Answer:
To solve the system of equations,
[tex]\begin{gathered} 5x-y=0 \\ \frac{y^2}{90}-\frac{x^2}{36}=1 \end{gathered}[/tex]Solving 1st equation we get,
[tex]y=5x[/tex][tex]\frac{y^2}{90}-\frac{x^2}{36}=1[/tex]Substitute y=5x in the above equation, we get
[tex]\frac{(5x)^2}{90}-\frac{x^2}{36}=1[/tex][tex]\frac{25x^2}{90}-\frac{x^2}{36}=1[/tex][tex]\frac{5x^2}{18}-\frac{x^2}{36}=1[/tex][tex]\frac{10x^2-x^2}{36}=1[/tex][tex]\frac{9x^2}{36}=1[/tex][tex]\frac{x^2}{4}=1[/tex][tex]x^2=4[/tex][tex]x=\pm2[/tex]when x=2, we get y=5x=5(2)=10
when x=-2, we get y=5x=5(-2)=-10
There are two solution for the given system.
[tex](2,10),(2,-10)[/tex]Answer is: x=2,y=10 and x=2,y=-10
The diameter of a bicycle wheel is 26 inches. What is its circumference? (Round to the nearest inch.)
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data
Bicycle wheel:
diameter = 26 in
circumference = ?
Step 02:
Circumference
C = π d
C = π * (26 in) = 81.68 in
The answer is:
C = 82 in
Answer:
82 inches.
Step-by-step explanation:
diameter = 26 inches
radius = 13 inches
circumference = 2πr
π = 22÷7 or 3.142
... 2 × 22÷7 × 13 = 81.714
(to nearest inch) = 82inches.
It cost Kaylee $7.26 to send 66 text messages. How much does each text cost to send? On the double number line below, fill in the given values, then use multiplication or division to find the missing value. dollars o text messages Answer: $ Submit Answer
This situation is represented by the equation
[tex]7.26=66x[/tex]Where x is the cost for each text sent, to find it, clear x from the equation
[tex]\begin{gathered} 7.26=66x \\ \frac{7.26}{66}=x \\ 0.11=x \\ x=0.11 \end{gathered}[/tex]It costs $0.11 to send a text
What is an equation for each translation of y=|x|?1. 4 units up2. 7 units down
To find an equation for each translation, keep in mind this:
f(x); g(x)=f(x)+a, the function is translated a units up.
f(x); g(x)=f(x)-a, the function is translated a units down.
Use this information to find the equation for each translation.
4 units up - Add 4 units to the function:
[tex]y=\lvert x\rvert+4[/tex]7 units down - Substract 7 units to the function:
[tex]y=\lvert x\rvert-7[/tex]Those are the answers for each translation.
Which is the graph of y = = where k is a constant?TO A.R०B.कOD. REE.
Step 1
Given;
[tex]\begin{gathered} y=\frac{k}{x} \\ where\text{ k is a constant.} \end{gathered}[/tex]Required; To find the graph of the function.
Step 2
Since the graph is that of a fraction it will be a graph that may have a discontinuity and the answer will be;
Complete the following: 1, Jabomplete the squares for each quadratic, list the center and radius, then graph each circle (a labeling its translated center: (a) r2 + 2x + y2 - 4y = 4 (c) 2x2 + 2y2 + 3x - 5y = 2 (e) r2 + y2 + 3x = 4 (g) x² + y2 + 4x = 0 (1) r² + y2 + 2mx - 2ny = 0 (b) x2 + y2 - 4x = 0 (d) x2 + y2 - 2x - 8y = 8 4x + 4y? - 16x + 24y = -27 (h) x + y? - 7y = 0 (i) x + y2 - 2ax + 2by = c Determine which of the following equations represents a circle with a real non-zero radiu a) r? + y + 10x = -30 (b) 3x2 + 3y? - 11x = -91 4x + 4y + 18-8y = -85 (d) 36x* + 36y- 36x + 48y = -16 the equation of the circle which accen 2 and is concentric
3x² + 3y² - 11x = -91
Divide through by 3
x² + y² - 11/3 x = -91/3
x² - 11/3 x + y² = -91/3
(x² - 11/3 x ) + y² = -91/3
[x² - 11/3 x +(-11/6)² ] + y² = -91/3 + (- 11/6)²
(x - 11/6)² + y² = -91/3 + 121 / 36
[tex](x-\frac{11}{6})^2+y^2=\frac{-1092+\text{ 121}}{36}[/tex][tex](x\text{ - }\frac{11}{6})^2+y^2=\frac{-971}{36}[/tex]Comparing this with (x-a)² + (y-b)² = r²
r² = -971/36
Taking the square root will give an immaginary number
The radius is NOT a real number
This equation does not have a real radius
Avery flipped a coin 24 times and recorded the results in the box below. Which of the following best describes the difference between the experimental and theoretical probability of flipping heads?Heads= h Tails= th, t, t, h, t, t, t, h, t, t, h, t, t, t, h, t, t, t, t, h, t, t, t, t.A. According to theoretical probability, Avery would have expected to land on heads ten more times than she did experimentally.B. According the theoretical probability, Avery would have expected to land on heads half as many times as she did experimentally.C. According to theoretical probability, Avery would have expected to land on heads twite as many times as she did experimentally.D. According to theoretical probability, Avery would have expected to land on heads 12 more times than she did experimentally.
In order to determine the best description, first count the number of times for heads h and tails t.
Based on the given information, the number of heads was h = 6 and the number of tails was t = 18.
The theoretical probability in this case results in a 50% of probability for both events, that is, for 24 times at which the coin was flipped, 12 would be tail and 12 would be head.
Then, based on the previous explanation, you can conclude that:
C. According to theoretical probability, Avery would have expected to land on heads twite as many times as she did experimentally.
For each value of y,determine whether it is a solution to y<7
Y < 7 indicates that any value below 7 is included
therefore, 5 is a solution, 12 is NOT a solution, 7 is NOT a solution and 4 is a solution.
A correlation cannot have the value:A) 0.0B) 0.4C) -1.01D) -0.5E) 0.99
The possible range of values for the correlation coefficient is -1.0 to 1.0. In other words, the values cannot exceed 1.0 or be less than -1.0. A correlation of -1.0 indicates a perfect negative correlation, and a correlation of 1.0 indicates a perfect positive correlation.
Therefore, the value that is not within the range -1.0 to 1.0 is -1.01
Answer: C)
Which of the following ordered pairs is a solution to the graph of the system of inequalities? Select all that apply(5, 2)(-3, -4)(0, -3)(0, 1)(-4, 1)
ANSWER
(5, -2) and (0, -3)
EXPLANATION
We want to find which of the ordered pairs is a solution to the system of inequalities.
Ordered pairs are written in the form (x, y), this means, whichever ordered pair is a solution, when inserted into the system of inequalities, should be true.
This means that the values of x and y must be true for both inequalities in the system.
The system of inequalities is:
[tex]\begin{cases}-2x-3\leq\text{ y} \\ y-1<\text{ }\frac{1}{2}x\end{cases}[/tex]A. (5, -2)
[tex]\begin{gathered} -2(5)\text{ - 3 }\leq-2\Rightarrow\text{ -10 - 3}\leq-2\Rightarrow\text{ -13 }\leq-2 \\ -2\text{ - 1 < }\frac{1}{2}(5)\Rightarrow\text{ -3 < }\frac{5}{2} \end{gathered}[/tex]Since both inequalities are correct, this is a solution.
B. (-3, -4)
[tex]-2(-3)\text{ - 3 }\leq-4\Rightarrow\text{ 6 - 3 }\leq-4\Rightarrow\text{ 3}\leq-4[/tex]Since the first inequality is already incorrect, we do not need to go further.
It is not a solution
C. (0, -3)
[tex]\begin{gathered} -2(0)\text{ - 3 }\leq\text{ -3 }\Rightarrow\text{ -3 }\leq\text{ -3} \\ -3\text{ - 1 < }\frac{1}{2}(0)\Rightarrow\text{ -4 < 0} \end{gathered}[/tex]Since both inequalities are correct, this is a solution.
D. (0, 1)
[tex]\begin{gathered} -2(0)\text{ - 3 }\leq\text{ 1 }\Rightarrow\text{ -3 }\leq\text{ 1} \\ 1\text{ - 1 < }\frac{1}{2}(0)\Rightarrow\text{ 0 < 0} \end{gathered}[/tex]Since 0 is not less than 0, this is not a solution.
E. (-4, 1)
[tex]-2(-4)\text{ - 3 }\leq\text{ 1}\Rightarrow\text{ 8 - 3 }\leq1\Rightarrow\text{ 5 }\leq1[/tex]Since 5 is not less than 1, this is not a solution.
Therefore, the solutions are (5, -2) and (0, -3)
The diagram shows two parallel lines cut by a transversal. One angle measure is shown.do8abº5499coFind the values of a, b, c, d, e, f, and g.
This question applies the rules of angles on a plane. The transversal that cuts the two parallel lines is a decisive one. From there you can determine which angles are opposite, alternate and so on.
Obeserve carefully that angle 54 and angle a lie on a straight line.
"Angles on a straight line sum up to 180 degrees."
Therefore,
Angle 54 + Angle A = 180
54 + A = 180
Subtract 54 from both sides of the equation
54 - 54 + A = 180 - 54
A = 126
Also note that;
"Opposite angles are equal in size."
Angle B is opposite to angle 54
Therefore angle B is 54 degrees.
Note also that angle C is opposite to angle A, therefore angle C equals angle A and that makes angle C = 126
If the two parallel lines are cut by a transversal, then it makes it easy to identify alternate angles. Alternate angles are formed on the inner sides of the two parallel lines but on the opposites sides of the tranversal. If you observe VERY CLOSELY, it usualltakes the form of a Z shape. You can equally determine alternate angles on the outer parts of the parallel lines in which case it becomes "exterior alternate angles."
"Alternate angles are equal."
Observe carefully and you'll see that angle B and angle D are interior alternate angles. That means B equals to D and therefore angle D = 54 degrees.
Similarly, angle A and angle G are alternate angles. Therefore angle G = 126 degrees.
Also angle F is opposite to angle D, and therefore angle F = 54 degrees.
Angle E is opposite to angle G, therefore angle E = 126 degrees
Help with this plsss !!!
The average rate of change of function f(x) over the interval 12 ≤ x ≤ 21 is 2/3
We use the formula of average rate of change of function over the interval [x1, x2],
r = [f(x2 - f(x1)]/ (x2 - x1)
We need to find the average rate of change of function f(x) over the interval 12 ≤ x ≤ 21
r = [f(21) - f(12)] / (21 - 12)
r = (31 - 25) / 9
r = 6/9
r = 2/3
Therefore, the average rate of change of function f(x) over the interval 12 ≤ x ≤ 21 is 2/3
Learn more about the average rate of change of function here:
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Can you help explain how to solve this for me?
SOLUTION:
[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex][tex]\begin{gathered} d=\sqrt{(-4-8)^2+(5--6)^2} \\ d=\sqrt{-12^2+11^2} \\ d=\sqrt{265}=16.28 \end{gathered}[/tex]b.
[tex]\begin{gathered} m=\frac{x_2+x_1}{2},\frac{y_2+y_1}{2} \\ m=(\frac{-6+5}{2},\frac{-4+8}{2}) \\ m=(-\frac{1}{2},2) \end{gathered}[/tex]Graph the function. Label the vertex and axis of symmetry. 1. f(x)=(x-2)^2
We have the following:
consider all of the 4 digit numbers that can be made from the digits 0 to 9 (assume that the numbers cannot start with 0) . What is the probability of choosing a random number from this group that is less than or equal to 8000? Enter a fraction or round your answer to 4 decimal places, if necessary.
First, we need to determine the total amount of numbers fulfilling the conditions:
- 4 digits
- Not starting with 0
For the first digit, we have then 9 possible numbers: 1, 2, 3, 4, 5, 6, 7, 8 and 9.
For the second, third and fourth, we have 10 possible numbers 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Then, to determine the amount of numbers available we just need to multiply the possibilities for each digit:
[tex]9\cdot10\cdot10\cdot10=9000[/tex]Then, randomly choosing one of the given numbers, we have 9000 possible outcomes. Those will be numbers from 1000 to 9999.
Now we just need to determine how many numbers among those 9000 are lower than or equal to 8000.
As the numbers start in 1000, we have 7001 cases where the randomly selected number is lower than or equal to 8000.
We obtain 7001 since 8000 - 1000 = 7000 but we need to consider also the number 1000.
The probability will be then:
[tex]\frac{7001}{9000}\approx0.7779[/tex]Write the slope of the line in slope-intercept form using y=mx+b
In order to find the equation for this line in the slope-intercept form, let's use two points of the line in the equation.
Using the points (-3, 3) and (0, -3), we have:
[tex]\begin{gathered} y=mx+b \\ (0,-3)\colon \\ -3=0\cdot m+b \\ b=-3 \\ \\ (-3,3)\colon \\ 3=-3m-3 \\ -3m=6 \\ m=-2 \end{gathered}[/tex]So the slope of this line is m = -2, the y-intercept is b = -3 and the equation is y = -2x - 3.
Two trees are leaning on each other in the forest. One tree is 19 feet long and makes a 32° angle with the ground. The second tree is 16 feet long.What is the approximate angle, x, that the second tree makes with the ground?
39º
1) Considering what's been given we can sketch this out:
From these trees leaning on each other, we can visualize a triangle (in black).
2) So now, since we need to find the other angle, then we need to apply the Law of Sines to find out the missing angle:
[tex]\begin{gathered} \frac{a}{\sin(A)}=\frac{b}{\sin (B)} \\ \frac{16}{\sin(32)}=\frac{19}{\sin (X)} \\ 16\cdot\sin (x)=19\cdot\sin (32) \\ \frac{16\sin(X)}{16}=\frac{19\sin (32)}{16} \\ \sin (X)=\frac{19\sin(32)}{16} \\ \end{gathered}[/tex]As we need the measure of the angle, (not any leg) then we need to use the arcsine of that quotient:
[tex]\begin{gathered} X=\sin ^{-1}(\frac{19\cdot\sin (32)}{16}) \\ X=38.996\approx39 \end{gathered}[/tex]3) Hence, the approximate measure of that angle X is 39º
[tex]( - 4x + 2y \leqslant 4 )(x + 4y \ \textgreater \ - 10)[/tex]how do I graph this
Given the system of inequalities
[tex]\begin{gathered} -4x+2y\leq4 \\ x+4y>-10 \end{gathered}[/tex]We start by getting the plot of one of these inequalities. Let's start on -4x + 2y ≤ 4. This equation can be rewritten as
[tex]\begin{gathered} 2y\leq4+4x \\ y\leq2+2x \end{gathered}[/tex]Initially, we consider the equation
[tex]y=2+2x[/tex]Plotting the equation, we have
Considering the values of these function at
[tex]y\leq2+2x[/tex]Let's use the same steps implemented above for the second inequality. We have
[tex]\begin{gathered} 4y>-10+x \\ y>\frac{x-10}{4} \end{gathered}[/tex]Plotting this we have,
The dashed line represents the values that are not included in the equation, as a present for inequalities with less than or greater than. The shaded region represents the solution
After plotting the two inequalities individually, we now superimpose the two graphs. We get
where the darker region represents the solution of the inequalities.
8. Su, who is 5 feet tall, is standing at point D in the drawing. The top of her head is at point E. A tree yard is at point B with the top of the tree at point C. Su stands so her shadow meets the end of the t shadow at point A. What is the length of side BC? C С + E 5 ft А 8 ft D 24 ft A) 20 feet B 15 fut B C) 22 feet D) 18 feet
Explanation:
We would be applying similar triangles theorem.
If you check the image, there is a small triangle and a big triangle
For similar triangles, the ratio of the corresponding sides are equal.
Trianglke AEB is similar to triangle ECB
AD corresponds to AB
ED corresponds to CB
AD/AB = ED/CB
AD = 8 ft,
AB = AD + DB = 8+24 = 32
ED = 5 ft
CB = ?
In the figure, ABCD and EFGF are rectangle. ABCD and EFGH are similar.(a) If the length of AB is a cm, try to use a to Indicate the length of EF(b) Find the ratio of the areas of ABCD and EFGH.(English isn't my native language. Please correct me if I have any grammatical mistakes.)
Given:
BC = 3 cm, FG = 4 cm
Required: bLength of EF and ratio of areas
Explanation:
(a) Since the rectangles ABCD and EFGH are similar, the correponding angles are proportional. Hence
[tex]\frac{AB}{EF}=\frac{BC}{FG}[/tex]Plug the given values.
[tex]\frac{AB}{EF}=\frac{3}{4}[/tex]If AB = a cm, then
[tex]\begin{gathered} \frac{a}{EF}=\frac{3}{4} \\ EF=\frac{4a}{3} \end{gathered}[/tex](b) Ara of ABCD
[tex]\begin{gathered} =\text{ Length}\times\text{ Width} \\ =3a\text{ cm}^2 \end{gathered}[/tex]Area of EFGH
[tex]\begin{gathered} =\text{ Length}\times\text{ Width} \\ =4\times\frac{4a}{3} \\ =\frac{16a}{3}\text{ cm}^2 \end{gathered}[/tex]Ratio of areas
[tex]\begin{gathered} =3a:\frac{16a}{3} \\ =9:16 \end{gathered}[/tex]Final Answer: The ratio of areas of ABCD to EFGH is 916.
Solve for the missing side lengths.V1045°A. Ou = 10/2, vV =10./33B. Ou-20v2, v =1033c. Ou = 20v2, v =10D. Ou = 10v2, v = 10
We have a right triangle, where we know that one of the angles (besides the right angle) has a measure of 45°.
Then, the other angle measure can be calculated as:
[tex]\begin{gathered} \alpha+45+90=180 \\ \alpha=180-90-45 \\ \alpha=45\degree \end{gathered}[/tex]Then, as the other angle measure is equal, we have an isosceles triangle.
Then, length v has to be equal to the side with length 10.
With the value of v we can calculate u with the Pythagorean theorem:
[tex]\begin{gathered} u^2=v^2+10^2 \\ u^2=10^2+10^2 \\ u^2=2\cdot10^2 \\ u=\sqrt[]{2}\cdot10 \\ u=10\sqrt[]{2} \end{gathered}[/tex]Answer: u = 10√2, v = 10
What is the length of the side of an equilateral triangle if the height is 9√3
An equilateral triangle is a triangle were all the sides have the same measurement, and all the angles are the same(60º).
The height of an equilateral triangle divides the triangle into two equal right triangles. The height represents the oposite side of the angle of 60º, and the hypotenuse has the length of the side of the equilateral triangle, if we find the hypotenuse we have our answer.
Using trigonometric relations on the right triangle, we can find the value for the hypotenuse. The ratio between the opposite side to an angle and the hypotenuse is equal to the sine of this angle. If we call the hypotenuse as h, we have the following relation
[tex]\sin (60^o)=\frac{9\sqrt[]{3}}{h}[/tex]The sine of 60º is a known value
[tex]\sin (60^o)=\frac{\sqrt[]{3}}{2}[/tex]Then, combining both expressions, we have
[tex]\frac{9\sqrt[]{3}}{h}=\frac{\sqrt[]{3}}{2}[/tex]Solving for h
[tex]\begin{gathered} \frac{9\sqrt[]{3}}{h}=\frac{\sqrt[]{3}}{2} \\ \frac{9}{h}=\frac{1}{2} \\ \frac{h}{9}=2 \\ h=18 \end{gathered}[/tex]The length of the side of an equilateral triangle if the height is 9√3 is equal to 18.
The graph below shows a company’s profit f(x) in dollars, depending on the price of goods x, in dollar’s, being sold by the company: Part A: What do the x-intercepts and maximum value of the graph represent in context of the disrobed situation?Part B: What are the intervals where the function is increasing and decreasing, and what do they represent about the sale and profit for the company in the situation described?Part C: What is an approximate average rate of change of the graph from x=1 to x=3, and what does this rate represent in context of the described situation?
We will have the following:
Part A:
The x-intercepts represent the prices of the goods than wen sold represent no net gain or loss.
The maximum value represents the price at which there will be a maximum profit.
Part B:
We will have that the increasing and decreasing intervals are respectively:
[tex]I_{\text{increaing}}=(-\, \infty,3)[/tex][tex]I_{\text{decreasing}}=(3,\infty)[/tex]They tells us respectively that:
Increasing: The greater the price the greater the profit.
Decreasing: The greater the price the smaller the profit.
Part C:
We determine the equation of the parabola. We can see that it's vertex is located at (3, 120), we can also see that the parabola passes by the origin (0, 0); so:
[tex]f(x)=a(x-3)^2+120\Rightarrow0=a(0-3)^2+120[/tex][tex]\Rightarrow0=9a+120\Rightarrow9a=-120\Rightarrow a=-\frac{40}{3}[/tex]So, the equation that represents the parabola is:
[tex]f(x)=-\frac{40}{3}(x-3)^2+120[/tex]Then, we will determine the average rate of change as follows:
[tex]\text{average rate of change}=\frac{f(b)-f(a)}{b-a}[/tex]So:
[tex]\text{average rate of change}=\frac{(-40/3((3)-3)^2+120)-(-40/3((1)-3)^2+120)}{3-1}[/tex][tex]\text{average rate of change}=\frac{80}{3}\Rightarrow average\text{ rate of change}\approx26.67[/tex]So, the avereage rate of change for the graph from x = 1 to x = 3 is exactly 80/3, that is approximately 26.67.
Can you please check 3 and 4 to see if I did them right?
We are given the following linear equation:
[tex]5x+15=45[/tex]The following problem is an example of a situation that can be modeled using the given equation.
Hayle did some chores this week, she got 5 dollars for each chore she did. Her dad forgot to pay her some days and gave her $15 dollars for the missing days. If she has a total of $45 dollars. How many chores did Hayle do?. The number of chores is represented by the variable "x".
I struggle with word problems please helpYou are ordering a new home theater system that consists of a TV, surround sound system, and DVD player. You can choose from 6 different TVs, 8 types of surround sound systems, and 20 types of DVD players. How many different home theater systems can you build?
We are given the following :
• Number of different Tvs = 6
,• Number of different surround system = 8
,• Number of different Dvds = 20
In order to determine How many different home theater systems you can build: just multiply the items as follows :
=6*8*20 = 960 You can build 960 home theater systems.
How many men and women should the sample include. What were the steps you took to solve?
We are asked to determine the sample size to determine the difference in the proportion of men and women who own smartphones with a confidence of 99% and an error of no more than 0.03. If we assume that both samples are equal then we can use the following formula:
[tex]n=\frac{Z^2_{\frac{\alpha}{2}}}{2E^2}[/tex]Where Z is the confidence and E is the error. Replacing the values we get:
[tex]n=\frac{(0.99)^2}{2(0.03)^2}[/tex]Solving the operations we get:
[tex]n=544.5\cong545[/tex]Therefore, each sample of men and women should be of 545.
Halley's Comet appears in the sky once every 75 years it last appeared in 1986 two students wrote Expressions to find out which years after 1986, will appear.Julian says 75× Casey says 1986+75xWhose expression is correct and why?A) Neither expression is correct, because the correct expression is 75 + 1986x.B)Both expressions are correct because both show that appears every 75 years.D) Julian's expression is correct, because it shows that the comet appears every 75 years. The hear it last appeared it doesn't matter.
The expression that will tell us the years after 1986 is Casey's:
[tex]1986+75x[/tex]The reason being: every 75 years the comet will do another "pass by" and we will have to add 75 each new time.
So, the option is B) since each can be used to determine the values, but Casey's is more "practical".
Last week, Thor ran 30 laps around the lake. Shaq ran 5/9 as many laps around the lake as Thor did. How many laps around the lake did Shaq run?
Explanation:
Thor: 30 laps
Shaq: 5/9 Thor's laps.
To find how many laps Shaq did, we have to multiply Thor's laps by 5/9:
[tex]30\times\frac{5}{9}=\frac{30\times5}{9}=\frac{150}{9}[/tex]And simplify the fraction:
[tex]\frac{150}{9}=\frac{50}{3}[/tex]It's an improper fraction. Written as a mixed number:
[tex]\frac{50}{3}=16\frac{2}{3}[/tex]Answer:
Shaq did 16 2/3 laps around the lake
The box-and-whisker plot shows the ages of employees at a video store. What fraction of the employees are 20 years or older Ages of Employees + 16 + 18 + 28 + Age 14 20 22 24 26 30 32 34 About of the employees are 20 years or older.
Based on the given box-and-whisker plot. Consider that 20 years concides with the second quartile or median of the sample.
It means that one half of the employees at the video store are 20 years or older.
Hence, the fraction of such employees related to the total number of employess is 1/2.
1 ptsQuestion 5Jane started jogging 5 miles from home, at a rate of 2 mph. Write the slope-intercept form of an equation for Jane's position relative to home.
Answer
[tex]y=2x+5[/tex]SOLUTION
Problem Statement
The question wants us to model the distance Jane is from her home given her initial starting point (5 miles from home) and her speed (2 mph)
Explanation
To solve the question, we simply need to model her jogging using the equation of a line.
The general equation of a line is given as:
[tex]\begin{gathered} y=mx+c \\ \text{where,} \\ m=\text{slope}=\text{this represents Jane's speed} \\ c=y-\text{intercept}=\text{this represents her initial position from her home} \\ x=\text{time taken for Jane to move} \\ y=\text{Jane's final position after moving for time, x} \end{gathered}[/tex]We have been told that her speed is 2 mph. Thus, m = 2. We have also been given her initial position from her house to be 5 miles.
Jane starts jogging 5 miles from her home, thus, her position relative to her home will continue to increase as she jogs on at 2 mph. Thus, c = 5 and NOT -5.
This means we can write the equation for her position is:
[tex]\begin{gathered} m=2,c=5 \\ \therefore y=2x+5 \end{gathered}[/tex]Final Answer
[tex]y=2x+5[/tex]Graph the function?Can you also make a chart or like try to edit onto the graph in the picture
For x = 0 , x = 2 and x = 1, we have the following values:
[tex]\begin{gathered} f(0)=2(\frac{1}{2})^0=2\cdot1=2 \\ f(2)=2(\frac{1}{2})^2=\frac{2}{4}=\frac{1}{2} \\ f(1)=2(\frac{1}{2})^1=\frac{2}{2}=1 \end{gathered}[/tex]thus, the graph would look like this taking these point as references: