While the equation x² = 49 suggests that x = 7, it also has a negative solution, x = -7. Both solutions can be verified by substitution into the original equation, proving they are correct through resulting identities.
The equation x² = 49 has two solutions in the realm of real numbers, not just x = 7. To find the values of x that satisfy the equation, we take the square root of both sides. Since the square root of a number has both a positive and a negative value, the solutions are x = 7 and x = -7. To verify these solutions, we can substitute them back into the original equation to confirm that they work, demonstrating identities such as 7² = 49 and (-7)² = 49.
What information is found in the left column of a two column proof?
Tina’s grandfather put $1000 in the bank for her when she was born. The account has been earning 5.85% interest compounded annually. Tina is now 18 years old. How much money is in the account?
part of the proceeds from a garage sale was $510 worth of $10 and $20 bills. if there were only 6 more $10 bills than $20, find the number for each denomination
A worksheet has an area of 48 square inches. Its perimeter is 28 inches. What are the dimensions of the worksheet?
convert 0.62 into a percent
There are approximately 2.2 million births in a country each year. Find the births per minute.
What’s a culprit??$$
I need help ASAP!!
What is 92 rounded to the nearest yard??
5 3/4 in a improper fraction?
Find constants a and b such that the function y = a sin(x) + b cos(x) satisfies the differential equation y'' + y' − 7y = sin(x).
The constants a = 1 and b = 1 satisfy the differential equation y'' + y' - 7y = sin(x).
Explanation:We can find the constants a and b by comparing the coefficients of the trigonometric functions in the given differential equation with the functions in the function y = a sin(x) + b cos(x).
In the differential equation, the coefficient of y'' is 1, the coefficient of y' is 1, and the coefficient of y is -7. In the function y = a sin(x) + b cos(x), the coefficient of sin(x) is a and the coefficient of cos(x) is b.
So, we equate the coefficients:
1 = a1 = b-7 = 0 (from the term -7y in the differential equation)Therefore, the constants a = 1 and b = 1 satisfy the differential equation.
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The value of a is [tex]\frac{-8}{\ 65}[/tex] and value of b is [tex]\frac{-1}{\ 65}[/tex].
Given the function (y = a sin(x) + b cos(x)), we need to find constants a and b such that the function satisfies the differential equation:[tex]\frac{-1}{\ 65}[/tex]
y'' + y' − 7y = sin(x)
First, find the first derivative y':
y' = a cos(x) - b sin(x)
Next, find the second derivative y'':
y'' = -a sin(x) - b cos(x)
Substitute y, y', and y'' into the differential equation:
(-a sin(x) - b cos(x)) + (a cos(x) - b sin(x)) − 7(a sin(x) + b cos(x)) = sin(x)
Combine like terms:
(-a − 7a)sin(x) + (a cos(x)) + (-b - 7b)cos(x) - b sin(x) = sin(x)
Simplify the equation:
-8a sin(x) + a cos(x) + (-8b cos(x)) - b sin(x) = sin(x)
Separate the equation into two parts, one for sin(x) and one for cos(x):
(-8a - b)sin(x) + (a - 8b)cos(x) = sin(x)
Match coefficients to make equal:
-8a - b = 1a - 8b = 0Solve the system of equations:
From a - 8b = 0, we get a = 8bSubstitute a = 8b into -8a - b = 1:-8(8b) - b = 1 -64b - b = 1-65b = 1[tex]b = \frac{-1}{\ 65}[/tex]Then a = 8b[tex]a = 8(\frac{-1}{\ 65})[/tex]The constants are [tex]a = \frac{-8}{\ 65}[/tex] and [tex]b = \frac{-1}{\ 65}[/tex].
The angle bisector of ∠ACD in rhombus ABCD makes a 64° angle with the diagonal BD . Find the measure of ∠BAD.
In a rhombus, the diagonals are perpendicular bisectors of each other, meaning they divide each other into two equal parts at a right angle. Let's denote the measure of ∠BAD as x degrees. The measure of ∠BAD is [tex]\( 26^\circ \)[/tex].
Since the angle bisector of ∠ACD makes a 64° angle with the diagonal BD, and BD bisects ∠ACD, then ∠ACB = 64° as well. Additionally, since ABCD is a rhombus, opposite angles are equal. Therefore, ∠ABD = ∠ACB = 64°.
Now, considering triangle ABD, we know that the sum of angles in a triangle is 180°. Thus:
[tex]\[ \angle BAD + \angle ABD + \angle ADB = 180^\circ \][/tex]
Substituting the known values:
[tex]\[ x + 64^\circ + 90^\circ = 180^\circ \][/tex]
[tex]\[ x + 154^\circ = 180^\circ \][/tex]
[tex]\[ x = 180^\circ - 154^\circ \][/tex]
[tex]\[ x = 26^\circ \][/tex]
So, the measure of ∠BAD is [tex]\( 26^\circ \)[/tex].
To summarize:
1. The angle bisector of ∠ACD in the rhombus makes a 64° angle with diagonal BD.
2. By the properties of a rhombus, ∠ACB = 64° as well.
3. Since opposite angles in a rhombus are equal, ∠ABD = ∠ACB = 64°.
4. In triangle ABD, the sum of angles is 180°.
5. Solving for ∠BAD, we find [tex]\( 26^\circ \).[/tex]
a rectangular field has dimensions of (2x-3) yards and (x+4) yards. the area of the field is 285 square yards. what are the length and width of the field?
The spiral in the picture is made from 45° right triangles. The smallest triangle (red) has a side length of 1. You can use the Pythagorean theorem to find the length of its hypotenuse. The next (orange) triangle has a side length exactly matching the hypotenuse of the smallest triangle, and so on with the larger triangles of other colors. The lengths of the hypotenuses of all the triangles (starting with the smallest) make a geometric sequence. What is the ratio of the sequence?
Which is the measure of the reference angle for 227 degrees
Amanda has a piece of ribbon that is 1/4 yard long. She wants to share ribbon with two friends so they can wear a ribbon for breast cancer awareness month. If each of the 3 friends get the same amount, how much ribbon will each person get?
Final answer:
To share the ribbon equally among the 3 friends, each person will get 1/12 yard of ribbon.
Explanation:
To determine how much ribbon each person will get, we need to divide the total length of the ribbon equally among all friends.
1/4 yard of ribbon divided by 3 friends gives each person 1/12 yard of ribbon.
Therefore, each person will get 1/12 yard of ribbon for breast cancer awareness month.
The coordinates (5, 2) and (–6, 2) are vertices of a hexagon. Explain how to find the length of the segment formed by these endpoints. How long is the segment?
The length of the segment formed by the two given endpoints with coordinates coordinates (5, 2) and (–6, 2) is;
d = 11
Formula for distance between two coordinates is given as;
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
We are given the coordinates (5, 2) and (–6, 2).
This means that;
x₁ = 5 and x₂ = -6
y₁ = 2 and y₂ = 2
Thus;
d = √[(-6 - 5)² + (2 - 2)²]
d = √(121 + 0)
d = √121
d = 11
In conclusion, the length of the segment formed by the two given endpoints is; d = 11
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The chemical formula for lithium phosphide is Li3P. Which best describes this ionic compound?
When lithium and phosphorus combine to form lithium phosphide, three lithium cations (Li+) combine with one phosphide anion (P₃-) to balance the charges and form a neutral compound, Li₃P.
What is Ionic compound?an ionic compound is a chemical compound composed of ions held together by electrostatic forces termed ionic bonding.
The correct description of the ionic compound lithium phosphide, with the chemical formula Li3P, is "Lithium ions have 1 positive charge, and phosphorus ions have 3 negative charges."
This is because lithium is a Group 1A element, which means it has one valence electron that it tends to lose in chemical reactions to form a cation with a charge of +1.
Phosphorus, on the other hand, is a Group 5A element, which means it has five valence electrons and tends to gain three electrons to form an anion with a charge of -3.
Thus, When lithium and phosphorus combine to form lithium phosphide, three lithium cations (Li+) combine with one phosphide anion (P₃-) to balance the charges and form a neutral compound, Li₃P.
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A company paid $48 for 2 cases of printer paper. Each case contains 12 packages of paper. Next month the company’s office manager needs to order 180 packages of the same paper. If the price per package does not change, what would be the total cost of next month’s order?
Given that Rectangle MNOP ~ rectangle STUV, what is the length of TU?
Answer:
The measure of TU is 15.
Step-by-step explanation:
mr samuel buys a sheet of stamps. there are 4 rows with 7 stamps in each row. how many stamps does mr samuel's buy?
convert the following equation into standard form
y= 3/14 - x/7
Answer:
[tex]\displaystyle 2x + 14y = 3[/tex]
Step-by-step explanation:
y = 3⁄14 - ⅐x
+ ⅐x + ⅐x
___________
⅐x + y = 3⁄14 ► [Multiply the equation by 14 to eradicate the fraction.]
14[⅐x + y = 3⁄14] → [tex]\displaystyle 2x + 14y = 3[/tex]
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the radius of a quarter circle is 1 mile. What is the quarter circle's perimeter?
In a region, there is a 0.95 probability chance that a randomly selected person of the population has brown eyes. Assume 13 people are randomly selected. Find the probability that the number of selected people that have brown eyes is 11 or more.
karla has read 85% of her book, which amounts to 238 pages. how long is the book?
Karla's book is 280 pages long because 85% of the book represents 238 pages, which means the entire book is 100% of that amount calculated by dividing 238 by 0.85. The additional questions can't be addressed accurately without more context or information.
Explanation:If Karla has read 85% of her book and that represents 238 pages, we can calculate the total length of the book. To find the full length of the book, we need to determine 100% of the book when 85% equals 238 pages. We can set up a proportion or use the formula Part = Percent \times Whole to solve for the whole, which is the total length of the book.
Let's use the formula to find the total length of the book:
First, we express 85% as a decimal, which is 0.85.Next, we rearrange the formula to solve for Whole: Whole = Part / Percent.Substitute the given part (238 pages) and the percent (0.85) into the formula: Whole = 238 / 0.85.When we calculate that, we get the total length of the book: Whole = 238 / 0.85 = 280 pages.Therefore, the total length of Karla's book is 280 pages.
Regarding the additional questions:
Without any calculations, it's not likely that the professor will need to read a total of more than 1,050 pages if we are only referencing the pages that Karla has read. However, if we are considering a separate piece of text or assignment for the professor, a calculation would be necessary to determine the likelihood.
The idea of dividing the total pages by the number of available days is a good strategy for managing reading assignments. This technique helps to break down the task into manageable daily portions and allows better time management.
As for the probability that the professor will need to read more than 1,050 pages, we cannot calculate this without additional information about the total number of assignments and the length of each assignment. Moreover, it is unlikely that the average length of the papers will be less than 12 pages if we consider typical lengths for academic papers, which often exceed this length.
The expression 4i gives the perimeter of a square with a side lengh of i units .
what is the perimeter of a square with a side lenth of 5/7 units ?
tossing a number cube numbered from 1 to 6 and getting an even number that is greater than or equal to 2
If two fair dice are rolled, find the probability that the sum of the dice is 77, given that the sum is greater than 33.
NEEEDDDDD HELPPPPP ASAPPPPP PLZZZZ!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Hector is training for a race that is 10 kilometers long. On Saturday he ran 2.5 km. He plans to add 0.5 km to his distance every Saturday. Hector's Calculations: 2.5 + 0.5 = 3.0 6.5 + 0.5 = 7.0 3.0 + 0.5 = 3.5 7.0 + 0.5 = 7.5 3.5 + 0.5 = 4.0 7.5 + 0.5 = 8.0 4.0 + 0.5 = 4.5 8.0 + 0.5 = 8.5 4.5 + 0.5 = 5.0 8.5 + 0.5 = 9.0 5.0 + 0.5 = 6.0 9.0 + 0.5 = 9.5 6.0 + 0.5 = 6.5 9.5 + 0.5 = 10.0 The equation equivalent to Hector’s calculations is 2.5 + 0.5x = 10. Use either of these methods to determine how many weeks it will take Hector to reach his goal of a 10 kilometer run.
The number of weeks it will take Hector to reach his goal of a 10-kilometer run will be 15 weeks.
What is a linear equation?A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
Hector is training for a race that is 10 kilometers long. On Saturday he ran 2.5 km. He plans to add 0.5 km to his distance every Saturday.
Let 'x' be the number of weeks. Then the equation is given as,
2.5 + 0.5x = 10
0.5x = 7.5
x = 15
The number of weeks it will take Hector to reach his goal of a 10-kilometer run will be 15 weeks.
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Here is the histogram of a data distribution. What is the shape of this distribution?
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Which of the following functions is graphed below?!?
The given graph is for a piecewise function consisting of two intervals:
(1) x ≤ 1 ⇒⇒⇒ the graph in this interval like a cubic function.
(2) x > 1 ⇒⇒⇒ the graph in this interval like a quadratic function
By comparing to the given options
The answer is option c.
Answer:
Option C is correct.
Explanation:
A piece-wise function is a function which is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain.
From the graph we can see that the function is made up of 2 pieces.
Solid dot means- Including
Open-dot means - not- including.
As we can see that the boundary lines are at x=1.
We have to look each line separately to determine each equation:
when [tex]x>1[/tex], it gives the function [tex]y=x^2+4[/tex] and
when [tex]x\leq 1[/tex], it gives the function [tex]y =x^3-2[/tex].
So, the piece-wise function is:
[tex]y=\left \{ {{x^3-2}, x\leq1 \atop {x^2+4}, x>1} \right.[/tex]