Answer:
b. [tex]\frac{1}{y^3}[/tex]
Step-by-step explanation:
Here we are given [tex]y^{-3}[/tex] and are asked to simplify it. We are going to apply the law of exponents which says that
[tex]a^{-n}=\frac{1}{a^n}[/tex]
Hence applying this rule in our problem we get, where a=y and n=3
[tex]y^{-3}=\frac{1}{y^3}[/tex]
Hence our C) Option is correct
Explain how the quotient of powers was used to simplify this expression. 5^4/25=5^2
Answer: The quotient of powers was used because 25=5^2 which means that 5^4/25 is the same as 5^4/5^2. 5^4/5^2= 5^2. You can check your answer by simplifying 5^4 which is 625 and 5^2 which is 25, then divide the two which is 625/25 which equals 25 (or 5^2)
Step-by-step explanation:
[tex]\bf ~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{5^4}{25}\implies \cfrac{5^4}{5^2}\implies \cfrac{5^4}{1}\cdot \cfrac{1}{5^2}\implies 5^4\cdot 5^{-2}\implies 5^{4-2}\implies 5^2[/tex]
PLEASE HELP!!! Given the functions, f(x) = 6x + 2 and g(x) = x - 7, perform the indicated operation. When applicable, state the domain restriction. (f/g)(x)
To find (f/g)(x) with f(x) = 6x + 2 and g(x) = x - 7, one must divide f(x) by g(x). The domain restriction occurs because division by zero is not defined, so we exclude the x value that makes g(x) zero, which is x = 7.
Explanation:To perform the indicated operation (f/g)(x) with the given functions f(x) = 6x + 2 and g(x) = x - 7, we need to divide the function f(x) by the function g(x). This operation is equivalent to finding the quotient of the two functions, which is expressed as:
(f/g)(x) = f(x)/g(x) = (6x + 2)/(x - 7)
The domain restriction occurs when the denominator, g(x), is equal to zero since division by zero is undefined. So we must find the value of x for which g(x) = 0. Since g(x) = x - 7, setting this equal to zero gives us:
x - 7 = 0 → x = 7
Therefore, the domain of the function (f/g)(x) is all real numbers except for x = 7, because at x = 7 the function is undefined. The domain of (f/g)(x) can be expressed as ℜ - {7}, where ℜ represents the set of all real numbers.
write a compound inequality that represents each situations all real numbers that are greater than -8 but less than 8
Answer:
[tex]-8 < x < 8[/tex]
Step-by-step explanation:
Your compound inequality will include two inequalities.
These are:
x > -8
x < 8
Put your lowest number first, ensuring that your sign is pointed in the correct direction.
[tex]-8 < x[/tex]
Next, enter your higher number, again making sure that your sign is pointing in the correct direction.
[tex]-8 < x < 8[/tex]
Answer:
-8 < r < 8
Step-by-step explanation:
Let r = real number
Greater than >
r>-8
less than <
r <8
We want a compound inequality so we combine these
-8 < r < 8
Find the reciprocal of the expression.
The quantity 10 multiplied by b end of quantity divided by the quantity 2 multiplied by b plus 8 end of quantity.
A. Negative the quantity of the quantity 10 multiplied by b end of quantity divided by the quantity 2 multiplied by b plus 8 end of quantity end of quantity.
B. The quantity 2 multiplied by b plus 8 end of quantity divided by the quantity 10 multiplied by b end of quantity.
C. The quantity 10 multiplied by b plus 8 end of quantity divided by the quantity 2 multiplied by b end of quantity.
D. Negative the quantity of the quantity 2 multiplied by b plus 8 end of quantity divided by the quantity 10 multiplied by b end of quantity end of quantity.
Answer:
Option B
Step-by-step explanation:
we know that
To find the reciprocal of a fraction, flip the fraction.
Remember that
A number multiplied by its reciprocal is equal to 1
In this problem we have
[tex]\frac{10b}{2b+8}[/tex]
Flip the fraction
[tex]\frac{2b+8}{10b}[/tex] -----> reciprocal
therefore
The reciprocal is
The quantity 2 multiplied by b plus 8 end of quantity divided by the quantity 10 multiplied by b end of quantity.
Tom has 8 toys each toy weighs either 20 grams or 40 grams or 50 grams he has a diffrent number of toys (at least one) of each weight What is the smallest possible total weight of Tom's toys
Answer:
110
Step-by-step explanation:
He said he had atleast 1 of each. Hope it helps.
The smallest possible total weight of Tom's toys is:
210 grams
Step-by-step explanation:It is given that:
Tom has 8 toys each toy weighs either 20 grams or 40 grams or 50 grams.
Also, he has a different number of toys (at least one) of each weight.
Now, the smallest possible weight of Tom's toy is such that:
He has one toy of 50 grams , one of 40 grams and the other's are of smallest weight i.e. 20 grams.
This means he has 6 toys of 20 grams.
One of 40 grams.
One of 50 grams.
Hence,
Total weight= 20×6+40+50
i.e.
Total weight= 120+90
i.e.
Total weight= 210 grams.
Find the area of quadrilateral ABCD. [Hint: the diagonal divides the quadrilateral into two triangles.]
A. 26.47 units²
B. 28.53 units²
C. 27.28 units²
D. 33.08 units²
Answer:
B) 28.53 unit²
Step-by-step explanation:
The diagonal AD divides the quadrilateral in two triangles:
Triangle ABDTriangle ACDArea of Quadrilateral will be equal to the sum of Areas of both triangles.
i.e.
Area of ABCD = Area of ABD + Area of ACD
Area of Triangle ABD:
Area of a triangle is given as:
[tex]Area = \frac{1}{2} \times base \times height[/tex]
Base = AB = 2.89
Height = AD = 8.6
Using these values, we get:
[tex]Area = \frac{1}{2} \times 2.89 \times 8.6 = 12.43[/tex]
Thus, Area of Triangle ABD is 12.43 square units
Area of Triangle ACD:
Base = AC = 4.3
Height = CD = 7.58
Using the values in formula of area, we get:
[tex]Area = \frac{1}{2} \times 4.3 \times 7.58 = 16.30[/tex]
Thus, Area of Triangle ACD is 16.30 square units
Area of Quadrilateral ABCD:
The Area of the quadrilateral will be = 12.43 + 16.30 = 28.73 units²
None of the option gives the exact answer, however, option B gives the closest most answer. So I'll go with option B) 28.53 unit²
how much is 2 plus 9
For this case we must represent the following expression algebraically, in addition to indicating its result:
"2 plus 9"
So, we have:
[tex]2 + 9 =[/tex]
By law of the signs of the sum, we have that equal signs are added and the same sign is placed:
[tex]2 + 9 = 11[/tex]
ANswer:
11
Write 1.47 as a mixed number or a fraction In simplest form
Answer:
[tex]\frac{147}{100}[/tex]
Step-by-step explanation:
This is the answer because 147 ÷ 100 = 1.47
Find x two secant lines
Anyone know the formula?
Answer:
26
Step-by-step explanation:
The formula is a half the positive difference of the measurements of the intercepted arcs.
That is you do .5(66-14) here.
I'm going to distribute first and instead of find the difference first.
.5(66)-.5(14)
33- 7
26
Or.... you could do the difference first which gives us .5(66-14)=.5(52)=26.
So that angle is 26 degrees.
Answer:
x = 26°
Step-by-step explanation:
A secant- secant angle is an angle whose vertex is outside the circle and whose sides are 2 secants of the circle. It's measure is
x = 0.5(66 - 14)° = 0.5 × 52° = 26°
What is the sum of the complex numbers below?
(5+7i)+(-2+6i)
A. -3 +13i
B. 3+13i
C. -3-13i
D. 3-13i
Answer:
B
Step-by-step explanation:
Given
(5 + 7i) + (- 2 + 6i ) ← remove parenthesis and collect like terms
= 5 + 7i - 2 + 6i
= 3 + 13i → B
The sum of the complex number is 3 + 13i.
Option B is the correct answer.
We have,
To find the sum of the complex numbers (5+7i) and (-2+6i), you can simply add the real parts together and add the imaginary parts together separately.
Real part: 5 + (-2) = 3
Imaginary part: 7i + 6i = 13i
Combining the real and imaginary parts, we get:
Sum = 3 + 13i
Therefore,
The sum of the complex number is 3 + 13i.
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if
[tex] \frac{a + ib}{c + id} [/tex]
is purely real complex number then prove that: ad=bc
Rewrite the given number as
[tex]\dfrac{a+ib}{c+id}=\dfrac{(a+ib)(c-id)}{(c+id)(c-id)}=\dfrac{ac+bd+i(bc-ad)}{c^2+d^2}[/tex]
If it's purely real, then the complex part should be 0, so that
[tex]\dfrac{bc-ad}{c^2+d^2}=0\implies bc-ad=0\implies\boxed{ad-bc}[/tex]
as required.
The equation of a circle in general form is x2+y2+20x+12y+15=0 . What is the equation of the circle in standard form?
ANSWER
[tex]{(x + 10)}^{2} + {(y + 6)}^{2} = 121[/tex]
EXPLANATION
The equation of the circle in general form is given as:
[tex] {x}^{2} + {y}^{2} + 20x + 12y + 15 = 0[/tex]
To obtain the standard form, we need to complete the squares.
We rearrange the terms to obtain:
[tex] {x}^{2} + 20x + {y}^{2} + 12y = - 15 [/tex]
Add the square of half the coefficient of the linear terms to both sides to get:
[tex]{x}^{2} + 20x +100 + {y}^{2} + 12y + 36 = - 15 + 100 + 36[/tex]
Factor the perfect square trinomial and simplify the RHS.
[tex]{(x + 10)}^{2} + {(y + 6)}^{2} = 121[/tex]
This is the equation of the circle in standard form.
A cube has a net with area 24 in squared. How long is an edge of the cube?
The length of the edge of the cube whose net area is 24 in sq is calculated as: 2 inches.
What is the length of a cube?
The net of a cube consists of six identical square faces. Let's denote the length of one side of the square as s.
The total surface area of the cube is the sum of the areas of its six faces. Since each face has an area of s², the total surface area (A) is given by:
[tex]\[ A = 6s^2 \][/tex]
You mentioned that the net has an area of 24 square inches. Therefore, we can set up the equation:
6s² = 24
Now, solve for s:
[tex]s^2 = \frac{24}{6}[/tex]
s² = 4
Take the square root of both sides:
[tex]s = \sqrt{4}[/tex]
s = 2
So, the length of each edge of the cube is 2 inches.
Which best describes how to find the length of an arc in a circle?
A. Divide the arc's degree measure by 360°, then multiply by the
circumference of the circle.
B. Divide the arc's degree measure by 360°, then multiply by the
diameter of the circle.
C. Multiply the arc's degree measure by 360°, then divide by the
circumference of the circle.
D. Multiply the arc's degree measure by 360°, then divide by the
diameter of the circle
Answer:
A. Divide the arc's degree measure by 360°, then multiply by the circumference of the circle.Step-by-step explanation:
The first choice describes the right way to find the length of an arc in a circle.
The length of an arc is defined as
[tex]L=2\pi r (\frac{\theta}{360\°} )[/tex]
Where [tex]2\pi r[/tex] represents the circumference of the circle and [tex]\theta[/tex] represents the arc's degree measure.
So, as you can observe through this formula, we need to divide the arc's degree measure by 360°, and then multiply this result with the circumference of the circle, that's the right way based on the definition of arc length.
Therefore, the right answer is A.
The length of an arc in the circle is L = Divide the arc's degree measure by 360°, then multiply by the circumference of the circle
What is Central Angle?The central angle is an angle with two arms and a vertex in the middle of a circle. The two arms of the circle's two radii intersect the circle's arc at two separate locations. It is an angle whose vertex is the center of a circle with the two radii lines as its arms, that intersect at two different points on the circle.
The central angle of a circle formula is as follows.
Central Angle = ( s x 360° ) / 2πr
where s is the length of the arc
r is the radius of the circle
Central Angle = 2 x Angle in other segment
Given data ,
Let the length of the arc of the circle be L
Now , Central Angle = ( L x 360° ) / 2πr
where s is the length of the arc
On simplifying the equation , we get
Divide by 360° on both sides , we get
L / 2πr = Central Angle θ / 360°
Multiply by 2πr ( circumference ) on both sides , we get
L = ( θ / 360° ) x 2πr
Hence , the length of an arc is L = ( θ / 360° ) x 2πr
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What is the solution to the system of equations graphed below?
- 3x+2
y = 5x + 28
Answer:
(-3.25, 11.75)Step-by-step explanation:
We only need two points to plot the graph of each equation.
[tex]y=-3x+2\\\\for\ x=0\to y=-3(0)+2=0+2=2\to(0,\ 2)\\for\ x=1\to y=-3(1)+3=-3+2=-1\to(1,\ -1)\\\\y=5x+28\\\\for\ x=-4\to y=5(-4)+28=-20+28=8\to(-4,\ 8)\\for\ x=-6\to y=5(-6)+28=-30+28=-2\to(-6,\ -2)[/tex]
Look at the picture.
Read the coordinates of the intersection of the line (solution).
A boat travels 33 miles downstream in 4 hours. The return trip takes the boat 7 hours. Find the speed of the boat in still water.
Answer:
Speed of the boat in still water = 6.125 miles/hour
Step-by-step explanation:
We are given that a boat travels 33 miles downstream in 4 hours and the return trip takes the boat 7 hours.
We are to find the speed of the boat in the still water.
Assuming [tex]S_b[/tex] to be the speed of the boat in still water and [tex]S_w[/tex] to be the speed of the water.
The speeds of the boat add up when the boat and water travel in the same direction.
[tex]Speed = \frac{distance}{time}[/tex]
[tex]S_b+S_w=\frac{d}{t_1}=\frac{33 miles}{4 hours} [/tex]
And the speed of the water is subtracted from the speed of the boat when the boat is moving upstream.
[tex]S_b-S_w=\frac{d}{t_2}=\frac{33 miles}{7 hours} [/tex]
Adding the two equations to get:
[tex]S_b+S_w=\frac{d}{t_1}[/tex]
+ [tex]S_b-S_w=\frac{d}{t_2} [/tex]
___________________________
[tex]2S_b=\frac{d}{t_1} +\frac{d}{t_2}[/tex]
Solving this equation for [tex]S_b[/tex] and substituting the given values for [tex]d,t_1, t_2[/tex]:
[tex]S_b=\frac{(t_1+t_2)d}{2t_1t_2}[/tex]
[tex]S_b=\frac{(4 hour + 7hour)33 mi}{2(4hour)(7hour)}[/tex]
[tex]S_b=\frac{(11 hour)(33mi)}{56hour^2}[/tex]
[tex]S_b=6.125 mi/hr[/tex]
Therefore, the speed of the boat in still water is 6.125 miles/hour.
Answer:
[tex]6.48\frac{mi}{h}[/tex]
Step-by-step explanation:
Let' call "b" the speed of the boat and "c" the speed of the river.
We know that:
[tex]V=\frac{d}{t}[/tex]
Where "V" is the speed, "d" is the distance and "t" is the time.
Then:
[tex]d=V*t[/tex]
We know that distance traveled downstream is 33 miles and the time is 4 hours. Then, we set up the folllowing equation:
[tex]4(b+c)=33[/tex]
For the return trip:
[tex]7(b-c)=33[/tex] (Remember that in the return trip the speed of the river is opposite to the boat)
By solving thr system of equations, we get:
- Make both equations equal to each other and solve for "c".
[tex]4(b+c)=7(b-c)\\\\4b+4c=7b-7c\\\\4c+7c=7b-4b\\\\11c=3b\\\\c=\frac{3b}{11}[/tex]
- Substitute "c" into any original equation and solve for "b":
[tex]4b+\frac{3b}{11} =33\\\\4b+\frac{12b}{11}=33\\\\\frac{56b}{11}=33\\\\b=6.48\frac{mi}{h}[/tex]
What type of angles are 1 and 5?
vertical
supplementary
corresponding
complementary
Answer:
corresponding
Step-by-step explanation:
Answer:
Corresponding
Step-by-step explanation:
I like to call corresponding angles, the copy and paste angles because you can copy and paste the top intersection over the bottom intersection; the angles that lay down on top of each other are the corresponding angles. 1 and 5 do this.
PLEASE HELP AS FAST AS POSSIBLE PLEASE HELP PLEASE
Evaluate 0.00008 ÷ 640,000,000.
A) 1.25 × 10 -15
B) 1.25 × 10 -14
C) 1.25 × 10 -13
D) 1.25 × 10 -12
Answer:
0.00008 ÷ 640,000,000 means
8*10^-5 ÷ 6.4*10^8
so let's collect to simplify the operation
(8÷6.4)*(10^-12) -5-7=-12
then the answer becomes 1.25×10^-14 that is B
Answer:
option C
Step-by-step explanation:
Evaluate 0.00008 ÷ 640,000,000.
0.00008 can be written in standard notation
Move the decimal point to the end
so it becomes [tex]8 \cdot 10^{-5}[/tex]
for 640,000,000 , remove all the zeros and write it in standard form
[tex]64 \cdot 10^7[/tex]
Now we divide both
[tex]\frac{8 \cdot 10^{-5}}{64 \cdot 10^7}[/tex]
Apply exponential property
a^m divide by a^n is a^ m-n
[tex]\frac{8}{64} =0.125[/tex]
[tex]\frac{10^{-5}}{10^7}=10^{-12}[/tex]
[tex]0.125 \cdot 10^{-12}= 1.25 \cdot 10^{-13}[/tex]
Solve for x: 5 over x equals 4 over quantity x plus 3
5
3
−3
−15
Answer:
x = - 15
Step-by-step explanation:
The equation is [tex]\frac{5}{x}=\frac{4}{x+3}[/tex]
We now cross mulitply and do algebra to figure the value of x (shown below):
[tex]\frac{5}{x}=\frac{4}{x+3}\\5(x+3)=4(x)\\5x+15=4x\\5x-4x=-15\\x=-15[/tex]
Hence x = -15
Answer:
D
Step-by-step explanation:
the number of three-digit numbers with distinct digits that be formed using the digits 1,2,3,5,8 and 9 is . The probability that both the first digit and the last digit of the three-digit number are even numbers .
Answer:
a)120
b)6.67%
Step-by-step explanation:
Given:
No. of digits given= 6
Digits given= 1,2,3,5,8,9
Number to be formed should be 3-digits, as we have to choose 3 digits from given 6-digits so the no. of combinations will be
6P3= 6!/3!
= 6*5*4*3*2*1/3*2*1
=6*5*4
=120
Now finding the probability that both the first digit and the last digit of the three-digit number are even numbers:
As the first and last digits can only be even
then the form of number can be
a)2n8 or
b)8n2
where n can be 1,3,5 or 9
4*2=8
so there can be 8 three-digit numbers with both the first digit and the last digit even numbers
And probability = 8/120
= 0.0667
=6.67%
The probability that both the first digit and the last digit of the three-digit number are even numbers is 6.67% !
1.
[tex]6\cdot5\cdot4=120[/tex]
2.
[tex]|\Omega|=120\\|A|=2\cdot4\cdot1=8\\\\P(A)=\dfrac{8}{120}=\dfrac{1}{15}\approx6.7\%[/tex]
Latesha’s mother puts $85 in Latesha’s lunch account at school. Each day Latesha uses $3 from her account for lunch. The table below represents this situation. Latesha’s Lunch Account Day Amount Left in Account ($) 0 $85 1 2 3 4 5 How much is left in Latesha’s lunch account after she has had lunch for 5 days?
A.$15
B.$67
C.$70
D.82
Answer:
=70$
Step-by-step explanation:
The total in her account at day zero =85$
Lunch for five days= 3$×5
=15$
Total in her account= Initial amount - Expenditure on lunch
=85$-15$
=70$
The balance in Latesha's Lunch Account after having lunch for five day=70$
Determine the scale factor of 5 to 10
there's a scale factor of two!
five times two is ten.
hope this helps! :) xx
The center of a circle is at the origin on a coordinate grid. The vertex of a parabola that opens upward is at (0, 9). If the circle intersects the parabola at the parabola’s vertex, which statement must be true?
The maximum number of solutions is one.
The maximum number of solutions is three.
The circle has a radius equal to 3.
The circle has a radius less than 9.
Answer:
"The maximum number of solutions is one."
Step-by-step explanation:
Hopefully the drawing helps visualize the problem.
The circle has a radius of 9 because the vertex is 9 units above the center of the circle.
The circle the parabola intersect only once and cannot intercept more than once.
The solution is "The maximum number of solutions is one."
Let's see if we can find an algebraic way:
The equation for the circle given as we know from the problem without further analysis is so far [tex]x^2+y^2=r^2[/tex].
The equation for the parabola without further analysis is [tex]y=ax^2+9[/tex].
We are going to plug [tex]ax^2+9[/tex] into [tex]x^2+y^2=r^2[/tex] for [tex]y[/tex].
[tex]x^2+y^2=r^2[/tex]
[tex]x^2+(ax^2+9)^2=r^2[/tex]
To expand [tex](ax^2+9)^2[/tex], I'm going to use the following formula:
[tex](u+v)^2=u^2+2uv+v^2[/tex].
[tex](ax^2+9)^2=a^2x^4+18ax^2+81[/tex].
[tex]x^2+y^2=r^2[/tex]
[tex]x^2+(ax^2+9)^2=r^2[/tex]
[tex]x^2+a^2x^4+18ax^2+81=r^2[/tex]
So this is a quadratic in terms of [tex]x^2[/tex]
Let's put everything to one side.
Subtract [tex]r^2[/tex] on both sides.
[tex]x^2+a^2x^4+18ax^2+81-r^2=0[/tex]
Reorder in standard form in terms of x:
[tex]a^2x^4+(18a+1)x^2+(81-r^2)=0[/tex]
The discriminant of the left hand side will tell us how many solutions we will have to the equation in terms of [tex]x^2[/tex].
The discriminant is [tex]B^2-4AC[/tex].
If you compare our equation to [tex]Au^2+Bu+C[/tex], you should determine [tex]A=a^2[/tex]
[tex]B=(18a+1)[/tex]
[tex]C=(81-r^2)[/tex]
The discriminant is
[tex]B^2-4AC[/tex]
[tex](18a+1)^2-4(a^2)(81-r^2)[/tex]
Multiply the (18a+1)^2 out using the formula I mentioned earlier which was:
[tex](u+v)^2=u^2+2uv+v^2[/tex]
[tex](324a^2+36a+1)-4a^2(81-r^2)[/tex]
Distribute the 4a^2 to the terms in the ( ) next to it:
[tex]324a^2+36a+1-324a^2+4a^2r^2[/tex]
[tex]36a+1+4a^2r^2[/tex]
We know that [tex]a>0[/tex] because the parabola is open up.
We know that [tex]r>0[/tex] because in order it to be a circle a radius has to exist.
So our discriminat is positive which means we have two solutions for [tex]x^2[/tex].
But how many do we have for just [tex]x[/tex].
We have to go further to see.
So the quadratic formula is:
[tex]\frac{-B \pm \sqrt{B^2-4AC}}{2A}[/tex]
We already have [tex]B^2-4AC}[/tex]
[tex]\frac{-(18a+1) \pm \sqrt{36a+1+4a^2r^2}}{2a^2}[/tex]
This is t he solution for [tex]x^2[/tex].
To find [tex]x[/tex] we must square root both sides.
[tex]x=\pm \sqrt{\frac{-(18a+1) \pm \sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex]
So there is only that one real solution (it actually includes 2 because of the plus or minus outside) here for x since the other one is square root of a negative number.
That is,
[tex]x=\pm \sqrt{\frac{-(18a+1) \pm \sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex]
means you have:
[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex]
or
[tex]x=\pm \sqrt{\frac{-(18a+1)-\sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex].
The second one is definitely includes a negative result in the square root.
18a+1 is positive since a is positive so -(18a+1) is negative
2a^2 is positive (a is not 0).
So you have (negative number-positive number)/positive which is a negative since the top is negative and you are dividing by a positive.
We have confirmed are max of one solution algebraically. (It is definitely not 3 solutions.)
If r=9, then there is one solution.
If r>9, then there is two solutions as this shows:
[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex]
r=9 since our circle intersects the parabola at (0,9).
Also if (0,9) is intersection, then
[tex]0^2+9^2=r^2[/tex] which implies r=9.
Plugging in 9 for r we get:
[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+4a^2(9)^2}}{2a^2}}[/tex]
[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+324a^2}}{2a^2}}[/tex]
[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{(18a+1)^2}}{2a^2}}[/tex]
[tex]x=\pm \sqrt{\frac{-(18a+1)+18a+1}{2a^2}}[/tex]
[tex]x=\pm \sqrt{\frac{0}{2a^2}}[/tex]
[tex]x=\pm 0[/tex]
[tex]x=0[/tex]
The equations intersect at x=0. Plugging into [tex]y=ax^2+9[/tex] we do get [tex]y=a(0)^2+9=9[/tex].
After this confirmation it would be interesting to see what happens with assume algebraically the solution should be (0,9).
This means we should have got x=0.
[tex]0=\frac{-(18a+1)+\sqrt{36a+1+4a^2r^2}}{2a^2}[/tex]
A fraction is only 0 when it's top is 0.
[tex]0=-(18a+1)+\sqrt{36a+1+4a^2r^2}[/tex]
Add 18a+1 on both sides:
[tex]18a+1=\sqrt{36a+1+4a^2r^2[/tex]
Square both sides:
[tex]324a^2+36a+1=36a+1+4a^2r^2[/tex]
Subtract 36a and 1 on both sides:
[tex]324a^2=4a^2r^2[/tex]
Divide both sides by [tex]4a^2[/tex]:
[tex]81=r^2[/tex]
Square root both sides:
[tex]9=r[/tex]
The radius is 9 as we stated earlier.
Let's go through the radius choices.
If the radius of the circle with center (0,0) is less than 9 then the circle wouldn't intersect the parabola. So It definitely couldn't be the last two choices.
Answer:
Option A.
Step-by-step explanation:
A circle was drawn with the center at origin (0, 0) and a point (0, 9) on the circle.
So the radius will be r = [tex]\sqrt{(0-0)+(0-9)^{2}}=9[/tex]
Equation of this circle will be in the form of x² + y² = r²
Here r represents radius.
So the equation of the circle will be x² + y² = 9²
Or x² + y² = 81
Now we will form the equation of the parabola having vertex at (0, 9)
y² = (x - h)² + k
where (h, k) is the vertex.
Equation of the parabola will be y² = (x - 0)² + 9
y² = x² + 9
Now we will replace the value of y² from this equation in the equation of circle to get the solution of this system of the equations.
x² + x² + 9 = 81
2x² = 81 - 9
2x² = 72
x²= 36
x = ±√36
x = ±6
Since circle and parabola both touch on a single point (0, 9) therefore, there will be only one solution that is x = 6.
For x = 6,
6² + y² = 9²
36 + y² = 81
y² = 81 - 36 = 45
y = √45 = 3√5
Option A. will be the answer.
one x-intercept for a parabola is at the point (2, 0). use the quadratic formula to find the other x-intercept for the parabola defined by y=x^2-3x+2
Answer:
Step-by-step explanation:
There are 3 ways to find the other x intercept.
1) Polynomial Long Division.
Divide x^2 - 3x + 2 by the binomial x - 2, because by the Factor Theorem if a is a root of a polynomial then x - a is a factor of said polynomial.
2) Just solving for x when y = 0, by using the quadratic formula.
[tex]x^2 - 3x + 2 = 0\\x_{12} = \frac{3 \pm \sqrt{9 - 4(1)(2)}}{2} = \frac{3 \pm 1}{2} = 2, 1[/tex].
So the other x - intercept is at (1, 0)
3) Using Vietta's Theorem regarding the solutions of a quadratic
Namely, the sum of the solutions of a quadratic equation is equal to the quotient between the negative coefficient of the linear term divided by the coefficient of the quadratic term.
[tex]x_1 + x_2 = \frac{-b}{a}[/tex]
And the product between the solutions of a quadratic equation is just the quotient between the constant term and the coefficient of the quadratic term.
[tex]x_1 \cdot x_2 = \frac{c}{a}[/tex]
These relations between the solutions give us a brief idea of what the solutions should be like.
A football coach is trying to decide: when a team ahead late in the game, which strategy is better?
Answer:
regular
Step-by-step explanation:
1. look at table
notice regular and prevent defense have different wins but same total. 2. wins have more for regular so that's your answerAnswer:
The correct option is A)
P(Win|Regular)=0.76
P(Win|Prevent )=0.58
You are more likely to win by playing regular defense.
Step-by-step explanation:
Consider the provided table.
We need to find which strategy is better.
If team play regular defense then they win 38 matches out of 50.
[tex]Probability=\frac{\text{Favorable outcomes}}{\text{Total number of outcomes}}[/tex]
[tex]P(Win|Regular)=\frac{38}{50}[/tex]
[tex]P(Win|Regular)=0.76[/tex]
If team play prevent defense then they win 29 matches out of 50.
Thus, the probability of win is:
[tex]P(Win|Prevent )=\frac{29}{50}[/tex]
[tex]P(Win|Prevent )=0.58[/tex]
Since, 0.76 is greater than 0.58
That means the probability of winning the game by playing regular defense is more as compare to playing prevent defense.
Hence, the conclusion is: You are more likely to win by playing regular defense.
Thus, the correct option is A)
P(Win|Regular)=0.76
P(Win|Prevent )=0.58
You are more likely to win by playing regular defense.
Need help with this problem h+-3=4 please
Answer:
h=7
Step-by-step explanation:
[tex]h+(-3)=4[/tex]
may be rewritten as
[tex]h-3=4[/tex]
as adding a negative is the same as subtracting a positive.
To solve, add 3 to both sides.
[tex]h-3=4\\h=7[/tex]
Answer:
h=7
Step-by-step explanation:
1) Add three to both sides
2) You should get h=7
Find the values of k for which the quadratic equation 2x^2 − (k + 2)x + k = 0 has real and equal roots.
Answer:
k = 2
Step-by-step explanation:
If the roots are real and equal then the condition for the discriminant is
b² - 4ac = 0
For 2x² - (k + 2)x + k = 0 ← in standard form
with a = 2, b = - (k + 2) and c = k, then
(- (k + 2))² - (4 × 2 × k ) = 0
k² + 4k + 4 - 8k = 0
k² - 4k + 4 = 0
(k - 2)² = 0
Equate factor to zero and solve for k
(k - 2)² = 0 ⇒ k - 2 = 0 ⇒ k = 2
Answer:
k = 2Step-by-step explanation:
A quadratic equation has two equal real roots if a discriminant is equal 0.
[tex]ax^2+bx+c=0[/tex]
Discriminant [tex]b^2-4ac[/tex]
We have the equation
[tex]2x^2-(k+2)x+k=0\to a=2,\ b=-(k+2),\ c=k[/tex]
Substitute:
[tex]b^2-4ac=\bigg(-(k+2)\bigg)^2-4(2)(k)\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\=k^2+2(k)(2)+2^2-8k=k^2+4k+4-8k=k^2-4k+4\\\\b^2-4ac=0\iff k^2-4k+4=0\\\\k^2-2k-2k+4=0\\\\k(k-2)-2(k-2)=0\\\\(k-2)(k-2)=0\\\\(k-2)^2=0\iff k-2=0\qquad\text{add 2 to both sides}\\\\k=2[/tex]
Bianca has a stamp collection of 5 cent stamps and 7 cent stamps. She has 3 less 7 cent stamps as 5 cent stamps. If the collection has a face value of 87 cents, how many of each does she have?
She has ____ 5 cent stamps and ____ 7 cent stamps.
Answer:
She has 9 5 cent stamps and 6 7 cent stamps.
Step-by-step explanation:
Let the number of five cent stamps be represented by F and the number of seven cent stamps be represented by S.
The difference between the number of five cent and seven cent stamps is 3
F-S=3
The sum of the collection from each type of stamp is 87 cents
5F+7S=87
Let us solve the equations simultaneously.
F-S=3
5F+7S=87
Using substitution method,
F= 3+S
5(3+S)+7S=87
15+5S+7S=87
12S=87-15
12S=72
S=6
F=3+S
=3+6=9
Therefore the number of five cent stamps is 9 and seven cent stamps is 6.
Answer:
Number of 5 cent stamps = 9
Number of 7 cent stamps = 6
Step-by-step explanation:
We are given that Bianca has a stamp collection of 5 cent stamps and 7 cent stamps in which there are 3 less 7 cent stamps as 5 cent stamps.
If the total face value of stamps is 87 cents, we are to find the number of stamps of each value.
Assuming [tex]t[/tex] to be the number of 5 cent stamps and [tex]s[/tex] to be the 7 cent stamps so we can write it as:
[tex]0.05t+0.07s=0.87[/tex] --- (1)
[tex]s=t-3[/tex] --- (2)
Substituting this value of [tex]s[/tex] from (2) in (1):
[tex]0.05t+0.07(t-3)=0.87[/tex]
[tex]0.05s+0.07t-0.21=0.87[/tex]
[tex]0.12t=1.08[/tex]
[tex]t=9[/tex]
Number of 5 cent stamps = 9
Number of 7 cent stamps = 9 - 3 = 6
if you are paid $5.50/hour for mowing yards, and you take 3 1/3 hours to mow a yard, how much money are you owed?
[tex]\bf \begin{array}{ccll} \$&hour\\ \cline{1-2} 5.5&1\\ x&3\frac{1}{3} \end{array}\implies \cfrac{5.5}{x}=\cfrac{1}{3\frac{1}{3}}\implies \cfrac{5.5}{x}=\cfrac{1}{\frac{3\cdot 3+1}{3}}\implies \cfrac{5.5}{x}=\cfrac{1}{\frac{10}{3}}\implies \cfrac{5.5}{x}=\cfrac{\frac{1}{1}}{\frac{10}{3}} \\\\\\ \cfrac{5.5}{x}=\cfrac{1}{1}\cdot \cfrac{3}{10}\implies \cfrac{5.5}{x}=\cfrac{3}{10}\implies 55=3x\implies \stackrel{\textit{about 18 bucks and 33 cents}}{\cfrac{55}{3}=x\implies 18\frac{1}{3}=x}[/tex]
Answer:
$18.3
Step-by-step explanation:
If you are paid $5.50/hour for mowing yards, and you take 3 1/3 hours to mow a yard, you should earn $18.3.
3 1/3 hours
$5.50 and hour
$5.50 x 3 = $16.5
$5.50 / 3 = $1.8
$16.5 + $1.8 = $18.3
Therefore, you are owed $18.3.
how to divide (x^2+5x-6)/(x-1)
Answer:
x+6
Step-by-step explanation:
Let's see if the numerator is factorable.
Since the coefficient of x^2 is 1 (a=1), all you have to do is find two numbers that multiply to be -6 (c) and add up to be 5 (b).
Those numbers are 6 and -1.
So the factored form of the numerator is (x+6)(x-1)
So when you divide (x+6)(x-1) by (x-1) you get (x+6) because (x-1)/(x-1)=1 for number x except x=1 (since that would lead to division by 0).
Anyways, this is what I'm saying:
[tex]\frac{(x+6)(x-1)}{(x-1)}=\frac{(x+6)\xout{(x-1)}}{\xout{(x-1)}}[/tex]
[tex]x+6[/tex]