Answer: [tex]f(x)=x^3 -51x^2 -3x+153[/tex]
Step-by-step explanation:
By the conjugate root theorem, the roots are [tex]\sqrt{3}, -\sqrt{3}, 51[/tex].
Letting the leading coefficient be 1,
[tex]f(x)=(x-\sqrt{3})(x+\sqrt{3})(x-51)\\\\=(x^2 -3)(x-51)\\\\=x^3 -51x^2 -3x+153[/tex]
Find the lateral area of the cylinder .The lateral area of the given cylinder is _ M2(Round to the nearest whole number as needed .)
The lateral area of a cylinder is:
[tex]LA=2\pi rh[/tex]r is the radius
h is the height
For the given cylinder:
As the diameter is 4m, the radius is half of the diameter:
[tex]r=\frac{4m}{2}=2m[/tex]h=12m
[tex]\begin{gathered} SA=2\pi(2m)(12m) \\ SA=48\pi m^2 \\ SA\approx151m^2 \end{gathered}[/tex]Then, the lateral area of the given cylinder is 151 square meters
Solve the inequality and write the solution using:
Inequality Notation:
The solution for the given inequality is x >7.
InequalityIt is an expression mathematical that represents a non-equal relationship between a number or another algebraic expression. Therefore, it is common the use following symbols: ≤ (less than or equal to), ≥ (greater than or equal to), < (less than), and > (greater than).
The solutions for inequalities can be given by: a graph in a number line or numbers.
For solving this exercise, it is necessary to find a number and a graph solution for the given inequality.
The given inequality is [tex]1-\frac{6}{7}x < -5[/tex] . Then,
Move the number 1 for the other side of inequality and simplify.[tex]-\frac{6}{7}x < -5 -1\\ \\ -\frac{6}{7}x < -6[/tex]
Multiply both sides by -1 (reverse the inequality )[tex]-\frac{6}{7}x < -6 *(-1)\\ \\ \frac{6}{7}x > 6[/tex]
Solve the inequality for x[tex]\frac{6}{7}x > 6\\ \\ 6x > 42\\ \\ x > \frac{42}{6} \\ \\ x > 7[/tex]
You should also show the results t > 7 in a number line. Thus, plot the number line. See the attached image.
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A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment n=10, p=0.2, x=2
The binomial probability of x successes is 0.302.
How to calculate the probability of x successes?Since we are dealing with a binomial probability experiment. We are going to use the binomial distribution formula for determining the probability of x successes:
P(x = r) = nCr . p^r . q^n-r
Given: n=10, p=0.2, x=2
The failures can be calculated using q = 1 - p = 1 - 0.2 = 0.8
P(x = 2) = 10C2 x 0.2² x 0.8¹⁰⁻²
= 10!/(10-2)! 2! x 0.2² x 0.8⁸
= 10!/(8!2!) x 0.2² x 0.8^8
= 10x9x8!/(8!2!) x 0.2² x 0.8⁸
= 45 x 0.2² x 0.8⁸
= 0.302
Therefore, the probability of x successes in 10 trials is 0.302
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an art teacher makes a batch of green paint by mixing 5/8 cup of yellow paint with 5/8 cup of blue paint if she mixes 29 batches how many cups will she have with green paint
1 lote = 5/8 cup yellow + 5/8 cup blue
29 lotes = 29(5/8) +29(5/8) cups
29 lotes = 58(5/8)= (58*5)/8=290/8=145/4
145/4 =35.25 cups of paint
Write the first six terms of each arithmetic sequence,Please see the photo
Answer: - 9, - 3, 3, 9, 15, 21
Explanation:
The given formula is
an = a(n - 1) + 6
a1 = - 9
where
n, n - 1 and 1 are subscripts
This is a recursive formula. Each term is defined with respect to the term before it.
From the information given,
first term = a1 = - 9
Second term = a2 = a(2 - 1) + 6 = a1 + 6 = - 9 + 6
a2 = - 3
Third term = a3 = a(3 - 1) + 6 = a2 + 6 = - 3 + 6
a3 = 3
Fourth term = a4 = a(4 - 1) + 6 = a3 + 6 = 3 + 6
a4 = 9
Fifth term = a5 = a(5 - 1) + 6 = a4 + 6 = 9 + 6
a5 = 15
Sixth term = a6 = a(6 - 1) + 6 = a5 + 6 = 15 + 6
a6 = 21
Thus, the first six terms are
- 9, - 3, 3, 9, 15, 21
PLEASE HELP ASAP! What is the standard form of the hyperbola that the receiver sits on if the transmitters behave as foci of the hyperbola?
A hyperbola is a particular kind of smooth curve that lies in a plane and is classified by its geometric characteristics or by equations for which it is the solution set.
What is hyperbola?A hyperbola is a particular kind of smooth curve that lies in a plane and is classified by its geometric characteristics or by equations for which it is the solution set. A hyperbola is made up of two mirror images of one another that resemble two infinite bows.These two sections are known as connected components or branches. A series of points in a plane that are equally spaced out from a directrix or focus is known as parabolas. The difference in distances between a group of points that are situated in a plane and two fixed points—which is a positive constant—is what is referred to as the hyperbola.Therefore, a hyperbola is a particular kind of smooth curve that lies in a plane and is classified by its geometric characteristics or by equations for which it is the solution set.
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A 35-foot wire is secured from the top of a flagpole to a stake in the ground. If the stake is 1 feet from the base of the flagpole, how tall is the flagpole?
The figure for the height of flagpole, wire and ground is,
Determine height of the pole by using the pythagoras theorem in triangle.
[tex]\begin{gathered} l^2=b^2+h^2 \\ (35)^2=(14)^2+h^2 \\ 1225-196=h^2 \\ h=\sqrt[]{1029} \\ =32.078 \\ \approx32.08 \end{gathered}[/tex]Thus, height of the flagpole is 32.08 feet.
5. How would you solve the system of equations y = 5x + 1 and -2x + 3y =-10 ? What is the solution? *
SOLUTION:
Step 1:
In this question, we are given the following:
Solve the system of equations y = 5x + 1 and -2x + 3y =-10 ?
What is the solution?
Step 2:
The solution to the systems of equations:
[tex]\begin{gathered} y\text{ = 5x + 1 -- equation 1} \\ -2x\text{ + 3y = -10 -- equation 2} \end{gathered}[/tex]check:
Given y = -4 , x = -1
Let us put the values into the equation:
y = 5x + 1 and -2x + 3y = -10
[tex]\begin{gathered} y\text{ = 5x + 1} \\ -4=5(-1)\text{ + 1} \\ -4=-5+1 \\ -4\text{ = - 4 (COR}\R ECT) \end{gathered}[/tex][tex]\begin{gathered} -2x+3y\text{ = -10} \\ -2(-1)+3(-4)_{}_{} \\ 2-12=-10\text{ (COR}\R ECT) \end{gathered}[/tex]CONCLUSION:
The solution to the system of equations are:
[tex]\begin{gathered} \text{x = -1} \\ y=-4 \end{gathered}[/tex]
6. Line 1 passes through the points (1,4) and (-2,5). Line 2 passes through the points (1,0) and (0,3). What is true about Line 1 and Line 2? (2 points) (A) (B) They are perpendicular. They are parallel. They both decrease. They both increase. (C) (D)
First, calculate the slope (m) of both lines.
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Line 1:
Point 1 = (x1,y1) = (1,4)
Point 2 = (x2,y2) = (-2,5)
Replacing:
[tex]m=\frac{5-4}{-2-1}=\frac{1}{-3}=-\frac{1}{3}[/tex]Line 2:
Point 1 = (x1,y1) = (1,0)
Point 2 = (x2,y2) = (0,3)
[tex]m=\frac{3-0}{0-1}=\frac{3}{-1}=-3[/tex]Lines to be parallel must have the same slope, and to be perpendicular, they must have negative reciprocal slope.
None of the slopes are equal or negative reciprocal. SO, A and B are false-
Now, for the increase/ decrease
We can see that both lines have a negative slope, so they both decrease.
Correct option: C
1a. 100 foot-long rope is cut into 3 pieces.The first piece of rope is 3 times as long asthe second piece of rope. The third piece istwice as long as the first piece of rope.What is the length of the longest piece ofrope?
To solve the exercise, it is easier to make a drawing, like this
So, you have
[tex]\begin{gathered} z=3y \\ y=y \\ x=2z \\ z+y+x=100 \end{gathered}[/tex]Now solving
[tex]\begin{gathered} x=2z \\ x=2(3y) \\ x=6y \end{gathered}[/tex][tex]\begin{gathered} z+y+x=100 \\ 3y+y+6y=100 \\ 10y=100 \\ \frac{10y}{10}=\frac{100}{10} \\ y=10\text{ ft} \end{gathered}[/tex][tex]\begin{gathered} x=6y \\ x=6(10) \\ x=60\text{ ft} \end{gathered}[/tex][tex]\begin{gathered} z=3y \\ z=3(10) \\ z=30\text{ ft} \end{gathered}[/tex]Therefore, the length of the longest piece is 60ft.
Solve for k 4k – 6/3k – 9 = 1/3
hello
to solve this simple equation, we need to follow some simple steps.
[tex]4k-\frac{6}{3}k-9=\frac{1}{3}[/tex]step 1
multiply through by 3
we are doing this to eliminate the fraction and it'll help us solve this easily
[tex]\begin{gathered} 4k(3)-\frac{6}{3}k(3)-9(3)=\frac{1}{3}(3) \\ 12k-6k-27=1 \end{gathered}[/tex]notice how the equation haas changed suddenly? well this was done to make the question simpler and faster to solve.
step 2
collect like terms and simplify
[tex]\begin{gathered} 12k-6k-27=1 \\ 12k-6k=1+27 \\ 6k=28 \\ \end{gathered}[/tex]step three
divide both sides by the coefficient of k which is 6
[tex]\begin{gathered} \frac{6k}{6}=\frac{28}{6} \\ k=\frac{14}{3} \end{gathered}[/tex]from the calculations above, the value of k is equal to 14/3
need help with image
Step by step explanation:
sum of co-exterior angle is 180°
(10x-48)+(6x)=180°
4x-48=180°
4x=180-48
4x=132
x=132/4
x=33
Martin and Isabelle go bowling. Each game costs $10, and they split that cost. Martin has his own bowling shoes, but Isabelle pays $3 to rent shoes.Which graph shows a proportional relationship? Explain why.
We have the following:
Martin's graph is good and correct, although it is not totally straight, but the relationship that it keeps is totally proportional.
On the other hand, Isabelle's graph, although it is totally straight, is wrong, because she must start from 3, which is the rental value of the shoes, and her graph starts at 0, therefore it is wrong, despite of which shows a proportional relationship.
Therefore the correct answer is Martin's graph.
Answer:
Step-by-step explanation:
1+——>1/12 write. Fraction to make each number sentence true, answer I got is 1/1
c) Set x to be the number we need to find; therefore, the inequality to be solved is
[tex]\begin{gathered} 1+x>1\frac{1}{2}=1+\frac{1}{2}=\frac{3}{2} \\ \Rightarrow1+x>\frac{3}{2} \\ \Rightarrow-1+1+x>-1+\frac{3}{2} \\ \Rightarrow x>\frac{1}{2} \end{gathered}[/tex]Therefore, any number greater than 1/2 (greater, not equal to) satisfies the inequality; particularly 1/1=1>1/2. Thus, 1/1 is a possible answer
the sum of interior angle measures of a polygon with n sides is 2340 degrees. find n15
the measure of each angle will be 2340/n then if n=15 the measure of each one of the angles will be 2340/15=156 degrees
15 = a/3 - 2
what is a?
Answer: a is 51
Step-by-step explanation:
Hope this help.
Answer:
a==51
Step-by-step explanation:
15=a/3-2
a/3-2+2=15+2
a/3=17
a=17*3
a=51
95-a(b+c) when a= 9, b = 3 and c=7.4 I don’t get how to solve this please put an explanation
Notice that in the statement of the exercise are the values of a, b and c. Then, to evaluate the given expression, we replace the given values of a, b, and c. So, we have:
[tex]\begin{gathered} a=9 \\ b=3 \\ c=7.4 \\ 95-a\mleft(b+c\mright) \\ \text{ We replace the given values} \\ 95-a(b+c)=95-9(3+7.4) \\ 95-a(b+c)=95-9(10.4) \\ 95-a(b+c)=95-93.6 \\ 95-a(b+c)=\boldsymbol{1.4} \end{gathered}[/tex]Therefore, the result of evaluating the given expression when a = 9, b = 3, and c = 7.4 is 1.4.
Write the Distance Formula
Replace c with d to write the distance formula. Use the Distance Formula to Find the Distance Between Two Points
Find the distance, d, between G and H using the distance formula.
The distance between any two points (x1,y₁) and (x2,y2) on a
coordinate plane can be found by using the distance formula. Let (x,y)= (-2,1) and (x2,y2) =(4,-3). Substitute these values into the
distance formula and evaluate.
The distance between the two points is [tex]2\sqrt{13} units[/tex]
What is distance formula?
Distance formula is the measurement of distance between 2 points. It calculates the straight line distance between the given points. The formula can be given as [tex]distance=\sqrt{(c-a)^{2} +(d-b)^{2} }[/tex] Where A(a, b) B(c, d) Are the coordinates.
We are given the coordinates as (-2, 1) and (4, -3)
We substitute the values in the distance formula we get
[tex]distance=\sqrt{(c-a)^{2} +(d-b)^{2} } \\distance=\sqrt{(4+2)^{2} +(-3-1)^{2} }\\ distance=\sqrt{36+16 } \\distance=\sqrt{52 } \\distance =2\sqrt{13}[/tex]
Hence the distance between two points is [tex]2\sqrt{13} units[/tex]
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vertical anges are always equal to each other
Given the statement:
Vertical angles are always equal to each other
The answer is: True
Because they are inclosed by the same lines
0> -2x^2+4x+4Solve each inequality by graphing. Sketch it.
To solve the inequality we need to find the x-values that are the roots of the quadratic equation, let's use the quadratic formula:
[tex]\begin{gathered} \text{For an equation in the form:} \\ ax^2+bx+c=0 \\ The\text{ quadratic formula is:} \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \text{Then a=-2, b=4 and c=4} \\ x=\frac{-4\pm\sqrt[]{4^2-4(-2)(4)}}{2(-2)} \\ x=\frac{-4\pm\sqrt[]{16+32}}{-4} \\ x=\frac{-4\pm\sqrt[]{48}}{-4} \\ x=\frac{-4\pm6.93}{-4} \\ \text{Then} \\ x1=\frac{-4+6.93}{-4}=\frac{2.93}{-4}=-0.732 \\ x2=\frac{-4-6.93}{-4}=\frac{-10.93}{-4}=2.732 \end{gathered}[/tex]Now, let's try values less or greater than these roots:
If x=-1:
[tex]\begin{gathered} 0>-2(-1)^2+4(-1)+4 \\ 0>-2\cdot1-4+4 \\ 0>-2\text{ This is right, then number less than -0.732 are solutions of the inequality} \end{gathered}[/tex]Now let's try x=3:
[tex]\begin{gathered} 0>-2(3)^2+4(3)+4 \\ 0>-2\cdot9+12+4 \\ 0>-18+16 \\ 0>-2\text{ This is correct two, then the values greater that 2.732 are solutions to the inequality too} \end{gathered}[/tex]Then, the graph of the inequality is:
The red-shaded area are the solution to the inequality, then in interval notation we have:
[tex](-\infty,-0.732)\cup(2.732,\infty)[/tex]In builder notation it would be:
[tex]x|x<-0.732orx>2.732[/tex]Find the greatest common factor of the following monomials. 28g^5h^2 12g^6h^5
The GCF of these monomials i.e, 28g^5h^2 and 12g^6h^5 is 4h^2g^5
What is monomials?
Monomial expressions include only one non-zero term. Numbers, variables, or multiples of numbers and variables are all examples of monomials.
First take the coefficient ie, 28 and 12 to find the GCF
The GCF of 28 and 12 is 4
Now, find out the GCF of the variables for that you take the lowest exponent from both the variables g and h
for g variable it will be g^5 and,
for h variable it will be h^2
Therefore, the GCF of these monomials is 4h^2g^5
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Find the area of the sector interms of pi.2460°Area = [?]
Answer:
Area= 24π.
Explanation:
The area of a sector is calculated using the formula below:
[tex]A=\frac{\theta}{360\degree}\times\pi r^2[/tex]From the diagram:
• The central angle, θ = 60°
Diameter of the circle = 24
• Therefore, Radius, r = 24/2 = 12
Substitute these values into the formula:
[tex]\begin{gathered} A=\frac{60\degree}{360\degree}\times\pi\times12^2 \\ =24\pi\text{ square units} \end{gathered}[/tex]The area of the sector in terms of pi is 24π square units.
Create three different proportions that can be used to find BC in the figure above. At least one proportion must include AC as one of the measures.
We are given two similar triangles which are;
[tex]\begin{gathered} \Delta AEB\text{ and }\Delta ADC \\ \end{gathered}[/tex]Note that the sides are not equal, but similar in the sense that the ratio of two sides in one triangle is equal to that of the two corresponding sides in the other triangle.
To calculate the length of side BC, we can use any of the following ratios (proportions);
[tex]\frac{AE}{ED}=\frac{AB}{BC}[/tex][tex]\frac{AB}{AC}=\frac{AE}{AD}[/tex][tex]\frac{AE}{AB}=\frac{AD}{AC}[/tex]Using the first ratio as stated above, we shall have;
[tex]\begin{gathered} \frac{AE}{ED}=\frac{AB}{BC} \\ \frac{8}{5}=\frac{6.5}{BC} \end{gathered}[/tex]Next we cross multiply and we have;
[tex]\begin{gathered} BC=\frac{6.5\times5}{8} \\ BC=4.0625 \end{gathered}[/tex]ANSWER:
[tex]BC=4.0625[/tex]find the area of the circle with a circumference of 30π. write your solution in terms of π
we know that
the circumference of a circle is giving by
[tex]C=2\pi r[/tex]we have
C=30pi
substitute
[tex]\begin{gathered} 30\pi=2\pi r \\ \text{simplify} \\ r=\frac{30}{2} \\ r=15\text{ units} \end{gathered}[/tex]Find the area of the circle
[tex]A=\pi r^2[/tex]substitute the value of r
[tex]\begin{gathered} A=\pi(15^2) \\ A=225\pi\text{ unit\textasciicircum{}2} \end{gathered}[/tex]the area is 225π square unitsWhat is the value of sinθ given that (3, −7) is a point on the terminal side of θ?
Solution
[tex]\begin{gathered} \text{ using pythagoras theorem} \\ \\ OB=\sqrt{OA^2+AB^2}=\sqrt{3^2+7^2}=\sqrt{58} \\ \\ \Rightarrow\sin\theta=\frac{AB}{OB}=-\frac{7}{\sqrt{58}}=-\frac{7\sqrt{58}}{58} \end{gathered}[/tex]mr dudzic has above ground swimming pool thatbis a circular cylinder. the diameter of the pool is 25 ft. and the height isb4.5 ft. in order to open he needs to shock it with chlorine. if one gallon of liquid chlorin treats 3000 gallons of water, how many full gallons will he need to buy. (1 foot^3=7.48 gallons)
The volume of the cylinder is
[tex]V=\pi\text{ }\times r^2\times h[/tex]The diameter of the cylinder is 25 feet, then
The radius of it = 1/2 x diameter
[tex]r=\frac{1}{2}\times25=12.5ft[/tex]Since the height is 4.5 ft
Substitute them in the rule above
[tex]\begin{gathered} V=3.14\times(12.5)^2\times4.5 \\ V=2207.8125ft^3 \end{gathered}[/tex]Now we will change the cubic feet to gallons
[tex]\because1ft^3=7.48\text{ gallons}[/tex]Then multiply the volume by 7.48 to find the number of gallons
[tex]7.48\times2207.8125=16514.4375gallons[/tex]Now let us divide the number of gallons by 3000 to find how many gallons of liquid chlorin he needs to buy
[tex]\frac{16514.4375}{3000}=5.5048125[/tex]Then he has to buy 6 full gallons
find the lowest common denominator of - not graded !
Given:
There are two equation given in the question.
Required:
We have to find the lowest common denominator of both equation.
Explanation:
[tex]\frac{p+3}{p^2+7p+10}and\frac{p+5}{p^2+5p+6}[/tex]are given equations
first of all we need to factorization both denominator
[tex]\begin{gathered} p^2+7p+10and\text{ }p^2+5p+6 \\ (p+5)(p+2)and\text{ \lparen p+3\rparen\lparen p+2\rparen} \end{gathered}[/tex]so here (p+2) is common in both so take (p+2) for one time only
so now the lowest common denominator is
[tex](p+5)(p+2)(p+3)[/tex]Final answer:
The lowest common denominator for given two equations is
[tex](p+5)(p+2)(p+3)[/tex]
J is the midpoint of CT if CJ=5x-3 and JT=2x+21 find CT
Since J is the midpoint of the CT segment, then:
[tex]\begin{gathered} CJ=JT \\ 5x-3=2x+21 \end{gathered}[/tex]Now, you can solve the equation for x:
[tex]\begin{gathered} 5x-3=2x+21 \\ \text{ Add 3 from both sides of the equation} \\ 5x-3+3=2x+21+3 \\ 5x=2x+24 \\ \text{ Subtract 2x from both sides of the equation} \\ 5x-2x=2x+24-2x \\ 3x=24 \\ \text{ Divide by 3 from both sides of the equation} \\ \frac{3x}{3}=\frac{24}{3} \\ x=8 \end{gathered}[/tex]Replace the value of x into the equation for segment CJ or segment JT to find out what its measure is. For example in the equation of the segment CJ:
[tex]\begin{gathered} CJ=5x-3 \\ x=8 \\ CJ=5(8)-3 \\ CJ=40-3 \\ CJ=37 \end{gathered}[/tex]Finally, you have
[tex]\begin{gathered} CJ=37 \\ CJ=JT \\ 37=JT \\ \text{ Then} \\ CT=CJ+JT \\ CT=37+37 \\ CT=74 \end{gathered}[/tex]Therefore, the measure of the segment CT is 74.
A box contains 6 red pens, 4 blue pens, 8 green pens, and some black pens. Leslie picks a pen and returns it to the box each time. the outcomes are: number of times a red pen is picked: 8number of times a blue pen is picked: 5 number of times a green pen is picked: 14number of times a black pen is picked: 3Question: if the theoretical probability of drawing a black pen is 1/10, how many black pens are in the box?
We have:
x = total pens
n = number of black pens
so:
[tex]x=6+4+8+n=18+n[/tex]and for black pen:
[tex]\begin{gathered} \frac{1}{10\text{ }}=0.1\text{ (probability)} \\ \text{then} \\ \frac{n}{18+n}=0.1 \\ n=0.1(18+n) \\ n=1.8+0.1n \\ n-0.1n=1.8+0.1n-0.1n \\ 0.9n=1.8 \\ \frac{0.9n}{0.9}=\frac{1.8}{0.9} \\ n=2 \end{gathered}[/tex]answer: 2 black pens
Enter your solution as an ordered pair, with no spaces and with parentheses. OR the answer could be: Infinitely many OR No Solution
Given the equation system:
[tex]\begin{gathered} 1)y=4x \\ 2)3x+2y=55 \end{gathered}[/tex]The first step is to replace the first equation in the second equation
[tex]3x+2(4x)=55[/tex]With this, we have a one unknown equation. Now we can calculate the value of x:
[tex]\begin{gathered} 3x+8x=55 \\ 11x=55 \\ \frac{11x}{11}=\frac{55}{11} \\ x=5 \end{gathered}[/tex]Now that we know the value of x, we can determine the value of y, by replacing x=5 in the first equation
[tex]\begin{gathered} y=4x \\ y=4\cdot5 \\ y=20 \end{gathered}[/tex]This system has only one solution and that is (5,20)