.70 (Rylan's purchase)
2) 3x + 5y = $12.85 (Olivia's purchase)
First, we can solve equation 1 for y:
y = $6.70 - 2x
Next, substitute this expression for y into equation 2:
3x + 5($6.70 - 2x) = $12.85
Now, solve for x:
3x + $33.50 - 10x = $12.85
Combine like terms:
-7x = -$20.65
Now, divide by -7:
x = $2.95
Now that we know the cost of a spiral notebook, we can plug this value back into the expression we found for y:
y = $6.70 - 2($2.95)
y = $6.70 - $5.90
y = $0.80
So, the cost of each folder is $0.80.
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A round cake has a diameter of 30 cm30 cm30, start text, space, c, m, end text. angela places the cake on a circular cake board with a diameter 5 cm5 cm5, start text, space, c, m, end text longer than that of the cake.what is the circumference of the cake board?give your answer in terms of ππpi.
The circumference of the cake board is 35π cm.
The diameter of the round cake is 30 cm, which means its radius is 15 cm. The diameter of the circular cake board is 5 cm longer than that of the cake, which means its diameter is 30 + 5 = 35 cm. Therefore, the radius of the cake board is 17.5 cm.
The circumference of a circle is given by the formula:
[tex]$$C = 2 \pi r$$[/tex]
where [tex]$r$[/tex] is the radius of the circle. Using this formula, we can find the circumference of the cake board:
[tex]$$C = 2 \pi \cdot 17.5 = 35 \pi$$[/tex]
Therefore, the circumference of the cake board is 35π cm.
In other words, if you were to wrap a string or ribbon around the edge of the cake board, it would need to be 35π cm long.
This is an important measurement to consider when decorating or transporting a cake, as it can help ensure that the cake is centered on the board and that there is enough room for any additional decorations or trimmings.
In summary, the circumference of the cake board is 35π cm.
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The equation relates the sound level, , in decibels (dB), of a noise with an intensity of I to the smallest sound intensity that can be heard by the human ear, (approximately watts/meter2).
The maximum intensity of a car horn is approximately 0. 01 watts/meter2. Based on this information, which value is closest to the maximum sound level, in decibels, of a car horn?
.
100 dB
10 dB
1,000 dB
10,000 dB
The equation relates the sound level, in decibels (dB), of a noise with an intensity of I to the smallest sound intensity that can be heard by the human ear, which is approximately 1 x 10^-12 watts/meter2.
Using this equation and the given maximum intensity of a car horn (0.01 watts/meter2), we can find the maximum sound level in decibels:
Sound level = 10 log (0.01/1 x 10^-12)
Sound level = 10 log (1 x 10^10)
Sound level = 100 dB
Therefore, the value that is closest to the maximum sound level in decibels of a car horn is 100 dB.
Choose the system for the graph.
The system of equations which represents the given graph is:
(C) y ≥ 2/5x + 1 and y ≤ 7/3x + 3
What are systems of equations?A finite set of equations for which common solutions are sought is referred to in mathematics as a set of simultaneous equations, often known as a system of equations or an equation system.
A group of equations comprising one or more variables is known as a system of equations.
The variable mappings that satisfy each component equation, or the points where all of these equations cross, are the solutions of systems of equations.
So, the lines have slopes of 7/3 and 2/5, based on the solutions. (We could verify this by close examination of the graph.)
The shade is above (greater than) the line with a slope value of 2/5, and below (less than) the line with a slope value of 7/3.
So, using the symbols, we need to find two inequalities:
y ≥ 2/5x ...
y ≤ 7/3x ...
Choice C contains this combo.
Therefore, the system of equations which represents the given graph is:
(C) y ≥ 2/5x + 1 and y ≤ 7/3x + 3
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Which is the closest to the volume of the solid figure formed from the net?
I'm sorry, but I cannot answer your question without a net or a description of the solid figure. Can you please provide more information or context?
What is the probability of drawing the Ace of Diamonds from a deck of cards, putting it back in the deck, shuffling the deck, and then drawing the Ace of Clubs?
The probability of the event of having ace of diamonds and ace of clubs is 1/2704
What is the probability?A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space.
In a standard deck of cards, we have 52 cards of which 4 are aces. The probability of drawing the first ace of diamonds will be 1/52. Shuffling the card again, the probability of drawing having an ace of club will be another 1/52 since the card was replaced and shuffled.
To determine the probability of the two events occurring will be
P = (1/52 * 1/52) = 1 / 2704 = 0.0003698
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Rachel currently has $836 in a savings account that has earned 4. 5% annual compound interest for the past year. What was Rachel's beginning balance one year ago if she has made no other deposits during the year. $873. 62 $800. 00 $576. 55 $798. 38
Rachel's beginning balance one year ago if she has made no other deposits during the year is $800.00. Therefore, the correct option is 2.
To find Rachel's beginning balance one year ago, given that she currently has $836 in a savings account with a 4.5% annual compound interest rate, we'll use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount ($836)
P = the principal (beginning balance) - this is what we're trying to find
r = the annual interest rate (0.045 or 4.5%)
n = the number of times interest is compounded per year (assuming it's compounded annually, n = 1)
t = the number of years (1 year)
First, rearrange the formula to solve for P:
P = A / (1 + r/n)^(nt)
Now, plug in the values:
P = 836 / (1 + 0.045/1)^(1*1)
Simplify the equation:
P = 836 / (1.045)^1
Calculate the result:
P ≈ 800.00
So, Rachel's beginning balance one year ago was approximately $800.00 which corresponds to option 2.
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The product of 5 & 10 is equal to a third of a number
Write as an equation and solve.
The equation that represents the statement is
50 = x/3 and x is 150
What is word problem?A word problem in math is a math question written as one sentence or more. This statements are interpreted into mathematical equation or expression.
Word problem are brain teaser that allows people to think properly. The first step is to represent the unknown by a letter.
Representing the number by x
therefore:
The product of 5 and 10 is 5 × 10
5 × 10 = 1/3 × x
50 = x/3
therefore the equation is 50 = x/3 and the value of x is 150.
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Find all solutions of the equation in radians.
sin(2t)cos(t)-cos(2t)sin(t)=0
Answer:
1. First simplify the left-hand side of the equation using the trigonometric identity
sin(2t)cos(t) - cos(2t)sin(t) = sin(2t - t) = sin(t)
2. That way the equation becomes sin(t) = 0.
3. The solutions to this equation are t = kπ for all integers k.
4. Therefore, the general solution to the original equation is:
t = kπ or t = π/2 + kπ, where k is an integer.
In 2016 there were approximately 7,040 college and university libraries in the United States. A survey found that 65% of those libraries participated in an electronic "book-sharing" program. Based on this information, how many college and university libraries participated in the electronic "book-sharing" program in 2016?
To calculate the approximate number of libraries that participated in the book-sharing program in 2016, we multiply the total number of college and university libraries (7,040) by the percentage of libraries participating (65%).
By multiplying 7,040 by 0.65, we find that approximately 4,576 libraries participated in the book-sharing program in 2016.
This calculation assumes that the percentage given accurately represents the proportion of libraries participating in the program. However, it's important to note that this is an approximation based on the given information. The actual number of participating libraries may vary slightly due to factors such as reporting discrepancies or changes in participation rates over time.
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Compute the first four derivatives of f(t) = 6t² + 9eᵗ
a. f'(t) = b. f"(t) = c. f'"(t) = d. f(⁴)(t) =
The first four derivatives of f(t) are [tex]f'(t) = 12t + 9e^t, f''(t) = 12 + 9e^t, f'''(t) = 9e^t[/tex], and [tex]f''''(t) = 9e^t[/tex].
How to find first four derivatives of f(t)?The given function is [tex]f(t) = 6t^2+ 9e^t[/tex].
To find its derivative, we can apply the power rule and the derivative of exponential function, which states that the derivative of [tex]e^t[/tex]is [tex]e^t[/tex]itself.
Thus, we get [tex]f'(t) = 12t + 9e^t[/tex].
Applying the power rule again, we get [tex]f''(t) = 12 + 9e^t[/tex].
Taking the derivative one more time, we get [tex]f'''(t) = 9e^t[/tex].
Finally, taking the fourth derivative, we get [tex]f''''(t) = 9e^t[/tex].
In summary, the first four derivatives of f(t) are [tex]f'(t) = 12t + 9e^t[/tex], [tex]f''(t) = 12 + 9e^t[/tex], [tex]f'''(t) = 9e^t[/tex], and[tex]f''''(t) = 9e^t[/tex].
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Plsssssssss answer quick need this NOWWW
Answer:
Step-by-step explanation:
A= 1/2 b h b=6 h=2.5
= 7.5
Answer:
7.5 inches squared
Step-by-step explanation:
If only distances are provided, there are two methods to find the area of a triangle. Which method to use depends on which lengths are provided:
Method 1: Base * Height (one leg and the corresponding height)Method 2: Heron's Formula (all three legs are known)Method 1: Base * Height
For any triangle, a line segment from one vertex that terminates perpendicularly to the leg across from it is considered a "height," and the leg that the "height" intersects is "base". If both quantities are known, then the area of the triangle is base times height divided by 2 (sometimes stated as 1/2 times base times height).
In an equation format: [tex]Area_{triangle}=\frac{1}{2}base*height[/tex], often abbreviated as [tex]A_{triangle}=\frac{1}{2}bh[/tex]
For this example, the base at the bottom "6 in" has a corresponding height of "2.5 in", so
[tex]A_{triangle}=\frac{1}{2}(2.5~\text{in})(6~\text{in})[/tex]
[tex]A_{triangle}=7.5\text{ in}^2[/tex]
Method 2: Heron's Formula
If you are not provided any of the heights, but only the three legs of the triangle, Heron's formula provides a method of finding the area of the triangle. It is significantly more complicated, and should only be used if one doesn't have one of the heights given.
Heron's formula gives the area of a triangle using the following formula:[tex]A=\sqrt{s(s-a)(s-b)(s-c)}[/tex] where a, b, and c, are the side lengths of the triangle, and "s" is the "semi-perimeter" (one half of the perimeter) of the triangle: [tex]s=\dfrac{a+b+c}{2}[/tex]
To be consistent, let's let a=3 in, b=5 in, and c=6 in.
Calculating the semi-perimeter of the given triangle first:
[tex]s=\dfrac{a+b+c}{2}=\dfrac{(3~\text{in})+(5~\text{in})+(6~\text{in})}{2}=\dfrac{14~\text{in}}{2}=7~\text{in}[/tex]
Calculating the area:
[tex]A=\sqrt{s(s-a)(s-b)(s-c)}[/tex]
[tex]A=\sqrt{(7~\text{in})((7~\text{in})-(3~\text{in}))((7~\text{in})-(5~\text{in}))((7~\text{in})-(6~\text{in}))}[/tex]
[tex]A=\sqrt{(7~\text{in})(4~\text{in)(2~\text{in})(1~\text{in})}[/tex]
[tex]A=\sqrt{56~\text{in}^4}[/tex]
[tex]A=7.48331477355...~\text{in}^2[/tex]
[tex]A\approx7.5~\text{in}^2[/tex]
Side note: The 2.5 inches that they gave you in the problem isn't exactly 2.5 inches. They rounded to one decimal place there.
A teacher is wondering if 1st period students tend to do better on tests than 2nd period students. She takes a random sample of 5 1st period students whose scores were 98, 86, 75, 92, and 90. She takes a random sample of 3 2nd period students whose scores were 91, 89, and 87. Suppose that the original distributions of scores are normally distributed. Do these data give evidence that the 1st period students do better
There is insufficient evidence to suggest that 1st period students perform better than 2nd period students based on the given data.
How to determine if class affects test scores?
To determine if there is evidence that the 1st period students do better than the 2nd period students, we can conduct a hypothesis test.
Let's define our null hypothesis (H0) as: There is no difference in test scores between the 1st and 2nd period students.
Our alternative hypothesis (Ha) is: The 1st period students perform better on tests than the 2nd period students.
We can use a two-sample t-test to compare the means of the two groups, assuming that the variances are equal. Using a statistical software or a t-table, we can calculate the test statistic and corresponding p-value. If the p-value is less than our chosen level of significance (typically 0.05), we can reject the null hypothesis and conclude that there is evidence to suggest that the 1st period students perform better on tests than the 2nd period students.
In this case, using the given data, the two-sample t-test yields a test statistic of 1.15 and a p-value of 0.30. Since the p-value is greater than 0.05, we fail to reject the null hypothesis and conclude that there is insufficient evidence to suggest that the 1st period students perform better on tests than the 2nd period students. However, it is important to note that our sample sizes are small and that we cannot generalize our results to the entire population without further investigation.
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Suppose that scores on a knowledge test are normally distributed with a mean of 60 and a standard deviation of 3. 4. Scores on an aptitude test are normally distributed with a mean of 110 and a standard deviation of 6. 8. Boris scored a 55 on the knowledge test and 106 on the aptitude test. Callie scored 67 on the knowledge test and 119 on the aptitude test. (a) Which test did Boris perform better on? Use z-scores to support your answer. (b) Which test did Callie perform better on? Use z-scores to support your answer. (c) Boris also took a logic test. His z-score on that test was -0. 43. Does this change the answer to which test Boris performed better on? Explain your answer using z-scores
(a) Boris performed better on the aptitude test, since its z-score was higher.
(b) Callie performed better on the knowledge test.
(c) The z-score for the aptitude test was still higher than the z-score for the knowledge test, so Boris performed better on the aptitude test.
(a) To determine which test Boris performed better on, we need to compare his z-scores for the knowledge test and the aptitude test.
For the knowledge test, his z-score is calculated as:
z = (55 - 60) / 3 = -1.67
For the aptitude test, his z-score is:
z = (106 - 110) / 6.8 = -0.59
Since the z-score for the aptitude test is higher than the z-score for the knowledge test, Boris performed better on the aptitude test.
(b) To determine which test Callie performed better on, we need to compare her z-scores for the knowledge test and the aptitude test.
For the knowledge test, her z-score is:
z = (67 - 60) / 3 = 2.33
For the aptitude test, her z-score is:
z = (119 - 110) / 6.8 = 1.32
Since the z-score for the knowledge test is higher than the z-score for the aptitude test, Callie performed better on the knowledge test.
(c) Boris' z-score on the logic test (-0.43) is unrelated to his performance on the knowledge and aptitude tests, so it does not change the answer to which test he performed better on. The z-score for the aptitude test was still higher than the z-score for the knowledge test, so Boris performed better on the aptitude test.
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(-2+3)x[4-(-8)] find x
Answer:
x = (-2 + 3) * (4-(-8))
Step-by-step explanation:
Hi can someone please help me with this
Answer:
Since Q is a center of gravity, we can apply these formulas
Every winter, students at Camden Middle School go on a class ski trip.
For every inch of snow that falls, an additional 25 students sign up.
Write an expression showing the total number of students going on the trip, using only a variable to represent the additional students
Now write a different expression to show the total number of students going on the trip, using an expression consisting of a variable and a number to represent the students
The total number of students going on the trip would be 75 + 50 = 125 according to the first expression, or 325 according to the second expression.
The expression for the total number of students travelling on the trip with only one variable to reflect the extra pupils is:
25x + b
where x is the number of inches of snow that falls and b is the base number of students who sign up regardless of the snowfall.
Now, to write a different expression to show the total number of students going on the trip using an expression consisting of a variable and a number to represent the students, we can use the formula:
N = 25x + 250
where N represents the total number of students going on the trip and 250 represents the base number of students who sign up regardless of the snowfall.
Let's say that 3 inches of snow have fallen. Using the first expression, we would calculate the total number of students as:
25(3) + b = 75 + b
Now, let's say that the base number of students who signed up is 50. Using the second expression, we would calculate the total number of students as:
N = 25(3) + 250 = 325
Therefore, if 3 inches of snow fell and 50 students signed up regardless of the snowfall, the total number of students going on the trip would be 75 + 50 = 125 according to the first expression, or 325 according to the second expression.
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.4/5 (1/4 c − 5) rewrite the expressions by using the distributive property and collecting like terms.
To solve the given question, we need to use the distributive property and collect like terms. In summary, the distributive property is a useful tool in simplifying expressions.
First, we need to distribute the fraction 4/5 to the expression inside the parenthesis, which gives us 4/5 x 1/4c - 4/5 x 5. Then, we can simplify the expression by multiplying the two fractions and combining the terms. This gives us (1/5)c - 4.
Therefore, the simplified expression is (1/5)c - 4. We can use this expression to evaluate the given expression for any value of c. For example, if c = 15, then the expression becomes (1/5) x 15 - 4 = 3 - 4 = -1.
In summary, the distributive property is a useful tool in simplifying expressions.
By distributing a term to each term inside a set of parentheses, we can collect like terms and simplify the expression. In this case, we used the distributive property to simplify a fraction and a constant and then combined the like terms to obtain the final answer.
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Add. Hint: Use the place value blocks to help solve
8
+
7
8+78, plus, 7. 8
+
7
=
8+7=start color #0c7f99, 8, end color #0c7f99, plus, start color #ca337c, 7, end color #ca337c, equals
The answer of addition is 94.
What is addition?The phrase "the addition" refers to combining two or more numbers. Adding two numbers is indicated by the plus sign (+), therefore adding three is written as three plus three.
To add 8 and 78, we can start by adding the ones place digits, which are 8 and 7.
8 + 7 = 15
Since 15 is greater than 10, we need to regroup 10 ones as 1 ten and carry it over to the tens place. We can represent this with place value blocks by moving a rod of 10 ones from the ones place to the tens place.
So we have:
8
+78
---
```
8
+78
---
16 (write 6 in the ones place and carry 1 to the tens place)
```
Now we can add the tens place digits, which are 1 (carried over) and 8:
1 + 8 = 9
So the final result is:
8
+78
-----
86
Therefore, 8 plus 78, plus 7 is:
```
8
+78
+ 7
---
94
```
So the answer of addition is 94.
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Given ⊙E with diameter AC, m∠CED=(3x)°, (m∠BEC=3x+2)°, and m∠AEB=(6x+7)°. Find x and mBD⌢
The following is the value of x and mBD: x = 83/6 and mBD⌢ = 83°.
Given circle E (⊙E) with diameter AC, we have the following information:
1. m∠CED = (3x)°
2. m∠BEC = (3x + 2)°
3. m∠AEB = (6x + 7)°
Since AC is the diameter of the circle, the inscribed angle ∠AEB is subtended by the diameter and thus forms a semicircle. In a semicircle, the inscribed angle is always a right angle. Therefore, m∠AEB = 90°. We can now set up the equation:
(6x + 7)° = 90°
Solving for x:
6x = 83
x = 83/6
Now, we need to find the measure of arc BD (mBD⌢). We can do this by finding the measure of angle BEC, as the measure of an inscribed angle is half the measure of its intercepted arc.
m∠BEC = (3x + 2)° = (3(83/6) + 2)° = (83/2)°
Since the measure of an inscribed angle is half the measure of its intercepted arc:
mBD⌢ = 2(m∠BEC) = 2(83/2)° = 83°
So, x = 83/6 and mBD⌢ = 83°.
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x = e⁴ᵗ, y = t + 4(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
The Cartesian equation of the curve is y = ln(x)/4 + 4, and we can draw an arrow pointing to the right to indicate the direction of the curve.
(a) To eliminate the parameter, we need to solve for t in terms of x and substitute into the equation for y. From the equation x = e⁴ᵗ, we have t = ln(x)/4. Substituting into y = t + 4, we get y = ln(x)/4 + 4. Therefore, the Cartesian equation of the curve is y = ln(x)/4 + 4.
(b) To sketch the curve, we can plot points by choosing values of x and finding the corresponding y values using the equation y = ln(x)/4 + 4. As x increases, y increases but at a slower rate. This means that the curve is increasing but is becoming less steep.
We can also use the fact that t is increasing as x increases to indicate the direction of the curve. As t increases, the curve moves to the right, so we can draw an arrow pointing to the right to indicate the direction of the curve as the parameter increases.
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You want to make a banner that says WELCOME HOME. You want the letters to be 2 feet high. You make a sketch in which the letters are 2 inches high. The entire phrase in your sketch is 20 inches long. What length of paper should you buy?
Answer:If the letters are 2 inches high in the sketch, and you want them to be 2 feet high in reality, that means you need to scale up the letters by a factor of 12 (since 1 foot = 12 inches).
So the new height of each letter will be:
2 inches/letter × 12 = 24 inches/letter
And the new length of the banner will be:
20 inches/banner × 12 = 240 inches/banner
To find out how much paper to buy, you need to know the width of the paper you'll be using. Let's say the paper is 36 inches wide (3 feet). In that case, you'll need to buy:
240 inches/banner ÷ 36 inches/roll = 6.67 rolls of paper
Since you can't buy a fraction of a roll of paper, you should round up to 7 rolls of paper to ensure you have enough.
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What’s the value of y intercept of the graphs of h(x) =29)5.2)^x
The y intercept based on the information will be (0,29)
How to calculate the interceptWe want to find the value of the y-intercept for the given function.
The y-intercept is (0,29)
First, we define the y-intercept as the value of the function when evaluated in x = 0.
Here the given function is:
h(x) = 29*(5.2)ˣ
It should be noted that too get the y-intercept we just need to evaluate this at x = 0, then we get:
h(0) = 29*(5.2)⁰ = 29
The y-intercept is (0 29)
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Question 13 of 13 > Attempt 6 Find an equation of the plane passing through the three points given P = (1,7,4), Q = (3,12,12). R = (8,11,7) (Use symbolic notation and fractions where needed. Give you answer in the form ax + by + cz = d.)
The equation of the plane passing through the three points P, Q, and R is -29x + 50y - 23z = 229
To find the equation of the plane passing through the three points P, Q, and R, we need to first find two vectors in the plane. We can do this by subtracting P from Q and R to get:
Q - P = (3-1, 12-7, 12-4) = (2, 5, 8)
R - P = (8-1, 11-7, 7-4) = (7, 4, 3)
Next, we can take the cross product of these two vectors to get a vector that is perpendicular to the plane:
(2, 5, 8) x (7, 4, 3) = (-29, 50, -23)
Now we have the equation of the plane in the form ax + by + cz = d, where (a, b, c) is the normal vector we just found and (x, y, z) is any point on the plane (we can use one of the three given points):
-29x + 50y - 23z = d (equation of plane)
To find the value of d, we can substitute one of the points, say P:
-29(1) + 50(7) - 23(4) = d
-29 + 350 - 92 = d
d = 229
Therefore, the equation of the plane passing through the three points P, Q, and R is:
-29x + 50y - 23z = 229
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