Using the discriminant to know about the nature of the solution of the quadratic equation x² -x = -1/4 tells us the fact as given by: Option c. one real solution
How to use discriminant to find the property of solutions of given quadratic equation?Let the quadratic equation given be of the form [tex]ax^2 + bx + c = 0[/tex], then
The quantity [tex]b^2 - 4ac[/tex] is called its discriminant.
The solution contains the term [tex]\sqrt{b^2 - 4ac}[/tex] which will be:
Real and distinct if the discriminant is positiveReal and equal if the discriminant is 0Non-real and distinct roots if the discriminant is negativeThere are two roots of a quadratic equations always(assuming existence of complex numbers). We say that the considered quadratic equation has 2 solution if roots are distinct, and have 1 solutions when both roots are same.
For this case, the given equation is:
[tex]x^2 - x = -1/4[/tex]
Converting this to the form [tex]ax^2 + bx + c = 0[/tex], we get:
[tex]x^2 - x + 1/4= 0\\or\\4x^2 -4x + 1 = 0[/tex]
Thus, we get:
a = 4, b = -4, c = 1
Putting these values in the expression for discriminant, we get:
[tex]D = b^2 - 4ac =(-4)^2 - 4(4)(1) = 16 - 16 = 0[/tex]
The discriminant is 0, so the considered quadratic equation is going to have both roots real and equal. Or in terms of distinct solutions, it is going to have one real solution (distinct).
Thus, using the discriminant to know about the nature of the solution of the quadratic equation x² -x = -1/4 tells us the fact as given by: Option c. one real solution
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Two grandparents want to pick up the mess that their granddaughter had made in her playroom. One can do it in 15 minutes working alone. The other, working alone, can clean it in 12 minutes. How long will it take them if they work together?
Answer:
6 2/3 minutes
Step-by-step explanation:
Their rates in "jobs per hour" are ...
(60 min/h)/(15 min/job) = 4 jobs/h
and
(60 min/h)/(12 min/job) = 5 jobs/h
So, their combined rate is ...
(4 jobs/h) + (5 jobs/h) = 9 jobs/h
The time required (in minutes) is ...
(60 min/h)/(9 jobs/h) = (60/9) min = 6 2/3 min
Working together, it will take them 6 2/3 minutes.
To find out how long it would take the two grandparents to clean the playroom together, we can use the concept of rates and set up an equation. Solving the equation, we find that it would take them 9 minutes to clean the playroom if they work together.
Explanation:To solve this problem, we can use the concept of rates to find the combined rate at which the two grandparents clean. Let's assign the variable x to represent the time it takes for them to clean together.
The rate at which the first grandparent cleans is 1/15th of the playroom per minute, while the rate at which the second grandparent cleans is 1/12th of the playroom per minute. The combined rate when they work together is the sum of their individual rates, which is given by the equation (1/15)+(1/12)=(1/x).
To solve this equation, we can find a common denominator of 60 to simplify the equation to 4/60+5/60=1/x. Adding the fractions gives us 9/60=1/x. Multiplying both sides of the equation by 60 gives us 9=x. Therefore, it would take the two grandparents 9 minutes to clean the playroom if they work together.
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MAJORRR HELP !!!!!
Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
Simplify each expression and match it with the equivalent value.
[tex]\frac{3}{4} = log_{2}(\sqrt[4]{8} )\\-4 = log_{3} \frac{1}{81} \\-6= -3log_{5} 25\\\frac{1}{3} = log_{6} (\sqrt[3]{6} )[/tex]
Here's how you solve it!
[tex]log_{2} \sqrt[4]{8}[/tex]
Write it in exponential form
[tex]log_{2} (2 \frac{3}{4} )[/tex]
Then simplify
[tex]\frac{3}{4}[/tex]
[tex]log_{3} \frac{1}{81}[/tex]
Write in exponential form
[tex]log_{3} (3^{-4} )[/tex]
Simplify
-4
[tex]-3log_{5} 25[/tex]
Write in exponential form
[tex]-3log_{5} (5^{2} )[/tex]
Simplify
-3 * 2 = -6
-6
[tex]log_{6} \sqrt[3]{6}[/tex]
Write in exponential form
[tex]log_{6} (6\frac{1}{3} )[/tex]
Simplify
[tex]\frac{1}{3}[/tex]
Hope this helps! :3
The problem involves simplifying mathematical expressions, through steps as prescribed by BIDMAS/PEDMAS rules. Start by addressing anything within parentheses, follow through with multiplication or division, and finally handle addition or subtraction.
Explanation:This question involves the process of mathematical simplification of expressions. To solve this, you will first need to perform any calculations within the parentheses, then handle any multiplication or division from left to right, lastly address any addition or subtraction, also from left to right (also known as the order of operations or BIDMAS/PEDMAS). For example, if you have an expression like '2(3+4)': First, process the operation within the parentheses, in this case, it's a sum so you have '2*7', resulting in '14'. This is considered the simplified version of your expression.
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Can u guys please find the perimeter and the area of this shape.
Answer:
P: 20pi A: 400-100pi
Find the area of this triangle. Round the sine value to the nearest hundredth. Round the area to the nearest tenth of a centimeter.
Answer:
18.8 cm²
Step-by-step explanation:
Sometimes, as here, when the problem is not carefully constructed, the answer you get depends on the method you choose for solving the problem.
Following directions
Using the formula ...
Area = (1/2)ab·sin(C)
we are given the values of "a" (BC=5.9 cm) and "b" (AC=7.2 cm), but we need to know the value of sin(C). The problem statement tells us to round this value to the nearest hundredth.
sin(C) = sin(118°) ≈ 0.882948 ≈ 0.88
Putting these values into the formula gives ...
Area = (1/2)(5.9 cm)(7.2 cm)(0.88) = 18.6912 cm² ≈ 18.7 cm² . . . rounded
You will observe that this answer does not match any offered choice.
__
Rounding only at the End
The preferred method of working these problems is to keep the full precision the calculator offers until the final answer is achieved. Then appropriate rounding is applied. Using this solution method, we get ...
Area = (1/2)(5.9 cm)(7.2 cm)(0.882948) ≈ 18.7538 cm² ≈ 18.8 cm²
This answer matches the first choice.
__
Using the 3 Side Lengths
Since the figure includes all three side lengths, we can compute a more precise value for angle C, or we can use Heron's formula for the area of the triangle. Each of these methods will give the same result.
From the Law of Cosines, the angle C is ...
C = arccos((a² +b² -c²)/(2ab)) = arccos(-38.79/84.96) ≈ 117.16585°
Note that this is almost 1 full degree less than the angle shown in the diagram. Then the area is ...
Area = (1/2)(5.9 cm)(7.2 cm)sin(117.16585°) ≈ 18.8970 cm² ≈ 18.9 cm²
This answer may be the most accurate yet, but does not match any offered choice.
Across a horizontal distance of 25 feet, a roller coaster has a steep drop. The height of the roller coaster at the bottom of the drop is -150 feet, compared to its height at the top of the drop. What is the average amount that the roller coaster's height changes over each horizontal foot?
Hence, the average rate of change in vertical height is:
-6
Step-by-step explanation:We know that the average amount that the roller coaster's height changes over each horizontal foot is basically the slope or the average rate of change of the height of the roller coaster to the horizontal distance.
i.e. it is the ratio of the vertical change i.e. the change in height of the roller coaster to the horizontal change.
Here the vertical change= -150 feet
and horizontal change = 25 feet
Hence,
Average rate of change is:
[tex]=\dfrac{-150}{25}\\\\=-6[/tex]
So, for every change in horizontal distance by 1 feet the vertical height drop by 6 feet.
Answer:
The average amount that the roller coaster's height changes over each horizontal foot is -6.
Further explanation:
The rate of linear function is known as the slope. And the slope can be defined as the ratio of vertical change (change in y) to the horizontal change (change in x).
Mathematically, we can write
[tex]\text{Slope}=\dfrac{\text{change in y}}{\text{change in x}}=\dfrac{\Delta y}{\Delta x}[/tex]
If slope is negative then function is decreasing.If slope is positive then function is increasing.Now, we have been given that
Roller coaster has a steep drop at a horizontal distance of 25 feet.
Thus, [tex]\Delta x=25\text{ feet}[/tex]
The height of the roller coaster at the bottom of the drop is -150 feet.
Thus, [tex]\Delta y=-150\text{ feet}[/tex]
Using the above- mentioned formula, the average rate of change is given by
[tex]\text{Average rate of change }=\dfrac{-150}{25}[/tex]
On simplifying the fraction
[tex]\text{Average rate of change }=\dfrac{-6}{1}=-6[/tex]
It means for every 1 foot of horizontal distance, the roller coaster moves down by 6 feet.
Please refer the attached graph to understand it better.
Therefore, we can conclude that the average amount that the roller coaster's height changes over each horizontal foot is -6.
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Keywords:
Average rate of change, slope, change of y over change of x, the ratio of two numbers be the same.
A(n) _______ angle of a triangle is equal to the sum of the two remote interior angles.
-Exterior
-Interior
-Complementary
-Vertical
Answer:
Option A (Exterior)
Step-by-step explanation:
To understand this question, it is important to understand the concept of the exterior angle. An exterior angle is an angle which is made by two intersecting lines outside of the shape. Basically, one of the two lines is extended outside the shape. The angle between the extended line and the other line which is not extended is the exterior angle. It is outside the shape. The interior angle is the angle which is made by the same two lines but inside the shape.
The sum of the interior angle and the exterior angle is 180 degrees. It is also interesting to note that the sum of the angles in the triangle is 180 degrees.
Suppose that the angles in the triangles are A, B, and C, and the associated exterior angle with the angle A is angle D. By the argument, A+B+C=180 degrees and A+D=180 degrees. Since 180 degrees = 180 degrees, therefore A+B+C = A+D. Angle A cancels on both sides and reduces to B+C=D. This proves that the exterior angle of a triangle is equal to the sum of the two remote interior angles!!!
A page in a photo album is 10inches wide by 12 inches tall. There is a 1-inch
margin around the page that cannot be used for pictures. The space between each
picture is at least 1/2 - inch. How many 3-inch tall pictures can you fit on the page in
one column? Use a diagram to help you solve the problem
10.
Answer:
3
Step-by-step explanation:
The diagram shows the answer: 3 pictures will fit vertically.
You can solve this algebraically as well. For n pictures, there will be n-1 spaces, so the total height of the page must satisfy ...
1 + 3n + 1/2(n -1) + 1 ≤ 12
3.5n + 1.5 ≤ 12 . . . . . . . . . . . simplify
3.5n ≤ 10.5 . . . . . . . . . . . . . .subtract 1.5
n ≤ 3 . . . . . . . . . . . . . . . . . . . divide by 3.5
Up to 3 pictures will fit in a column.
Marty's Tee Shirt & Jacket Company is to produce a new line of jackets with an embroidery of a Great Pyrenees dog on the front. There are fixed costs of $ 680 to set up for production, and variable costs of $ 41 per jacket. Write an equation that can be used to determine the total cost, C(x), encountered by Marty's Company in producing x jackets.
Answer:
C(x)= 41x + 680
Step-by-step explanation:
If the fixed cost is 680, that will apply regardless of how many jackets the company makes for you. The number of jackets is unknown. However, we know that the cost of producing a single jacket is 41, so we can represent that expression as 41x. Putting those things together gives us a function of the cost:
C(x) = 41x + 680
The equation to determine the total cost encountered by Marty's Tee Shirt & Jacket Company in producing x jackets is C(x) = 680 + 41x.
Explanation:To determine the total cost, C(x), encountered by Marty's Tee Shirt & Jacket Company in producing x jackets, we need to consider both the fixed costs and the variable costs. The fixed costs, which are $680, are incurred regardless of the number of jackets produced. The variable costs, which are $41 per jacket, increase with each additional jacket produced. So the equation to calculate the total cost is:
C(x) = fixed costs + (variable costs per jacket) * x
Substituting the given values, the equation becomes:
C(x) = 680 + 41x
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Square EFGH stretches vertically by a factor of 2.5 to create rectangle E?F?G?H?. The square stretches with respect to the x-axis. If point H is located at (-2, 0), what are the coordinates of H? ?
Answer with explanation:
Pre-image =Rectangle EFGH
Image = Rectangle E'F'G'H'
Stretch Factor = 2.5
Coordinates of Point H= (-2,0)
If Coordinate of any point is (x,y) and it is stretched by a factor of k , then coordinate of that point after stretching = (k x , k y).
So, Coordinates of Point H' will be=(-2×2.5,0×2.5)
= (-5,0)
Answer: (-5,0)
Step-by-step explanation:
Given : Square EFGH stretches vertically by a factor of 2.5 to create rectangle E?F?G?H?.
The square stretches with respect to the x-axis such that the point H is located at (-2, 0).
Since , we know that to find the coordinate of image , we multiply the scale factor to the coordinate of pre-image.
Then , the coordinate of H? is given by :-
[tex](-2\times2.5, 0\times2.5)=(-5,0)[/tex]
Two water pumps, working simultaneously at their respective constant rates, took exactly 4 hours to fill a certain swimming pool. If the constant rate of one pump was 1.5 times the constant rate of the other, how many hours would it have taken the faster pump to fill the pool if it had worked alone at its constant rate?
Answer: [tex]\dfrac{20}{3}\text{ hours}[/tex]
Step-by-step explanation:
Let x be the speed of slower pump and 1.5x be the speed of faster pump to fill the swimming pool .
Then , According to the given question, we have the following equation:-
[tex]x+1.5x=\dfrac{1}{4}\\\\\rightarrow\ 2.5x=\dfrac{1}{4}\\\\\Rightarrow\ x=\dfrac{1}{10}=[/tex]
Now, the time taken by faster pump to fill the pool is given by :-
[tex]t=\dfrac{1}{1.5x}=\dfrac{10}{1.5}=\dfrac{20}{3}\text{ hours}[/tex]
Hence, the faster pump would take [tex]\dfrac{20}{3}\text{ hours}[/tex] to fill the pool if it had worked alone at its constant rate.
Solve for x 6^3-x=6^2
Answer:
D x=1
Step-by-step explanation:
6^(3-x)=6^2
Since the bases are the same, the exponents have to be the same
3-x = 2
Subtract 3 from each side
3-x-3 = 2-3
-x = -1
Multiply each side by -1
x = 1
Answer: Option D
[tex]x=1[/tex]
Step-by-step explanation:
We have the following exponential equation
[tex]6^{3-x}=6^2[/tex]
We must solve the equation for the variable x
Note that the exponential expressions [tex]6^{3-x}[/tex] and [tex]6 ^ 2[/tex] have the same base: 6
So if [tex]6^{3-x}=6^2[/tex] this means that [tex]3-x = 2[/tex]
Then we have that:
[tex]3-x = 2[/tex]
[tex]x = 3-2\\x=1[/tex]
You have decided to buy a new car, but you are concerned about the value of the car depreciating over time. You do some research on the model you are looking at and obtain the following information: Suggested retail price - $18,790 Depreciation per year - $1385 (It is assumed that this value is constant.) The following table represents the value of the car after n years of ownership.
Answer:
Option B After 14 years the car is worth $0
Step-by-step explanation:
we have
[tex]V=-1,385n+18,790[/tex]
where
V is the value of the cars
n is the number of years
Determine the n-intercept of the graph
we know that
The n-intercept is the value of n (number of years) when the value of V (value of the car) is equal to 0
so
For V=0
substitute and solve for n
[tex]0=-1,385n+18,790[/tex]
[tex]1,385n=18,790[/tex]
[tex]n=18,790/1,385[/tex]
[tex]n=14\ years[/tex]
That means
After 14 years the car is worth $0
Answer:
B
Step-by-step explanation:
Which description most accurately fits the definition of a combination?
An arrangement of beads on a necklace with a clasp.
An arrangement of objects on a key ring.
A selection or listing of objects in which the order of the objects is important.
A selection or listing of objects in which the order of the objects is not important.
Answer:
The correct option for the provided problem is D. A selection or listing of objects in which the order of the objects is not important.
Step-by-step explanation:
Consider the provided information.
Selecting all the parts of a set of objects without considering its order in which the objects are selecting is known as combination.
Now consider the provided options:
Options A, B, and C are not valid as the description does not fits accurately.
Thus, the correct option for the provided problem is D. A selection or listing of objects in which the order of the objects is not important.
Answer:
A selection or listing of objects in which the order of the objects is not important
Step-by-step explanation:
For a short time after a wave is created by wind, the height of the wave can be modeled using y = a sin 2πt/T, where a is the amplitude and T is the period of the wave in seconds.
How many times over the first 5 seconds does the graph predict the wave to be 2 feet high?
(SHOW WORK)
The graph hits [tex]\fbox{\begin\\\ \dfrac{10}{T}+2\\\end{minispace}}[/tex] times over 2 feet for [tex]a>2[/tex].
Further explanation:
The height of the wave is given by the equation as follows:
[tex]y=asin\left(\dfrac{2\pi t}{T}\right)[/tex] ......(1)
Here, [tex]a[/tex] is amplitude, [tex]T[/tex] is period of wave in second and [tex]t[/tex] time in seconds.
The height [tex]y[/tex] of the wave is given as 2 feet and time [tex]t[/tex] is given as 5 seconds.
Substitute 2 for [tex]y[/tex] and 5 for [tex]t[/tex] in equation (1).
[tex]2=asin\left(\dfrac{2\pi \times5}{T}\right)\\2=asin\left(\dfrac{10\pi}{T}\right)\\\dfrac{2}{a}=sin\left(\dfrac{10\pi}{T}\right)[/tex]
The above eqution is valid only for [tex]a\geq 2[/tex] because the maximum value of the term [tex]sin(10\pi /T)[/tex] is 1.
If [tex]T[/tex] is the time period then in [tex]T[/tex] seconds the graph will hit at least 2 times over 2 feet for [tex]a>2[/tex].
T seconds[tex]\rightarrow[/tex]2 hits
1 seconds [tex]\rightarrow[/tex] [tex]\dfrac{2}{T}[/tex] hits
5 seconds [tex]\rightarrow\dfrac{2\times5}{T}[/tex]
5 seconds [tex]\rightarrow[/tex] [tex]\dfrac{10}{T}[/tex]
If [tex]T[/tex] is time period in 5 seconds then the graph will hit [tex][10/T][/tex] times in interval 0 to [tex]2\pi[/tex].
Thus, the graph hits [tex]\fbox{\begin\\\ \dfrac{10}{T}+2\\\end{minispace}}[/tex] times over 2 feet for [tex]a>2[/tex].
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Answer details:
Grade: High school.
Subjects: Mathematics.
Chapter: function.
Keywords: Function, wave equation, height, amplitude, equation, period, periodic function, y=asin(2pit/T), frequency, magnitude, feet, height, time period, seconds, inequality, maximum value, range, harmonic motion, oscillation, springs, strings, sonometer.
Identify the equation of the circle Y that passes through (2,6) and has center (3,4).
Answer:
(x − 3)² + (y − 4)² = 5
Step-by-step explanation:
The equation of a circle is:
(x − h)² + (y − k)² = r²
where (h, k) is the center and r is the radius.
First use the distance formula to find the radius:
d² = (x₂ − x₁)² + (y₂ − y₁)²
r² = (2 − 3)² + (6 − 4)²
r² = 1 + 4
r² = 5
Given that (h, k) = (3, 4):
(x − 3)² + (y − 4)² = 5
Answer:
Step-by-step explanation:
Inserting the coordinates of the center (3, 4) into the standard equation of a circle with center at (h, k) and radius r, we get:
(x - 3)^2 + (y - 4)^2 = r^2
Next, we substitute 2 for x, 6 for y and solve the resulting equation for r^2:
(2 - 3)^2 + (6 - 4)^2 = r^2, or
1 + 4 = r^2.
Thus, the radius is √5. Subbing this result into the equation found above, (x - 3)^2 + (y - 4)^2 = r^2, we get:
(x - 3)^2 + (y - 4)^2 = (√5)^2 = 5, which matches the last of the four possible answer choices.
SOMEONE PLEASE HELP ME FIND THE ANSWER
Answer:
The measure of arc BC = 124°
Step-by-step explanation:
From the figure we can write,
measure of arc AB + measure of arc BC + measure of arc AC = 360
measure of arc AB = 146°
measure of arc BC = 90°
Therefore measure arc BC = 360 - (146 + 90)
= 360 - 236
= 124°
The measure of arc BC = 124°
Answer: 124 degrees
Step-by-step explanation: There is a 90 degree angle in the top right of the circle. There is a 146 degree angle. Add these two angles.
90 + 146 = 236
These two angles combined are 236 degrees. We are trying to find BC, which is the rest of the circle. There are 360 degrees in a circle. Subtract 360 from 236.
360 - 236 = 124
BC = 124 degrees.
Jamie and Imani each play softball. Imani has won 5 fewer games than Jamie. Is it possible for Jamie to have won 11 games if the sum of the games Imani and Jamie have won together is 30?
A.) Yes; Jamie could have won 11 games because 2x − 5 = 30.
B.) Yes; Jamie could have won 11 games because 11 − 5 is less than 30.
C.) No; Jamie could not have won 11 games because 2x − 5 ≠ 30.
D.) No; Jamie could not have won 11 games because 2x − 11 ≠ 30.
Answer: Option C
No; Jamie could not have won 11 games because [tex]2x - 5 \neq 30[/tex]
Step-by-step explanation:
Let's call x the number of games that Jamie has won
Let's call y the number of games that Imani has won
We know that Imani has won 5 more games than Jamie.
Then we can say that:
[tex]y= x - 5[/tex]
We know that the total number of games that Jamie and Imani have won together is 30.
So
[tex]x + y = 30[/tex]
We want to know if it is possible that [tex]x = 11[/tex].
Then we substitute the first equation in the second and get the following:
[tex]x + x - 5 =30\\2x - 5 = 30[/tex]
Now replace [tex]x = 11[/tex] in the equation and check if equality is met.
[tex]2 (11) - 5 = 30\\22 - 5 = 30\\17 \neq 30[/tex]
Equality is not met, then the correct answer is option C
Answer: is c (no Jamie could not have won 11 games because 2x-5=/30
Step-by-step explanation:
Write an equation for the problem and then solve.
The area of a triangle is 48 square meters. If the length of the base is 24 meters, what is the height of the triangle?
Answer: height of the triangle = _meters
Answer:
4 m
Step-by-step explanation:
Use the formula for the area of a triangle. Fill in the given numbers and solve for the unknown.
A = (1/2)bh
48 m² = (1/2)(24 m)h . . . . . put in the given numbers
(48 m²)/(12 m) = h = 4 m . . . . divide by the coefficient of h
The height of the triangle is 4 meters.
The formula for the area of a triangle is , where b is the length of the base and h is the height. Find the height of a triangle that has an area of 30 square units and a base measuring 12 units. 3 units 5 units 8 units 9 unitsThe formula for the area of a triangle is , where b is the length of the base and h is the height. Find the height of a triangle that has an area of 30 square units and a base measuring 12 units. 3 units 5 units 8 units 9 units
Answer: 5 units
Step-by-step explanation:
The formula to find the area of a triangle is given by :-
[tex]\text{Area}=\dfrac{1}{2}\text{ base * height}[/tex]
Given : The area of a triangle = 30 square units
The length of the base of the triangle = 12 units
Let h be the height of the triangle .
Then , we have
[tex]30=\dfrac{1}{2}12\times h\\\\\Rightarrow\ h=\dfrac{30}{6}\\\\\Rightarrow\ h=5\text{ units}[/tex]
Hence, the height of a triangle = 5 units
Which relation is not a function?
[Control] A. ((6.5).(-6, 5). (5.-6)
[Control] B. ((6,-5). (-6, 5). (5.-6))
[Control] C. ((-6,-5). (6.-5. (5.-6)}
[Control] D. ((-6,5).(-6.-6).(-6.-5))
Answer:
D.
Step-by-step explanation:
That would be D because there is a repetition of x = -6.
-6 maps to -6, 5 and -5 which is not allowed in a function.
Functions can be one-to-one or many-to-one but not one-to-many.
Choose the inequality that could be used to solve the following problem.
Three times a number is no less than negative six.
3x<-6
3x<-6
3x>-6
3x>-6
Answer:
3x ≥ -6
Step-by-step explanation:
"No less than" means "greater than or equal to". An appropriate translation of the problem statement is ...
3x ≥ -6
Answer:
3x ≥ -6
Step-by-step explanation:
The the inequality that could be used to solve three times a number is no less than negative six is 3x ≥ -6.
In the parabola y = (x + 12 + 2, what is the vertex?
Answer:
The vertex is the point (-6,-34)
Step-by-step explanation:
we know that
The equation of a vertical parabola into vertex form is equal to
[tex]y=a(x-h)^{2}+k[/tex]
where
(h,k) is the vertex of the parabola
In this problem we have
[tex]y=x^{2}+12x+2[/tex]
Convert in vertex form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]y-2=x^{2}+12x[/tex]
Complete the square . Remember to balance the equation by adding the same constants to each side.
[tex]y-2+36=x^{2}+12x+36[/tex]
[tex]y+34=x^{2}+12x+36[/tex]
Rewrite as perfect squares
[tex]y+34=(x+6)^{2}[/tex]
[tex]y=(x+6)^{2}-34[/tex]
The vertex is the point (-6,-34)
solve and graph each inequality -2y+7<1 or 4y+3<-5
Answer:
3 < yy < -2Step-by-step explanation:
1. -2y+7 < 1
Add 2y-1:
6 < 2y
Divide by 2:
3 < y
__
2. 4y +3 < -5
Subtract 3:
4y < -8
Divide by 4:
y < -2
_____
These are graphed on the number line with open circles because y=-2 and y=3 are not part of the solution set.
Answer:
y < -2 or y > 3Step-by-step explanation:
[tex](1)\\\\-2y+7<1\qquad\text{subtract 7 from both sides}\\-2y+7-7<1-7\\-2y<-6\qquad\text{change the signs}\\2y>6\qquad\text{divide both sides by 2}\\\boxed{y>3}\\\\(2)\\\\4y+3<-5\qquad\text{subtract 3 from both sides}\\4y+3-3<-5-3\\4y<-8\qquad\text{divide both sides by 4}\\\boxed{y<-2}\\\\\text{From (1) and (2) we have:}\ y<-2\ or\ y>3[/tex]
[tex]<,\ >-\text{op}\text{en circle}\\\leq,\ \geq-\text{closed circle}[/tex]
In the figure below, segments YZ and XY are both segments that are tangent to circle E. Segments XY and YZ are congruent.
Answer:
True
Step-by-step explanation:
Segments drawn to a circle from the same outside point are congruent.
Segments YZ and XY are tangent to circle E draw from outside point Y. The segments are congruent, so the statement is true.
HELPPPPP!!!!
An investment in a savings account grows to three times the initial value after t years.
If the rate of interest is 5%, compounded continuously, t = years.
Answer:
t = 21.97 years
Step-by-step explanation:
The formula for the continuous compounding if given by:
A = p*e^(rt); where A is the amount after compounding, p is the principle, e is the mathematical constant (2.718281), r is the rate of interest, and t is the time in years.
It is given that p = $x, r = 5%, and A = $3x. In this part, t is unknown. Therefore: 3x = x*e^(0.05t). This implies 3 = e^(0.05t). Taking natural logarithm on both sides yields ln(3) = ln(e^(0.05t)). A logarithmic property is that the power of the logarithmic expression can be shifted on the left side of the whole expression, thus multiplying it with the expression. Therefore, ln(3) = 0.05t*ln(e). Since ln(e) = 1, and making t the subject gives t = ln(3)/0.05. This means that t = 21.97 years (rounded to the nearest 2 decimal places)!!!
Answer:
t = 22 years
Step-by-step explanation:
* Lets explain the compound continuous interest
- Compound continuous interest can be calculated using the formula:
A = P e^rt
# A = the future value of the investment, including interest
# P = the principal investment amount (the initial amount)
# r = the interest rate
# t = the time the money is invested for
- The formula gives you the future value of an investment,
which is compound continuous interest plus the
principal.
* Now lets solve the problem
∵ The initial investment amount is P
∵ The future amount after t years is three times the initial value
∴ A = 3P
∵ The rate of interest is 5%
∴ r = 5/100 = 0.05
- Lets use the rule above to find t
∵ A = P e^rt
∴ 3P = P e^(0.05t)
- Divide both sides by P
∴ 3 = e^(0.05t)
- Insert ㏑ for both sides
∴ ㏑(3) = ㏑(e^0.05t)
- Remember ㏑(e^n) = n ㏑(e) and ㏑(e) = 1, then ㏑(e^n) = n
∴ ㏑(3) = 0.05t
- Divide both sides by 0.05
∴ t = ㏑(3)/0.05 = 21.97 ≅ 22
* t = 22 years
The volumes of soda in quart soda bottles can be described by a Normal model with a mean of 32.3 oz and a standard deviation of 1.2 oz. What percentage of bottles can we expect to have a volume less than 32 oz?
Answer: We can expect about 40.13% of bottles to have a volume less than 32 oz.
Step-by-step explanation:
Given : The volumes of soda in quart soda bottles can be described by a Normal model with
[tex]\mu=\text{32.3 oz}\\\\\sigma=\text{1.2 oz}[/tex]
Let X be the random variable that represents the volume of a randomly selected bottle.
z-score :[tex]\dfrac{x-\mu}{\sigma}[/tex]
For x = 32 oz
[tex]z=\dfrac{32-32.3}{1.2}=-0.25[/tex]
The probability of bottles have a volume less than 32 oz is given by :-
[tex]P(X<32)=P(z<-0.25)=0.4012937[/tex] [Using standard normal table]
In percent, [tex]0.4012937\times100=40.12937\%\approx40.13\%[/tex]
Hence, we can expect about 40.13% of bottles to have a volume less than 32 oz.
A railing needs to be build with 470.89 metric ton of iron the factory purchased only 0.38 part of required iron . How much iron is needed to complete the railing?
Answer:
291.9518 T are required for completion
Step-by-step explanation:
The remaining 0.62 part is ...
0.62 × 470.89 T = 291.9518 T
Answer:
291.9518 metric Ton
Step-by-step explanation:
Hello
according to the data provided by the problem.
Total Iron needed to build the railing (A)= 470.89 Ton
Total Iron purchased by the factory =0.38 of total
Total Iron purchased by the factory =0.38 *470.89
Total Iron purchased by the factory (B)=178.9382metric Ton
the difference between the total iron needed and the iron supplied by the factory will be the iron we need to get
A-B=iron we need to get(c)
C=A-B
C=470.89-178.9382
C=291.9518 metric Ton
Have a great day.
Please help me ): I don’t know what to do
Answer:
Question 1: the slope is -6
Question 2: the first choice is the one you want
Step-by-step explanation:
For the first one, I can't tell what fraction is on the left side with the y, but it doesn't matter. To me it looks like 1/2, but like I said, it won't change or affect our answer regarding the slope. That number has nothing to do with the slope.
In order to determine the slope of that line that is currently in point-slope form, we need to change it to slope-intercept form. Another expression for slope-intercept form is to solve it for y. Doing that:
[tex]y - \frac{1}{2}=-6x-42[/tex]
Now we can add 1/2 to both sides. That gives us the slope-intercept form of the line:
[tex]y=-6x- \frac{83}{2}[/tex]
The form is y = mx + b, where the number in the "m" place is the slope. Our slope is -6.
For the second one, we will sub in the x coordinate in a pair for x in the equation of the line and do the same for y to see if the left side equals the right side. The answer is [tex](\frac{2}{9},-7)[/tex] and I'll show you why. I will also show you how another point DOESN'T work in the equation. Filling in 2/9 for x and -7 for y:
[tex]-7+7=-3( \frac{2}{9} -\frac{2}{9})[/tex] which simplifies to
0 = -3(0) so
0 = 0 and this is true.
The other point I am going to use in exactly the same process is (-3, -7) since it doesn't have fractions in it. First I'm going to distribute the -3 into the parenthesis to get:
[tex]y+7= -3 x + \frac{6}{9}[/tex]
Subbing in -3 for x and -7 for y:
[tex]-7+7=-3( -3) +\frac{6}{9}[/tex]
As you can see, the left side equals 0 but the right side does not. If the lft side doesn't equal the right side, then the expression is not true, so the point is not on the line.
can someone please help prove b.,c., and d.? i need help!!!
Answer:
Proofs are in the explanation.
Step-by-step explanation:
b) My first thought is to divide top and bottom on the left hand side by [tex]\cos(\alpha)[/tex].
I see this would give me 1 on top and where that sine is, it would give me tangent since sine/cosine=tangent.
Let's do it and see:
[tex]\frac{\cos(\alpha)}{\cos(\alpha)-\sin(\alpha)} \cdot \frac{\frac{1}{\cos(\alpha)}}{\frac{1}{\cos(\alpha)}}[/tex]
[tex]=\frac{\frac{\cos(\alpha)}{\cos(\alpha)}}{\frac{\cos(\alpha)}{\cos(\alpha)}-\frac{\sin(\alpha)}{\cos(\alpha)}}[/tex]
[tex]=\frac{1}{1-\tan(\alpha)}[/tex]
c) My first idea here is to expand the cos(x+y) using the sum identity for cosine.
So let's do that:
[tex]\frac{\cos(x)\cos(y)-\sin(x)\sin(y)}{\cos(x)\sin(y)}[/tex]
Separating the fraction:
[tex]\frac{\cos(x)\cos(y)}{\cos(x)\sin(y)}-\frac{\sin(x)\sin(y)}{\cos(x)\sin(y)}[/tex]
The cos(x) cancel's in the first fraction and the sin(y) cancels in the second fraction:
[tex]\frac{\cos(y)}{\sin(y)}-\frac{\sin(x)}{\cos(x)}[/tex]
[tex]\cot(y)-\tan(x)[/tex]
d) This one makes me think it is definitely essential that we use properties of logarithms.
The left hand side can be condense into one logarithm using the product law:
[tex]\ln|(1+\cos(\theta))(1-\cos(\theta))|[/tex]
We are multiplying conjugates inside that natural log so we only need to multiply the first and the last:
[tex]\ln|1-\cos^2(\theta)|[/tex]
I can rewrite [tex]1-\cos^2(\theta)[/tex] using the Pythagorean Identity:
[tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex]:
[tex]\ln|\sin^2(\theta)|[/tex]
Now by power rule for logarithms:
[tex]2\ln|\sin(\theta)|[/tex]
find the missing angle and side measures of abc, given that A=25, C=90, and CB=16
Answer:
B = 65°AB = 37.859AC = 34.312Step-by-step explanation:
The given side is opposite the given acute angle in this right triangle, so the applicable relation is ...
Sin(25°) = CB/AB
Solving for AB, we get ...
AB = CB/sin(25°) ≈ 37.859
__
The relation involving the other leg of the triangle is ...
Tan(25°) = CB/AC
Solving for AC, we get ...
AC = CB/tan(25°) ≈ 34.312
__
Of course, the missing angle is the complement of angle A, so is 90-25 = 65 degrees.