Answers
Given the system of equations:
[tex]\begin{gathered} x^2+y^2=68 \\ y=4x \end{gathered}[/tex]We can solve it by substituting the second equation into the first as follows:
[tex]\begin{gathered} x^2+(4x)^2=68 \\ \\ Operating: \\ \\ x^2+16x^2=68 \\ 17x^2=68 \end{gathered}[/tex]Dividing by 17:
[tex]x^2=\frac{68}{17}=4[/tex]Applying square root on both sides:
[tex]\begin{gathered} x=\pm\sqrt{4} \\ x=\pm2 \end{gathered}[/tex]There are two solutions for x and they produce two solutions for y.
For x = 2
y = 4*2 = 8
For x = -2
y = 4*(-2) = -8
Thus, the solutions are:
(2,8) and (-2, -8)
Related Questions
Petrolyn motor oil is a combination of natural oil and synthetic oil. It contains 5 liters of natural oil for every 4 liters of synthetic oil. In order to make 531 litersof Petrolyn oll, how many liters of synthetic oil are needed?
Answers
The ratio 4 : 5 means that in every 9 liters of oil, we will have 4L of synthetic oil and 5L of natural oil.
Divide the 531 by 9 to get how many times we have to amplify the ratio:
[tex]\frac{531}{9}=59[/tex]Multiply the ratio by 59:
[tex]4\colon5\rightarrow(4)(59)\colon(5)(59)\rightarrow236\colon295[/tex]Meaning that for the 531L of oil, 236L would be synthetic and 295L natural.
Answer: 236 Liters.
given AD is congruent to AC and AB is congruent to AE, which could be used to prove?
Answers
Answer
Option B is correct.
SAS | 2 sides and the angle between them in one triangle are congruent to the 2 sides and the angle between them in the other triangle, then the triangles are congruent.
Explanation
We have been told that the two triangles have two sets of sides that are congruent to each other.
And we can see that the angle between those congruent sides for the two triangles is exactly the same for the two triangles.
So, it is easy to see that thes two triangles have 2 sides that are congruent and the angle between these two respective sides are also congruent.
Hope this Helps!!!
Mrs walters had a bag full of candy she wanted to share with 18 students. If she had 335 pieces of candy how many pieces will each student get
Answers
Find the quantities indicated in the picture (Type an integer or decimal rounded to the nearest TENTH as needed.)
Answers
Remember that 3, 4 and 5 is a Pythagorean triple, since:
[tex]3^2+4^2=5^2[/tex]Since one side of the given right triangle has a length of 3 and the hypotenuse has a length of 5, then, the remaining leg b must have a length of 4.
Therefore:
[tex]b=4[/tex]The angles A and B can be found using trigonometric identities.
Remember that the sine of an angle equals the quotient of the lengths of the side opposite to it and the hypotenuse of the right triangle.
The side opposite to A has a length of 3 and the length of the side opposite to B is 4. Then:
[tex]\begin{gathered} \sin (A)=\frac{3}{5} \\ \sin (B)=\frac{4}{5} \end{gathered}[/tex]Use the inverse sine function to find A and B:
[tex]\begin{gathered} \Rightarrow A=\sin ^{-1}(\frac{3}{5})=36.86989765\ldotsº \\ \Rightarrow B=\sin ^{-1}(\frac{4}{5})=53.13010235\ldotsº \end{gathered}[/tex]Then, to the nearest tenth:
[tex]\begin{gathered} A=36.9º \\ B=53.1º \end{gathered}[/tex]Therefore, the answers are:
[tex]undefined[/tex]Equation of the line that passes through points (8,7) and (0,0)
Answers
Equation of the line:
y = mx+b
where:
m= slope
b= y-intercept
First, we have to find the slope:
m = (y2-y1) / (x2-x1)
Since we have:
(x1,y1) = (8,7)
(x2,y2)= (0,0)
Replacing:
m = (0-7)/ (0-8) = -7/-8 = 7/8
Now, that we have the slope:
y = 7/8 x +b
We can place the point (8,7) in the equation and solve for b:
7 = 7/8 (8) +b
7=7 +b
7-7=b
b=0
Since the y-intercept=0
The final equation is:
y= 7/8x
Solve fory.y = 6O O2y = 5y = 6.67оо3y = 94Previous
Answers
Here the chords are intersecting outside hence
[tex]\begin{gathered} 2\times(2+10)=3\times(3+y) \\ 2\times12=3(3+y) \\ 2\times4=(3+y) \\ 8=3+y \\ y=8-3 \\ y=5 \end{gathered}[/tex]Hence the answer is y=5
I really need help on this and I would really appreciate if anyone would want to help me please and thank you.
Answers
Given the equation of the parabola:
[tex]y=x^2+6x-12[/tex]To find the vertex of the parabola,
we will substitute with the value (-b/2a) into the function y
[tex]\begin{gathered} a=1 \\ b=6 \\ c=-12 \\ \\ x=-\frac{b}{2a}=-\frac{6}{2\cdot1}=-3 \\ y=(-3)^2+6\cdot-3-12=9-18-12=-21 \end{gathered}[/tex]so, the coordiantes of the vertex :
x = -3
y = -21
what is 9932.8 rounded to the nearest integer
Answers
ANSWER
9933
EXPLANATION
We have the number 9932.8.
We want to round it to the nearest integer.
An integer is a number that can be written without decimal or fraction.
To do that, we follow the following steps:
1. Identify the number after the decimal
2. If the number is greater than or equal to 5, round up to 1 and add to the number before the decimal.
3. If the number is less than 5, round down to 0.
Since the number after the decimal is 8, we therefore have that:
[tex]9932.8\text{ }\approx\text{ 9933}[/tex]Rick's average score on his first three tests in math is 80. What must he score on his next test to raise his average to 84?
Answers
SOLUTION
Now, we don't know the scores for his first three tests. But we are told that the average score for the first three tests was 80.
So, let the scores of the first three tests be a, b, and c. That means
[tex]\frac{a+b+c}{3}=80[/tex]Also, let's assume the total score for his first three tests was x, This means that
[tex]\begin{gathered} a+b+c=x \\ or \\ x=a+b+c \end{gathered}[/tex]Comparing with the first equation it means that
[tex]\begin{gathered} \frac{a+b+c}{3}=80 \\ \frac{x}{3}=80 \\ x=3\times80 \\ x=240 \end{gathered}[/tex]Now we are asked "What must he score on his next test to raise his average to 84?"
So this means the total tests becomes 4. Hence
[tex]\begin{gathered} \frac{a+b+c+d}{4}=84 \\ \frac{x+d}{4}=84 \\ \frac{240+d}{4}=84 \\ 240+d=84\times4 \\ 240+d=336 \\ d=336-240 \\ d=96 \end{gathered}[/tex]So he must score 96 to raise his average score to 84.
Hence, the answer is 96
please help me work through this, thank you very much!
Answers
Given
[tex]plane-height=650m[/tex]To Determine: The angle function
Solution
The information can be represented as shown below
From the diagram below
[tex]\begin{gathered} tan\theta=\frac{650}{x} \\ \theta(x)=tan^{-1}(\frac{650}{x}) \end{gathered}[/tex]Michelle can wash dry and fold 5 loads of laundry in 3 1/2 hours. what is the average amount of time it takes Michelle to do one load of laundry
Answers
Which measurement is closest to the shortest distance in miles from Natasha's house to the library?
Answers
Given:
The objective is to find the shortest distance between house and library.
Consider the given triangle as,
Here, A represents the house, B the grocery and C the library.
Since it is a right angled triangle, the distance between the house and the library can be calculated using Pythagoras theorem.
[tex]\text{Hypotenuse}^2=Opposite^2+Adjacent^2[/tex]Apply the given values in the above formula,
[tex]\begin{gathered} AC^2=17^2+0.9^2 \\ AC^2=289+8.1 \\ AC^2=297.1 \\ AC=\sqrt[]{297.1} \\ AC=17.237\text{ miles} \end{gathered}[/tex]If Natasha walks through Grocery store,
[tex]\begin{gathered} AC^{\prime}=AB+BC \\ AC^{\prime}=0.9+17 \\ AC^{\prime}=17.9\text{ miles} \end{gathered}[/tex]By comparing the two ways, ACHence, the hypotenuse distance AC, between house and library is the closest distance.
Glenda borrowed $4,500 at a simple interest rate of 7% for 3 years to
buy a car. How much simple interest did Glenda pay?
Answers
Answer: I = $ 1,102.50
Step-by-step explanation: First, converting R percent to r a decimal
r = R/100 = 7%/100 = 0.07 per year,
then, solving our equation
I = 4500 × 0.07 × 3.5 = 1102.5
I = $ 1,102.50
The simple interest accumulated
on a principal of $ 4,500.00
at a rate of 7% per year
for 3.5 years is $ 1,102.50.
Consider the expression 6+(x+3)^2. Tabulate at least SIX different values of the expression.
Answers
Considering the expression 6+(x+3)^2. the table of at least SIX different values of the expression is
x y
0 15
1 22
2 31
3 42
4 55
5 70
How to determine the he table of at least SIX different values of the expressionThe table is completed by substituting the values of x in the given expression as follows
6 + ( x + 3 )^2
for x = 0, y = 6 + ( 0 + 3) ^2 = 15
for x = 1, y = 6 + ( 1 + 3) ^2 = 22
for x = 2, y = 6 + ( 2 + 3) ^2 = 31
for x = 3, y = 6 + ( 3 + 3) ^2 = 42
for x = 4, y = 6 + ( 4 + 3) ^2 = 55
for x = 5, y = 6 + ( 5 + 3) ^2 = 70
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find the perimeter of the triangle whose vertices are (-10,-3), (2,-3), and (2,2). write the exact answer. do not round.
Answers
We have to calculate the perimeter of a triangle of which we know the vertices.
The perimeter is the sum of the length of the three sides, which can be calculated as the distance between the vertices.
The vertices are V1=(-10,-3), V2=(2,-3), and V3=(2,2).
We then calculate the distance between each of the vertices.
We start with V1 and V2:
[tex]\begin{gathered} d_{12}=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2} \\ d_{12}=\sqrt[]{(-3-(-3))^2+(2-(-10)^2} \\ d_{12}=\sqrt[]{(-3+3)^2+(2+10)^2} \\ d_{12}=\sqrt[]{0^2+12^2} \\ d_{12}=12 \end{gathered}[/tex]We know calculate the distance between V1 and V3:
[tex]\begin{gathered} d_{13}=\sqrt[]{(y_3-y_1)^2+(x_3-x_1)^2} \\ d_{13}=\sqrt[]{(2-(-3))^2+(2-(-10))^2} \\ d_{13}=\sqrt[]{5^2+12^2} \\ d_{13}=\sqrt[]{25+144} \\ d_{13}=\sqrt[]{169} \\ d_{13}=13 \end{gathered}[/tex]Finally, we calculate the distance between V1 and V3:
[tex]\begin{gathered} d_{23}=\sqrt[]{(y_3-y_2)^2+(x_3-x_2)^2} \\ d_{23}=\sqrt[]{(2-(-3))^2+(2-2)^2} \\ d_{23}=\sqrt[]{5^2+0^2} \\ d_{23}=5 \end{gathered}[/tex]Then, the perimeter can be calcualted as:
[tex]\begin{gathered} P=d_{12}+d_{13}+d_{23} \\ P=12+13+5 \\ P=30 \end{gathered}[/tex]Answer: the perimeter is 30 units.
Allison earned a score of 150 on Exam A that had a mean of 100 and a standard deviation of 25. She is about to take Exam B that has a mean of 200 and a standard deviation of 40. How well must Allison score on Exam B in order to do equivalently well as she did on Exam A? Assume that scores on each exam are normally distributed.
Answers
Allison must score 280 on Exam B to do equivalently well as she did on Exam A
Explanations:Note that:
[tex]\begin{gathered} z-\text{score = }\frac{x-\mu}{\sigma} \\ \text{where }\mu\text{ represents the mean} \\ \sigma\text{ represents the standard deviation} \end{gathered}[/tex][tex]\begin{gathered} \text{For Exam A:} \\ x\text{ = 150} \\ \mu\text{ = 100} \\ \sigma\text{ = 25} \\ z-\text{score = }\frac{150-100}{25} \\ z-\text{score = 2} \end{gathered}[/tex]Since we want Allison to perform similarly in Exam A and Exam B, their z-scores will be the same
Therefore for exam B:
[tex]\begin{gathered} \mu\text{ = 200} \\ \sigma\text{ = 40} \\ z-\text{score = 2} \\ z-\text{score = }\frac{x-\mu}{\sigma} \\ 2\text{ = }\frac{x-200}{40} \\ 2(40)\text{ = x - 200} \\ 80\text{ = x - 200} \\ 80\text{ + 200 = x} \\ x\text{ = 280} \end{gathered}[/tex]Allison must score 280 on Exam B to do equivalently well as she did on Exam A
Solve the inequality 3.5 >b + 1.8. Then graph the solution.
Answers
Collect like terms
[tex]\begin{gathered} 3.5-1.8\ge b \\ 1.7\ge b \\ b\leq\text{ 1.7} \end{gathered}[/tex]evaluate B-( - 1/8) + c where b =2 and c=- 7/4
Answers
Answer: 3/8
Step-by-step explanation:
Given:
[tex]B-(-\frac{1}{8} )+c[/tex]
replace variables with their given values: b = 2 and C = 7/4
[tex]2-(-\frac{1}{8})+\frac{-7}{4}[/tex]
to make subtracting and addition easier, make each number has the same common denominator.
[tex]\frac{16}{8} -(-\frac{1}{8})+(\frac{-14}{8})[/tex]
Finally, solve equation.
***remember that subtracting a negative is the same as just adding and adding by a negative is the same as simply subtracting.
[tex]\frac{16}{8} -(-\frac{1}{8})+(\frac{-14}{8})=\frac{16}{8} +\frac{1}{8}-\frac{14}{8}[/tex]
= 3/8
Answer:
3/8
Step-by-step explanation:
2 - (-1/8) + (-7/4)
= 17/8 - 7/4
= 17/8 + -7/4
= 3/8
The Thompson family and the Kim family each used their sprinklers last summer. The Thompson family's sprinkler was used for 25 hours. The Kim family'ssprinkler was used for 35 hours. There was a combined total output of 1075 L of water. What was the water output rate for each sprinkler if the sum of the tworates was 35 L per hour?Thompson family's sprinkler:Kim family's sprinkler:
Answers
Let x be the rate of water output by the Thompson family and let y be the rate of water output by the Kim family.
We know that the Thompson family sprinkler was used for 25 hours, Kim's family sprinkler was used for 35 hours and that there was a combined total output of 1075 L of water; then we have the equation:
[tex]25x+35y=1075[/tex]We also know that the combined water output was 35 L per hour, then:
[tex]x+y=35[/tex]Hence we have the system of equations:
[tex]\begin{gathered} 25x+35y=1075 \\ x+y=35 \end{gathered}[/tex]To solve this system we solve the second equation for y:
[tex]\begin{gathered} x+y=35 \\ y=35-x \end{gathered}[/tex]And we plug this value in the first equation and solve for x:
[tex]\begin{gathered} 25x+35(35-x)=1075 \\ 25x+1225-35x=1075 \\ -10x=1075-1225 \\ -10x=-150 \\ x=\frac{-150}{-10} \\ x=15 \end{gathered}[/tex]Once we have the value of x we plug it in the expression of y:
[tex]\begin{gathered} y=35-15 \\ y=20 \end{gathered}[/tex]Therefore we have that:
[tex]\begin{gathered} x=15 \\ y=20 \end{gathered}[/tex]which means:
Thompson family's sprinkler: 15 L per hour
Kim family's sprinkler: 20 L per hour.
P(-3,-5) and Q(1.–3) represent points in a coordinate plane. Find the midpoint of Pe.
Answers
By formula,
Midpoint between two points PQ =
[tex](\frac{x_2+x_1}{2},\text{ }\frac{y_2+y_1}{2})[/tex][tex]\begin{gathered} (\frac{1+-3}{2},\text{ }\frac{-3+\text{ -5}}{2}) \\ \\ \frac{-2}{2},\text{ }\frac{-8}{2}\text{ = (-1,-4)} \\ \\ \end{gathered}[/tex]So, (-1,-4) (option 3)
Find the average rate of change of the function in the graph shown below between x=−1 and x=1.
Answers
Answer:
Step-by-step explanation:
The last description actually clarifies the given equation. The equation should be written as: f(x) = 2ˣ +1. The x should be in the exponent's place.
The average rate of change, in other words, is the slope of the curve at certain points. In equation, the slope is equal to Δy/Δx. It means that the slope is the change in the y coordinates over the change in the x coordinate. So, we know the denominator to be: 2-0 = 2. To determine the numerator, we substitute x=0 and x=2 to the original equation to obtain their respective y-coordinate pairs.
f(0)= 2⁰+1 = 2
f(2) = 2² + 1 = 5
A translation is a type of transformation in which a figure is flipped,TrueFalse
Answers
Need help with this review question. I need to know how to find the measurements from the cyclic quadrilateral
Answers
Given a quadrilateral ABCD
A cyclic quadrilateral has all its vertices on the circumference of the circle
Also cyclic quadrilateral
has the opposites angles add up to 180°
then
[tex]\angle a+\angle c=180[/tex][tex]\angle b+\angle d=180[/tex]then
Option A
A=90
B=90
C=90
D=90
since A+C= 180
and B+D = 180
measures from Option A could come from a cyclic quadrilateral
Option B
A=80
B=80
C=100
D=100
Since A+C = 80+100 = 180
and B+D = 80 + 100 = 180
measures from Option B could come from a cyclic quadrilateral
Option C
A=70
B=110
C=70
D=110
Since A+C=70+70 = 140
And B+D =110+110=220
measures from Option C could NOT come from a cyclic quadrilateral
Option D
A=60
B=50
C=120
D=130
A+C= 60+120 = 180
B+D= 50+130 = 180
measures from Option D could come from a cyclic quadrilateral
Option E
A=50
B=40
C=120
D=150
A+C=50+120= 170
B+D=40+150 = 190
measures from Option E could NOT come from a cyclic quadrilateral
Then correct options are
Options
A,B and D
Data Set A has a Choose... interquartile range than Data Set B. This means that the values in Data Set A tend to be Choose... the median.
Answers
The median of the given data set will be 35.
What do we mean by media?In statistics and probability theory, the median is the number that separates the upper and lower half of a population, a probability distribution, or a sample of data. For a data set, it might be referred to as "the middle" value.
So, The variability metrics for each class are listed below:
The further classifications: Class A; Class B;
Range: 30 Range: 30IQR: 12.5 IQR: 20.5MAD: 7.2 MAD: 9.2Greater variability in the data set is suggested by class B's wider interquartile range and mean absolute deviations.
Set A's median will be:
median = (20 + 32+ 36+ 37 + 50) / 5median = 175 / 5median = 35Therefore, the median of the given data set will be 35.
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Which of the following logarithmic expressions have been evaluated correctly?
Answers
Given:
Logarithmic expressions in options.
Required:
Select correct calculated option.
Explanation:
1). ln 1 = 0
2).
[tex]log_29=3.1699250014[/tex]3)
[tex]log\frac{1}{100}=-2_[/tex]4).
[tex]log_3(-1)=NaN[/tex]5).
[tex]log_5\text{ }\frac{1}{125}=-3[/tex]Answer:
Hence, option A and E are correct.
In an elementary school, 20% of the teachers teach advanced writing skills. If there are 25writing teachers, how many teachers are there in the school?
Answers
Answer:
125 teachers
Explanation:
We were given that:
20% of teachers teach advanced writing skills = 20/100 = 0.2
Number of writing teachers = 25
The total number of teachers = x
We will obtain the number of teachers in the school as shown below:
[tex]\begin{gathered} \frac{No.of.writing.teachers}{Total.number.of.teachers}\times100\text{\%}=20\text{\%} \\ \frac{25}{x}\times100\text{\%}=20\text{\%} \\ \frac{25\times100\text{\%}}{x}=20\text{\%} \\ \text{Cross multiply, we have:} \\ x\cdot20\text{\% }=25\times100\text{\%} \\ \text{Divide both sides by 20\%, we have:} \\ \frac{x\cdot20\text{\%}}{20\text{\%}}=\frac{25\times100\text{\%}}{20\text{\%}} \\ x=\frac{2500}{20} \\ x=125 \\ \\ \therefore x=125 \end{gathered}[/tex]Hence, the total number of teachers in the school is 125
Construct a polar equation for the conic section with the focus at the origin and the following eccentricity and directrix.Conic Eccentricity Directrix1ellipsex= -75e =
Answers
In order to find the polar equation of the ellipse, first let's find the rectangular equation.
Since the directrix is a vertical line, the ellipse is horizontal, and the model equation is:
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]Where the center is located at (h, k), the directrix is x = -a/e and the eccentricity is e = c/a.
So, if the eccentricity is e = 1/5 and the directrix is x = -7, we have:
[tex]\begin{gathered} \frac{c}{a}=\frac{1}{5}\rightarrow a=5c\\ \\ -\frac{a}{e}=-7\\ \\ \frac{a}{\frac{c}{a}}=7\\ \\ \frac{a^2}{c}=7\\ \\ \frac{25c^2}{c}=7\\ \\ 25c=7\\ \\ c=\frac{7}{25}\\ \\ a=5\cdot\frac{7}{25}=\frac{7}{5} \end{gathered}[/tex]Now, let's calculate the value of b with the formula below:
[tex]\begin{gathered} c^2=a^2-b^2\\ \\ \frac{49}{625}=\frac{49}{25}-b^2\\ \\ b^2=\frac{25\cdot49}{625}-\frac{49}{625}\\ \\ b^2=\frac{24\cdot49}{625}\\ \\ b^2=\frac{1176}{625} \end{gathered}[/tex]Assuming h = 0 and k = 0, the rectangular equation is:
[tex]\frac{x^2}{\frac{49}{25}}+\frac{y^2}{\frac{1176}{625}}=1[/tex]Now, to convert to polar form, we can do the following steps:
[tex]\begin{gathered} \frac{25x^2}{49}+\frac{625y^2}{1176}=1\\ \\ 600x^2+625y^2=1176\\ \\ 600(r\cos\theta)^2+625(r\sin\theta)^2=1176\\ \\ 600r^2\cos^2\theta+625r^2\sin^2\theta=1176\\ \\ r^2(600\cos^2\theta+625\sin^2\theta)=1176\\ \\ r^2=\frac{1176}{600\cos^2\theta+625\sin^2\theta}\\ \\ r=\sqrt{\frac{1176}{600\cos^2\theta+625\sin^2\theta}}\\ \\ r=\sqrt{\frac{1176}{600+25\sin^2\theta}} \end{gathered}[/tex]Another way of writing this equation in polar form is:
[tex]r=\frac{ep}{1+\sin^2\theta}[/tex]Where p is the distance between the focus and the directrix.
Since the foci are located at (±c, 0) = (±7/25, 0) and the directrix is x = -7, the distance is:
[tex]p=7-\frac{7}{25}=\frac{175}{25}-\frac{7}{25}=\frac{168}{25}[/tex]So the equation is:
[tex]\begin{gathered} r=\frac{\frac{1}{5}\cdot\frac{168}{25}}{1+\sin^2\theta}\\ \\ r=\frac{\frac{168}{125}}{1+\sin^2\theta}\\ \\ r=\frac{1.344}{1+\sin^2\theta} \end{gathered}[/tex]Hello Just Want to make sure my answer is correct
Answers
So,
Let's remember that:
The three point postulate states that:
Through any three noncollinear points, there exists exactly one plane.
The Plane-Point Postulate states that:
A plane contains at least three noncollinear points.
As you can notice, the diagram illustrates that:
Given that a plane exists, then, there are three collinear points.
That's the three point postulate.
the width of a rectangle is 8 inches less than its length, and the area is 9 square inches. what are the length and width of the rectangle?
Answers
The given situation can be written in an algebraic way:
Say x the width of the rectangle and y its height.
- The width of a rectangle is 8 inches less than its length:
x = y - 8
- The area of the rectangle is 9 square inches:
xy = 9
In order to find the values of y and x, you first replace the expression
x = y - 8 into the expression xy = 9, just as follow:
[tex]\begin{gathered} xy=9 \\ (y-8)y=9 \end{gathered}[/tex]you apply distribution property, and order the equation in such a way that you obtain the general form of a quadratic equation:
[tex]\begin{gathered} (y-8)y=9 \\ y^2-8y=9 \\ y^2-8y-9=0 \end{gathered}[/tex]Next, you use the quadratic formula to solve the previous equation for y:
[tex]y=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]here you have a = 1, b = -8 and c = 9. By replacing these values you obtain:
[tex]\begin{gathered} y=\frac{-(-8)\pm\sqrt[]{(-8)^2-4(1)(-9)}}{2(1)}=\frac{8\pm\sqrt[]{64+36}}{2} \\ y=\frac{8\pm\sqrt[]{100}}{2}=\frac{8\pm10}{2}=\frac{8}{2}\pm\frac{10}{2}=4\pm5 \end{gathered}[/tex]Hence, you have two solutions for y:
y1 = 4 + 5 = 9
y2 = 4 - 5 = -1
You select only the positive solution, because negative lengths do not exist in real life. Hence, you have y = 9.
Finally, you replace the value of y into the expression x = y - 8 to obtain x:
[tex]\begin{gathered} x=y-8 \\ x=9-8 \\ x=1 \end{gathered}[/tex]Hence, the width and length of the given recgtangle are:
width = 1 in
length = 9 in
Find the percent increase in volume when 1 foot is added to each dimension of the prism. Round your answer to the nearest tenth of a percent.7 ft10 ft86 ft
Answers
Solution
Step 1
The volume of a triangular prism = Cross-sectional area x Length
Step 2
[tex]\begin{gathered} Cross\text{ sectional area = area of the triangle} \\ Base\text{ = 6ft} \\ Height\text{ = 7ft} \\ Cross\text{ sectional area = }\frac{1}{2}\times\text{ 7 }\times\text{ 6 = 21 ft}^2 \\ Volume\text{ = 21 }\times\text{ 10 = 210 ft}^3 \end{gathered}[/tex]Step 3:
When 1 foot is added to each dimension of the prism.
The new dimensions becomes Base = 7, Height = 8 and length = 11
[tex]\begin{gathered} \text{Cross-sectional area = }\frac{1}{2}\text{ }\times\text{ 7 }\times\text{ 8 = 28 ft}^2 \\ Length\text{ = 11 ft} \\ Volume\text{ = 28 }\times\text{ 11 = 308 ft}^3 \end{gathered}[/tex]Step 4
Find the percent increase in volume
[tex]\begin{gathered} \text{Percent increase in volume = }\frac{308\text{ - 210}}{210}\text{ }\times\text{ 100\%} \\ \text{= }\frac{98}{210}\text{ }\times100 \\ \text{= 46.7} \end{gathered}[/tex]Final answer
46.7