We are given the following polynomial
[tex]4y^2-28y+49[/tex]Let us factor out the above polynomial.
We need to break the middle term into two numbers such that their sum is equal to -28 and their product is equal to 196
How about -14 and -14?
-14 -14 = -28
-14×-14 = 196
[tex]\begin{gathered} 4y^2-28y+49 \\ 4y^2-14y-14y+49 \\ (4y^2-14y)+(-14y+49) \\ 2y(2y-7)-7(2y-7) \\ (2y-7)(2y-7) \\ (2y-7)^2 \end{gathered}[/tex]As you can see, the polynomial is completely factored.
Also, notice that the polynomial is not a prime since it has more than 2 factors.
The Smith family sprinkler was used for 25 hours and the Figueroa family sprinkler was used for 35 hours. The combined total output was 1850L of water. What was the water output for each sprinkler if the sum of the two rates was 60L per hour?
From the text, we get two equations that relates the rate of the Smith family sprinkler and the Figueroa family sprinkler. Let's call the rate of water of the Smith family sprinkler "S" and the rate of water of the Figueroa family sprinkler "F".
Since the the Smith family sprinkler was used for 25 hours and the Figueroa family sprinkler was used for 35 hours, and the combined output was 1850L, we have the following equation:
[tex]25S+35F=1850[/tex]And since the sum of both rates equals 60L/h, we have the following:
[tex]S+F=60[/tex]Now, we just have to solve this system.
[tex]\begin{cases}25S+35F=1850 \\ S+F=60\end{cases}[/tex]First, we can simplify the first equation by dividing every term by 5.
This lead us to this system
[tex]\begin{cases}5S+7F=370 \\ S+F=60\end{cases}[/tex]If we multiply the second equation by five, and subract it from the first equation, we get an equation only with F.
[tex]\begin{gathered} 5\times(S+F)=5\times60 \\ \Rightarrow5S+5F=300 \end{gathered}[/tex][tex]\begin{gathered} 5S+7F-(5S+5F)=370-(300) \\ \Rightarrow2F=70 \\ \Rightarrow F=\frac{70}{2}=35 \end{gathered}[/tex]This tell us that the rate of the water output for the Figueroa family sprinkler is 35L/h. With this value, we can just substitute in any equation of our system and get the other rate.
[tex]\begin{gathered} S+35=60 \\ \Rightarrow S=25 \end{gathered}[/tex]And this is the rate for the Smith family sprinkler. 25L/h.
Hello, I need help solving question 9 a please, thanks
Solution
9
a)
[tex]\begin{gathered} \cos \left(61\right)+\sin \left(29\right) \\ \\ \mathrm{Use\:the\:following\:identity}:\quad \cos \left(x\right)=\sin \left(90^{\circ \:}-x\right) \\ \\ \cos \left(61^{\circ \:}\right)=\sin \left(90^{\circ \:}-61^{\circ \:}\right) \\ \\ =\sin \left(90^{\circ \:}-61^{\circ \:}\right)+\sin \left(29^{\circ \:}\right) \\ \\ =2\sin \left(29^{\circ \:}\right) \end{gathered}[/tex]b)
[tex]\begin{gathered} \sec\left(\theta\right)-\text{ cosec\lparen90- }\theta) \\ \\ =\sec \left(θ\right)-\frac{1}{\cos \left(θ\right)} \\ \\ \mathrm{Use\:the\:basic\:trigonometric\:identity}:\quad \frac{1}{\cos \left(x\right)}=\sec \left(x\right) \\ \\ =\sec \left(θ\right)-\sec \left(θ\right) \\ \\ \mathrm{Add\:similar\:elements:}\:\sec \left(θ\right)-\sec \left(θ\right)=0 \\ 0 \end{gathered}[/tex]if y varies directly with x by a constant of 3/2 what is the value of x when y equals 14
If y varies directly with x by a constant of 3/2, it means that the relation between these variables is:
[tex]y=\frac{3}{2}x[/tex]To determine the value of x when y = 14, replace this value of y into the previous equation, and solve for x, just as follow:
[tex]14=\frac{3}{2}x[/tex]multiply by 2 and divide by 3 both sides:
[tex]\begin{gathered} 14(\frac{2}{3})=x \\ \frac{28}{3}=x \end{gathered}[/tex]Hence, the value of x qhen y=14 is x - 28/3
Solve the following of equations and state whether the system is dependent, independent, or inconsistent by graphing1/2x+3/4y=12x-3y=4
Given:
[tex]\begin{gathered} \frac{1}{2}x+\frac{3}{4}y=1 \\ \\ 2x-3y=4 \end{gathered}[/tex]Find-:
The system is dependent, independent, or inconsistent
Explanation-:
[tex]\begin{gathered} \frac{1}{2}x+\frac{3}{4}y=1 \\ \\ \frac{2x}{4}+\frac{3y}{4}=1 \\ \\ \frac{2x+3y}{4}=1 \\ \\ 2x+3y=4................(1) \end{gathered}[/tex][tex]2x-3y=4............(2)[/tex]Add both equations then,
[tex]\begin{gathered} 2x+3y+2x-3y=4+4 \\ \\ 4x=8 \\ \\ x=\frac{8}{4} \\ \\ x=2 \end{gathered}[/tex]So value of y is:
[tex]\begin{gathered} 2x-3y=4 \\ \\ 2(2)-3y=4 \\ \\ 4-3y=4 \\ \\ 3y=4-4 \\ \\ y=\frac{0}{3} \\ \\ y=0 \end{gathered}[/tex]Value of y is 0
The solution for this system of equation is the point (2,0) and this set of equation is consistent since it has one solution at exactly one point. Thus this is consistent and independent.
The intensity I (in watts per square meter) of a television signal at a distance of k kilometers is described by the equation I =180/d^2. Find the intensity if the signal at a distance of 5 kilometers.
3.6 x 10^-5 watts per m²
1) Since the formula of the intensity of a beam of light, is
[tex]I=\frac{k}{d^2}\Rightarrow I=\frac{180}{d^2}[/tex]2) And the distance given is 5 km, then we need to convert that to meters 5 km=5000 meters then we can write:
[tex]\begin{gathered} I=\frac{180}{5000^2} \\ I=3.6\times10^{-5} \end{gathered}[/tex]3) Then the Intensity of that TV 3.6 x 10^-5 watts, per m²
All I need is the answer please and thank you
Answer
(4, -3)
Explanation
The coordinate of the missing vertex of rectangle ABCD is C(4, -3).
an election is over and a new city council member has been elected. the new councilman received 3 votes for every vote received by his opponent. the new councilman received 2712 votes. how many votes did his opponent receive?
It is given that,
The new councilman received 2712 votes.
The new councilman received 3 votes for every vote received by his opponent.
To find the number of votes did his opponent receive:
Let x be the number of votes received by his opponent.
So, we have
[tex]undefined[/tex]While hanging Christmas lights for neighbors, Terrell counted the number of broken lights on each string Number of broken lights Number of strings 10 3 42 4 93 6 101 4 135 3 < is the number of broken lights that a randomly chosen string had. What is the expected value of X? Vrite your answer as a decimal.
The expected value is defined by
[tex]E=\Sigma x\cdot P(x)[/tex]This means we have to find the probability of each number of broken lights:
[tex]\begin{gathered} P_1=\frac{3}{20} \\ P_2=\frac{4}{20}=\frac{1}{5} \\ P_3=\frac{6}{20}=\frac{3}{10} \\ P_4=\frac{4}{20}=\frac{1}{5} \\ P_5=\frac{3}{20} \end{gathered}[/tex]Then, we multiply each probability by its x-value, and we add them to find the expected value:
[tex]\begin{gathered} E=10\cdot\frac{3}{20}+42\cdot\frac{1}{5}+93\cdot\frac{3}{10}+101\cdot\frac{1}{5}+135\cdot\frac{3}{20} \\ E=1.5+8.4+27.9+20.2+20.25 \\ E=78.25 \end{gathered}[/tex]Hence, the expected value is 78.25.Solve.{2d+e=8 d−e=4Use the linear combination method. The solution is (4, 0).There are infinitely many solutions.There is no solution.The solution is (5, −2).
EXPLANATION
Given the system of equations:
2d + e
nit Test nit Test Active TIME REMAININ 01:39:05 The weights of five grapefruits are 7.47 ounces, 7.23 ounces, 6.46 ounces, 7.48 ounces, and 6.81 ounces. Using the clustering estimation technique, what is the approximate total weight of the grapefruits? O 34 ounces O 35 ounces 35.45 ounces 35.5 ounces
The clustering estimation technique is used when the data sets all cluster around a certain number (close in value to a certain number).
Upon close observation, the weights of the five grapefruits all cluster around the number 7. All the numbers can be estimated as 7, hence the approximate total weight of the five grapefruits is simply,
5 x 7 = 35 ounces
Two polygons are similar. The perimeter of the larger polygon is 120 yards and the ratio of thecorresponding side lengths is 1/6. Find the perimeter of the other polygon.The perimeter of the other polygon isyards.
The perimeter of the larger polygon is 120 yards.
The scale factor is 1/6.
The perimeter is proportional to the length of the sides, so if the smaller polygon has sides that are 1/6 of the corresponding sides in the larger polygon, then the perimeter is also 1/6 of the perimeter of the larger polygon:
[tex]P_s=\frac{1}{6}P_l=\frac{1}{6}\cdot120=20[/tex]The perimeter is scaled in the same proportion as the sides.
Answer: The perimeter of the other polygon is 20 yards.
find the trigonometric ratio for each of the following bellow by using the triangle
sin
[tex]\sin (P)=\frac{7}{25}[/tex]cos
[tex]\cos (P)=\frac{24}{25}[/tex]tan
[tex]\tan (P)=\frac{7}{24}[/tex]tan
[tex]\tan (Q)=\frac{24}{7}[/tex]which lengths make sense for possible values of b
The first condition is for the perimeter of the equation so:
[tex]P=2a+b=15.7[/tex]Now the secon condition is that the sum of two sides should be:
[tex]\begin{gathered} a+b>b \\ b+b>a \end{gathered}[/tex]so the negative option and 0 don't make sence dor a distance, then we try with the value of b=0.5
So in the first equation will be:
[tex]\begin{gathered} 2a+0.5=15.7 \\ 2a=15.2 \\ a=7.6 \end{gathered}[/tex]So the condition:
[tex]\begin{gathered} b+b>a \\ 0.5+0.5>7.6 \end{gathered}[/tex]this is false, so the solution is
Can someone tell me the answer
Answer:
Explanation:
Here, we are to match the correct answer given the dimensions of each of the shapes
We proceed as follows:
a) The volume of the rectangular prism
Mathematically, the volume is the product of the base rectangle and the height
We have this as:
[tex]\frac{3}{5}\times\frac{1}{5}\times\frac{2}{5}=\text{ }\frac{6}{125}cm^3[/tex]b) The volume of the rectangular prism
This is same as above
It is the product of the side lengths
Mathematically, we have this as:
[tex]\frac{3}{5}\times\frac{4}{5}\times\frac{3}{5}\text{ = }\frac{36}{125}cm^3[/tex]c) A cube with the given side length
Mathematically, the volume is the cube of the gievn side length as follows:
[tex]\frac{2}{5}\times\frac{2}{5}\times\frac{2}{5}=\frac{8}{125}cm^3[/tex]d) Cube with side length of 1/5 cm
This is the cube of the side length as follows:
[tex]\frac{1}{5}\times\frac{1}{5}\times\frac{1}{5}\text{ = }\frac{1}{125}cm^3[/tex]Charmaine must choose a number between 55 and 101 that is a multiple of 2, 6, and 8. Write all the numbers that she could choose. If there is more than onenumber, separate them with commas.
Consider that any number multiple of 6 and 8 is also multiple of 2.
The following are the list of number multiple of 2, 6 and 8, between 55 and 101:
72, 96
a woman's volleyball league has 23 teams. What is the total number of games to be played if each team plays every other team twice?
Let:
N = Total number of teams
Since each team plays every other team twice, the total T games will be:
[tex]\begin{gathered} T=N(N-1) \\ T=23(23-1) \\ T=23(22) \\ T=506 \end{gathered}[/tex]The graph of a function is shown on the coordinate plane below. Which relationship represents a function with a lesser rate of change (slope) than the function graphed?
1. Identify the rate of change of the function given in the graph:
use two points in the graph in the next formula:
[tex]undefined[/tex]the relationship between dividing four and multiplying by 1/4 is that ?
If you divide a number by 4 and multiply it by 1/4 you will obtain the same result, because "1/4" is one fourth of the number which is the same as dividing it by four. So these two operations are equal.
Example:
[tex]\begin{gathered} 100\colon4\text{ = 25} \\ 100\cdot\frac{1}{4}\text{ = 25} \end{gathered}[/tex]how do solve for x in a equation
A science class has 7 girls and 1 boy in the seventh grade and 3 girls and 9 boys in the eighth grade. The teacher randomly selects a seventh and an eighth-grader from the class for a competition. What is the probability that the students she selects are both girls?Write your answer as a fraction in the simplest form.
Answer:
7/32
Explanation:
The seventh-grade class has 7 girls and 1 boy.
[tex]P(\text{selecting a girl from seventh grade)}=\frac{7}{7+1}=\frac{7}{8}[/tex]The eighth-grade class has 3 girls and 9 boys.
[tex]P(\text{selecting a girl from eighth grade)}=\frac{3}{3+9}=\frac{3}{12}[/tex]Therefore, the probability that the students she selects are both girls:
[tex]P(\text{both are girls)}=\frac{7}{8}\times\frac{3}{12}=\frac{7}{32}[/tex]The probability is 7/32.
25. Write the equation for lines 'a' and 'b'Line 'a':Line 'b'
As you can see, line a intersects the y axis at point ( 0, 1 ) and the x axis at point ( -2, 0).
Basically, the slope (m) is rise over run, that is, m = (y2 - y1) / (x2 - x1) = 0-1 / -2-0 = 1/2. And our y=intercept, b is 1 (since the line intersects that y axis at 1). Using the point slope form , y = mx +b, the equation of line a is:
[tex]\begin{gathered} y=\frac{1}{2}x+1 \\ 2y=x+2 \\ x-2y+2=0 \end{gathered}[/tex]As you can see, line b intersects the y axis at point ( 0, -2 ) and the x axis at point ( -1, 0).
Basically, the slope (m) is rise over run, that is, m = (y2 - y1) / (x2 - x1) = 0-(-2) / -1-0 = 2 / -1 = -2. And our y=intercept, b is -2 (since the line intersects that y axis at -2). Using the point slope form , y = mx +b, the equation of line a is:
[tex]\begin{gathered} y=(-2)x+(-2) \\ y=-2x-2 \\ 2x+y+2=0 \end{gathered}[/tex]Answer:
[tex]\begin{gathered} a\colon x-2y+2=0 \\ b\colon2x+y+2=0 \end{gathered}[/tex]The initial directions are in the pic below. I have to send an additional pic with the choices. All wouldn’t fit.
Recall that a translation of n units to the left and m units down transforms the point (x,y) as follows:
[tex](x-n,y-m)\text{.}[/tex]Therefore the given transformation is as follows:
[tex](x,y)\rightarrow(x-4,y-3).[/tex]Answer: Second option.
54.)What function rule models the values in the table?Х481216Y4.8121620a.) Y = -x + 4b.)Y = x + 4c.) Y = 4x + 4
Step 1
Use the point form equation to get the momo. This equation is given as
[tex]\begin{gathered} \frac{y_2-y_1}{x_2-x_1}=\frac{y-y_1}{x-x_1} \\ \text{Where} \\ y_2=20 \\ y_1=16 \\ x_2=16 \\ x_1=12 \end{gathered}[/tex]Step 2
Substitute in and find the equation.
[tex]\begin{gathered} \frac{20-16}{16-12}=\frac{y-16}{x-12} \\ \frac{4}{4}=\text{ }\frac{y-16}{x-12} \\ x-12=\text{ y-16} \\ y\text{ = x-12 +16} \\ y\text{ = x+4} \end{gathered}[/tex]Hence the required function that models the values of the table is y = x+4
Solve the equation, give exact solutions. Then approximate the solution to the nearest hundredth if necessary
Find the slant height of the cone need homework help
Slant Height of a Cone
We are given a cone where the height BC = 6 in and the diameter is 4 in.
The radius is half the diameter, thus AB = 4/2 = 2 in
It's required to find the slant height of the cone, i.e., the value of AC.
Note ABC is a right triangle where the right angle is at vertex B. Thus the hypotenuse is AC and the legs are AB and BC.
Applying the Pythagorean's Theorem:
[tex]AC^2=AB^2+BC^2[/tex]Substituting values:
[tex]\begin{gathered} AC^2=2^2+6^2 \\ \text{Calculating:} \\ AC^2=4+36 \\ AC^2=40 \end{gathered}[/tex]Taking the square root:
[tex]\begin{gathered} AC=\sqrt[]{40} \\ AC=\sqrt[]{4\cdot10} \\ AC=\sqrt[]{4}\cdot\sqrt[]{10} \\ AC=2\text{ }\sqrt[]{10} \end{gathered}[/tex]At what interest compounded continuously must you invest $4900 to have$10740.65 money in 9 years? First round the interest rate r to 4 decimalplaces and then re-write it as aa percentage with 2 decimal places.
ANSWER
[tex]\begin{gathered} r=0.0872 \\ r=8.72\% \end{gathered}[/tex]EXPLANATION
We want to find the rate at which the amount was continuously compounded.
The formula for the total amount for a continuously compounded principal is:
[tex]A=Pe^{rt}[/tex]where A = amount
P = principal
r = rate
t = time (in years)
Substituting the given values into the equation:
[tex]\begin{gathered} 10740.65=4900\cdot e^{r\cdot9} \\ 10740.65=4900\cdot e^{9r} \end{gathered}[/tex]Divide both sides by 4900:
[tex]\begin{gathered} \frac{10740.65}{4900}=e^{9r} \\ e^{9r}=2.1920 \end{gathered}[/tex]Find the natural logarithm of both sides of the equation:
[tex]\begin{gathered} \ln (e^{9r})=\ln 2.1920 \\ 9r=0.7848 \end{gathered}[/tex]Divide both sides by 9:
[tex]\begin{gathered} r=\frac{0.7848}{9} \\ r=0.0872 \end{gathered}[/tex]Convert to decimal number:
[tex]\begin{gathered} r=0.0872\cdot100 \\ r=8.72\% \end{gathered}[/tex]That is the interest rate.
in triangle ABC, AD is a straight line. What is the sum of the degrees in angle x and angle y?
the answer is D, because
[tex]\begin{gathered} \angle x+\angle y+180-110=180 \\ \angle x+\angle y=110 \end{gathered}[/tex]Question 1 (3 points)Evaluate the function when x = -2,0,3p(x) = -8x-2p(-2) =p(0) =p(3) =Blank 1:Blank 2:Blank 3:
Answer:
p(-2) = 14
p(0) = -2
p(3) = -26
Explanation:
To find the value of p(-2) we need to replace x by -2 on p(x) as:
[tex]\begin{gathered} p(x)=-8x-2 \\ p(-2)=-8\cdot(-2)-2 \\ p(-2)=16-2 \\ p(-2)=14 \end{gathered}[/tex]At the same way, p(0) and p(3) are calculated as:
[tex]\begin{gathered} p(x)=-8x-2 \\ p(0)=-8\cdot0-2 \\ p(0)=0-2 \\ p(0)=-2 \end{gathered}[/tex][tex]\begin{gathered} p(x)=-8x-2 \\ p(3)=-8\cdot3-2 \\ p(3)=-24-2 \\ p(3)=-26 \end{gathered}[/tex]Therefore, the answer is:
p(-2) = 14
p(0) = -2
p(3) = -26
factor trinomialsx^2+2x-80
We will factor as follows:
[tex]x^2+2x-80=(x+10)(x-8)[/tex]***
What we try to find in the values that are added [Or subtracted from x] is the following:
*When both numbers are multiplied between them and added together the solution musth be the value in the middle of the trinomial.
*W
Find the sum of the measures of the interior angles and the sum of the measures ofthe exterior angles of the polygon.Sum of the measures of the interior angles:Sum of the measures of the exterior angles:
Given:
Find-:
a) Sum of the measure of the interior angle
b) Sum of the measure of the exterior angle.
Sol:
Given figure is a hexagon and each Interior angle of the hexagon is 120
So the Sum of interior angles is:
[tex]\begin{gathered} =120+120+120+120+120+120 \\ \\ =720 \end{gathered}[/tex]An exterior angle of the hexagon is:
[tex]\begin{gathered} =180-120 \\ \\ =60 \end{gathered}[/tex]Sum of an exterior angle is:
[tex]\begin{gathered} =60+60+60+60+60+60 \\ \\ =360 \end{gathered}[/tex]