She can estimate that 50 seventh-graders will be boarding the bus to go to school.
The unitary technique entails finding the value by multiplying the single value and then solving the problem using the initial value of a single unit.
By using the unitary technique, we can determine the value of many units from the value of a single unit as well as the value of multiple units from the value of a single unit. We typically utilise this technique for math calculations.
10 out of the 32 children in the first-period class that we are given ride the bus to school. There are 160 students in seventh grade.
Therefore, we have;
160/32=5
10 x 5 =50
Thus, the answer is 50
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Please Help me with this problem!!! ASAP
The average rate of change is -5 per month
How to determine the average rate of change?The interval is given as:
July to December
From the graph, we have:
July = 110
December = 30
The average rate of a function over the interval (a, b) is calculated as:
Rate = [f(b) - f(a)]/[b - a]
So, we have:
Rate = (30 - 110)/(December- July)
This gives
Rate = (30 - 110)/(12 - 7)
Evaluate
Rate = -16
Hence, the average rate of change is -5 per month
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100 POINTS PLEASE HELP!!!
The coordinate plane below represents a community. Points A through F are houses in the community.
graph of coordinate plane. Point A is at negative 5, 5. Point B is at negative 4, negative 2. Point C is at 2, 1. Point D is at negative 2, 4. Point E is at 2, 4. Point F is at 3, negative 4.
Part A: Using the graph above, create a system of inequalities that only contains points C and F in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above. (7 points)
Part B: Explain how to verify that the points C and F are solutions to the system of inequalities created in Part A. (5 points)
Part C: Erica wants to live in the area defined by y < 7x − 4. Explain how you can identify the houses in which Erica is interested in living. (2 points)
Answer:
[tex]\sf A) \quad \begin{cases}\sf y > -5x+5\\\sf y < -5x+12\end{cases}[/tex]
B) see below
C) points C, E and F
Step-by-step explanation:
Given points:
A = (-5, 5)B = (-4, -2)C = (2, 1)D = (-2, 4)E = (2, 4)F = (3, -4)Part AA system of inequalities is a set of two or more inequalities in one or more variables.
To create a system of inequalities that only contains C and F in the overlapping shaded region, create a linear equation where points C, F and E are to the right of the line and a linear equation where points C, F, A, B and D are to the left of the line.
The easiest way to do this is to find the slope of the line that passes through points C and F, then add values to move the lines either side of the points.
[tex]\sf slope\:(m)=\dfrac{change\:in\:y}{change\:in\:x}=\dfrac{y_F-y_C}{x_F-x_C}=\dfrac{-4-1}{3-2}=-5[/tex]
Therefore:
[tex]\sf y = -5x + 5[/tex] → points C, F and E are to the right of the line.
[tex]\sf y=-5x+12[/tex] → points C, F, A, B and D are the left of the line.
Therefore, the system of inequalities that only contains points C and F in the overlapping shaded regions is:
[tex]\begin{cases}\sf y > -5x+5\\\sf y < -5x+12\end{cases}[/tex]
To graph the system of inequalities:
Plot 2 points on each of the lines.Draw a dashed line through each pairs of points.Shade the intersected region that is above the line y > -5x + 5 and below the line y < -5x + 12.Part BTo verify that the points C and F are solutions to the system of inequalities created in Part A, substitute the x-values of both points into the system of inequalities. If the y-values satisfy both inequalities, then the points are solutions to the system.
Point C (2, 1)
[tex]\implies \sf x=2 \implies 1 > -5(2)+5 \implies 1 > -5\quad verified[/tex]
[tex]\implies \sf x=2 \implies 1 < -5(2)+12\implies 1 < 2 \quad verified[/tex]
Point F (3, -4)
[tex]\implies \sf x=3 \implies -4 > -5(3)+5 \implies -4 > -10\quad verified[/tex]
[tex]\implies \sf x=3 \implies -4 < -5(3)+12\implies -4 < -3 \quad verified[/tex]
Part CMethod 1
Graph the line y = 7x - 4 (making the line dashed since it is y < 7x - 4).
Shade below the dashed line.
Points that are contained in the shaded region are the houses in which Erica is interested in living: points C, E and F.
Method 2
Substitute the x-value of each point into the given inequality y < 7x - 4.
Any point where the y-value satisfies the inequality is a house that Erica is interested in living.
[tex]\sf Point\: A: \quad x=-5 \implies 5 < 7(-5)-4 \implies -5 < -39 \implies no[/tex]
[tex]\sf Point\: B: \quad x=-4 \implies -2 < 7(-4)-4 \implies -2 < -32 \implies no[/tex]
[tex]\sf Point\: C: \quad x=2 \implies 1 < 7(2)-4 \implies 1 < 10 \implies yes[/tex]
[tex]\sf Point\: D: \quad x=-2 \implies 4 < 7(-2)-4 \implies 4 < -18 \implies no[/tex]
[tex]\sf Point\: E: \quad x=2 \implies 4 < 7(2)-4 \implies 4 < 10 \implies yes[/tex]
[tex]\sf Point\: F: \quad x=3 \implies -4 < 7(3)-4 \implies -4 < 17 \implies yes[/tex]
Amira's mother was in her garden, tying tomato plants to wooden stakes: it took her 10 minutes to stake each plant and she placed the plants 2 feet apart. IF she started at one end of the row at 8:00 am, how far from the first plant was the tomato plant she finished tying up at 9:00 am?
Answer:
12ft
Step-by-step explanation:
It takes 10 minutes for each plant and she spent 1 hour in total or 60 minutes: 60/10 = 6 plants
If each of the plants is 2 feet apart and there are 6 plants: 2ft*6 = 12ft
The distribution of results from a cholesterol test has a mean of 180 and a standard deviation of 20. A sample size of 40 is drawn randomly. Find the probability that the sum of the 40 values is less than 7,100. (Round your answer to four decimal places.)
Using the normal distribution, there is a 0.2148 = 21.48% probability that the sum of the 40 values is less than 7,100.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].For this problem, these parameters are given as follows:
[tex]\mu = 180, \sigma = 20, n = 40, s = \frac{20}{\sqrt{40}} = 3.1623[/tex]
A sum of 7100 is equivalent to a sample mean of 7100/40 = 177.5, which means that the probability is the p-value of Z when X = 177.5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{177.5 - 180}{3.1623}[/tex]
Z = -0.79
Z = -0.79 has a p-value of 0.2148.
There is a 0.2148 = 21.48% probability that the sum of the 40 values is less than 7,100.
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I need help I'm stuck on this
a) The length of p is 21.4 cm
b) The measure of angle P is 45.6°
TrigonometryFrom the question, we are to determine the length of p
From the Pythagorean theorem, we can write that
30² = 21² + p²
p² = 30² - 21²
p² = 900 - 441
p² = 459
p = √459
p = 21.4 cm
∴ The length of p is 21.4 cm
b)
Measure of angle P
Using SOH CAH TOA
[tex]cos P= \frac{21}{30}[/tex]
cos P = 0.7
P = cos⁻¹(0.7)
P = 45.6°
Hence, the measure of angle P is 45.6°
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Find a linear inequality with the following solution set. Each grid line represents one unit. [asy] size(200); fill((-2,-5)--(5,-5)--(5,5)--(3,5)--cycle,yellow); real ticklen=3; real tickspace=2; real ticklength=0.1cm; real axisarrowsize=0.14cm; pen axispen=black+1.3bp; real vectorarrowsize=0.2cm; real tickdown=-0.5; real tickdownlength=-0.15inch; real tickdownbase=0.3; real wholetickdown=tickdown; void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) { import graph; real i; if(complexplane) { label("$\textnormal{Re}$",(xright,0),SE); label("$\textnormal{Im}$",(0,ytop),NW); } else { label("$x$",(xright+0.4,-0.5)); label("$y$",(-0.5,ytop+0.2)); } ylimits(ybottom,ytop); xlimits( xleft, xright); real[] TicksArrx,TicksArry; for(i=xleft+xstep; i 0.1) { TicksArrx.push(i); } } for(i=ybottom+ystep; i 0.1) { TicksArry.push(i); } } if(usegrid) { xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.1),extend=true),p=invisible);//,above=true); yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.1),extend=true), p=invisible);//,Arrows); } if(useticks) { xequals(0, ymin=ybottom, ymax=ytop, p=black, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize)); yequals(0, xmin=xleft, xmax=xright, p=black, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize)); } else { xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize)); yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize)); } }; draw((-2,-5)--(3,5),dashed+red, Arrows(size=axisarrowsize)); rr_cartesian_axes(-5,5,-5,5); f
The linear inequality of the graph is: -x + 2y + 1 > 0
How to determine the linear inequality?First, we calculate the slope of the dashed line using:
[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
Two points on the graph are:
(1, 0) and (3, 1)
The slope (m) is:
[tex]m = \frac{1 - 0}{3 - 1}[/tex]
This gives
m = 0.5
The equation of the line is calculated as:
[tex]y = m(x -x_1) + y_1[/tex]
So, we have;
[tex]y = 0.5(x -1) + 0[/tex]
This gives
[tex]y = 0.5x -0.5[/tex]
Multiply through by 2
[tex]2y = x - 1[/tex]
Now, we convert the equation to an inequality.
The line on the graph is a dashed line. This means that the inequality is either > or <.
Also, the upper region of the graph that is shaded means that the inequality is >.
So, the equation becomes
2y > x - 1
Rewrite as:
-x + 2y + 1 > 0
So, the linear inequality is: -x + 2y + 1 > 0
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Complete question
Find a linear inequality with the following solution set. Each grid line represents one unit. (Give your answer in the form ax+by+c>0 or ax+by+c [tex]\geq[/tex] 0 where a, b, and c are integers with no common factor greater than 1.)
The function f(x) = (0.2)x
O increases for x > 0
O increases for all x
O decreases for all x
O decreases for x > 0
Answer:
Option (3)
Step-by-step explanation:
Since the base of the exponential is less than 1, the function is decreasing for all x.
PLS HELP!!!!!!!!!!!!! math related
Answer: B
Step-by-step explanation:
When you evaluate f(-1)=2x^4+x^2-5 we will get -2 in basic notation (-1,-2). We need to find the root of the function to find the other roots or factors, where y=0.
A scale drawing of a house addition shows a scale factor of 1 in. = 3.3 ft. Josh decides to make the house addition smaller, and he changes the scale of the drawing to 1 in. = 1.1 ft.
What is the change in the scale factor from the old scale to the new scale?
The change in the scale factor from the old scale to the new scale is 3.
What is the change in the scale factor?A scale drawing is a reduced form in terms of dimensions of an original image / building / object. The scale drawing is usually reduced at a constant dimension.
The change in the scale factor = larger scale / smaller scale
3.3 / 1.1 = 3
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Chocolate costing $\$10$ per pound is mixed with nuts costing $\$4$ per pound to make a mixture costing $\$6$ per pound. What fraction of the mixture's weight is chocolate
1/3 portion of the mixture, costing $\$6$ contains chocolate.
Calculation of the weight of the chocolate in the mixture:Assume that, in a mixture:
chocolate = x pound
nuts = y pound
mixture= x+y
The desired value is found by dividing the weight of the chocolate by the overall weight, or x/(x+y).
From the given data,
cost of x pound chocolate at $\$10$per pound = 10x
cost of y pound nuts at $\$4$ per pound = 4y
cost of x+y pound mixture at $\$6$ = 6(x+y)
As per condition,
⇒10x + 4y = 6(x+y)
⇒10x + 4y = 6x+ 6y
⇒4x = 2y
⇒y = 2x
The fraction of the chocolate in the mixture is
[tex]x/(x+y)[/tex] = [tex]x/(x+2x)[/tex] = [tex]x/3x[/tex] = 1/3
Therefore, chocolate makes 1/3 of the mixture's weight.
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Answer:
1/3
Step-by-step explanation:
I legit forgot how to do this I need help with this
Answer: 3/4
Step-by-step explanation:
Tangent is opp/adj
Therefore tan (x) = 18/24, which reduces to 3/4
Help me solve this question!
Answer:
1st quartile is 136
The second quartile is also the median 142
The third quartile is 162
The interquartile range is the difference between the 3rd and first quartile or 26
Step-by-step explanation:
d all the time and I like the time you
Answer:
Step-by-step explanation:
1a+1b(a+b-c)+1b+1c(b=c-a)+ 1a+1c(c+a-b)
1a+1b-c+1b +1c-a+1a+1c-b
1a+2b-c+a+1c-b
2a+1b+c this is the answer I think lol
Please help me it would mean a lot!! due right now. Please show work thank youuuu
Answer:
area of A'B'C'D' = 18.4 units²
Step-by-step explanation:
given 2 similar figures with
ratio of corresponding sides = a : b , then
ratio of areas = a² : b²
here ratio of sides = 5 : 2
ratio of areas = 5² : 2² = 25 : 4
let x be the area of A'B'C'D' , then by proportion
[tex]\frac{25}{115}[/tex] = [tex]\frac{4}{x}[/tex] ( cross- multiply )
25x = 460 ( divide both sides by 25 )
x = 18.4
area of A'B'C'D' = 18.4 units²
Solve the logarithmic equation. write the answer in exact, simplified form. log(-c) + log8= log (7c-75)
The value of c is 5 when we simplify the logarithmic equation log(-c) + log8= log (7c-75). This can be obtained using the properties of the logarithm.
What is the value of c ?Given that,
log(-c) + log8 = log (7c-75)
log (-8c) = log (7c-75) (∵ log a + log b = log ab)
- 8c = 7c - 75
(8+7)c = 75
15c = 75
⇒ c = 5
Hence the value of c is 5 when we simplify the logarithmic equation log(-c) + log8= log (7c-75).
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SELECT ALL THAT APPLY. In a population of 250 students, 60% are Whites, 20% are Latinos, 15% are Blacks, and 5% others. In a proportionate stratified random sample of 120, ______.
In a proportionate stratified sample of 120, there are 72 Whites, 24 Latinos, 18 Blacks and 6 others.
Proportionate Stratified Sample
A proportionate stratified sample is one in which the size of the strata in the sample is proportional to the size of the strata in the population; in other words, the chance of selecting a unit from a stratum depends on the relative size of that stratum in the population.
Calculating the Proportionate Stratified Sample
The given percentage of -
Whites = 60%
Latinos = 20%
Blacks = 5%
Strength of the sample = 120
⇒ Number of Whites = 60% of 120
= 0.6 × 120
=72
Number of Latinos = 20% of 120
= 0.2 × 120
=24
Proportionate Stratified Sample of Blacks and Others
Count of Blacks = 15 % of 120
= 0.15 × 120
= 18
Count of others = 5% of 120
= 0.05 × 120
6
Thus, in a Proportionate Stratified Sample of 120, 72 are Whites, 24 are Latinos, 18 are Blacks, and 6 are others.
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What is the solution to |x – 5| + 2 < 20?
Answer:
Step-by-step explanation:
hello :
|x – 5| + 2 < 20 means : |x – 5| + 2-2 < 20-2
|x – 5| < 18
-18< x - 5 < 18
-18+5< x - 5+5 < 18+5
-13< x< 23 ( solutions)
On a plat map are two adjacent rectangular lots. Lot A has a square footage of 46,060, and Lot B has a square footage of 43,005. The common boundary they share is 235 feet long. What is the width of Lot B
The Width of plot B is 183 ft.
According to the statement
we have given that the Plot A has a square footage of 46,060, and Plot B has a square footage of 43,005. The common boundary they share is 235 feet long.
And we have to find the width of the Plot B
We know that
Area of plot A = 46,060 square footage.
Area of plot B = 43005 square footage.
Common area share by both Plot = 235 feet long.
So, Now
The width of Plot B = Area of plot B / Common area
Substitute the values in it then
width of Plot B = Area of plot B / Common area
width of Plot B = 43005 / 235
width of Plot B = 183 feet.
So, The Width of plot B is 183 ft.
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Help checking my answer (solving inequalities)
(-4x + 2) / 4 >= 6
-4x + 2 >= 24
-4x >= 22
x <= 22/4 OR x <= 5 1/2 OR x <= 5.5
Hope this helps!
A medication order states, normal saline (ns) iv to infuse at 125 ml/hr. How many ml will infuse in 8 hours? (record answer as a whole number. Do not use a trailing zero. )
The amount of medicine infused in 8 hours is 1000ml.
According to the statement
we have given that the medicine normal saline (ns) iv to infuse at 125 ml/hr.
and we have to find the total amount of medicine infuse in the 8 hours in the whole number value.
So,
An infusion is the creation of a new substance by steeping another substance in a liquid, usually water
we have given a rate of infusion of medicine per hour which is 125 ml/hr.
And
Amount of medicine infuse in 1 hour is 125 ml
And the Amount of medicine infuse in 8 hours = 125*8
Amount of medicine infuse in 8 hours = 1000 ml.
So, The amount of medicine infused in 8 hours is 1000ml.
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If a school opens at 10 am and closes at 3.30pm. if the lunch interval is of 30 minutes, find the ratio of lunch interval to the total times of class periods.
The ratio of lunch interval to the total times of class periods is 1:10
Given that timing of school opens is 10am , the timing of school closes is 3:30pm and the lunch interval is 30 minutes
We need to find the ratio of lunch interval to the total times of class periods.
Now,
Timing of a school(10.a.mto3.30p.m)=5hours30minutes
Timing for lunch interval=30minutes
Total time of the class periods=5hours30minutes−30minutes
=5hours
=60×5=300 minutes.
Ratio of lunch interval to total time of the class period=30minutes:300 minutes=1:10
Hence the ratio of lunch interval to the total times of class periods is 1:10
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Solve for B.
R = x(A + B)
i dont know how to do this
The dimensions of the rectangle are: length= 18 in and width =1 in.
QuadrilateralsThere are different types of quadrilaterals, for example, square, rectangle, rhombus, trapezoid, and parallelogram. Each type is defined accordingly to its length of sides and angles. For example, in a rectangle, the opposite sides are equal and parallel and their interior angles are equal to 90°.
The area of a rectangle can be found for the formula : l*w, where b = length and w =width. The question gives that the area is 18 in².
For this question, the length exceeds its width by 17 inches - l=w+17. Thus, from the value of area given, you can find the values of the length and width of the rectangle.
A=l*w
18=(w+17)*w
18=w²+17w
w²+17w-18=0
Next step will be solve the previous equation ( W²+17W-18=0)
[tex]w_{1,\:2}=\frac{-17\pm \sqrt{17^2-4\cdot \:1\cdot \left(-18\right)}}{2\cdot \:1}\\ \\ w_{1,\:2}=\frac{-17\pm \:19}{2\cdot \:1}[/tex]
Therefore,
[tex]w_1=\frac{-17+19}{2\cdot \:1}=1\\ \\ \\ w_2=\frac{-17-19}{2\cdot \:1}=-18[/tex]
For dimensions, only positive numbers must be used. Then, the width is equal to 1 inch.
As, the area (l*w) is 18 in², you have.
18=l*w
18=l*1
l=18 in
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The magnitude of vector λa is 5. Find the value of λ, if: a = (−6, 8)
The value of λ in the vector is 0.2
What are vectors?Vectors are the opposite of scalar quantities and they are quantities that have directions and magnitude
How to determine the value of λ?The vector a is given as:
a = (−6, 8)
The magnitude of vector a is calculated using
|a| = √(x^2 + y^2)
So, the equation becomes
|a| = √((-6)^2 + 8^2)
Evaluate the exponent
|a| = √(36 + 64)
Evaluate the sum
|a| = √100
Take the square root of both sides
|a| = 10
Given that
λa = 5
This means that
λ * 10 = 5
Divide both sides by 10
λ = 0.2
Hence, the value of λ is 0.2
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PLEASE HELPPPPP WILL GIVE BRAINLIEST
consider this equation
5/8x = 1/2x + 2
generate a plan to solve for the variable describe the steps you’ll use
Answer:
the value of the variable x is 16
Step-by-step explanation:
[tex]given \: expression \\ \frac{5}{8}x = \frac{1}{2} x + 2 \\ substract \: \frac{1}{2} x from \: both \: sides \\ \frac{5}{8}x - \frac{1}{2} x = \frac{1}{2} x - \frac{1}{2} x + 2 - \frac{1}{2} x \\ factor \: out \: common \: x \\ x( \frac{5}{8} - \frac{1}{2} ) = \frac{1}{8} \\ simplify \\ \frac{1}{2} x + 2 - \frac{1}{2} x \: \frac{1}{2} x - \frac{1}{2} x = 0 = 2 \\ \frac{1}{8} x = 2 \\ multiply \: both \: sides \: by \: 8 \\ 8. \frac{1}{8} x = 2.8 \\ simplify \\ x = 16.[/tex]
Aubree's parents invested $500 into a mutual fund that paid 6.5% interest each year compounded annually when she was born. Find the value of the mutual fund in 18 years.
The value of the fund in 18 years as required by the task is; $1553.5.
What is the value of the fund in 18 years?Since, the interest rate is compounded annually as described, the value of the fund in 18 years can be calculated by means of the compound interest formula as;
Value = 500(1.065)^18.
Hence, we have; Value = $1553.5.
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The angle measurements in the diagram are represented by the following expressions.
Answer:
since those 2 lines are parrelel that means the angles are the same so
8x+6=4x+38
minus 6
8x=4x+32
minus 4x
4x=32
divide by 4
x=8
Hope This Helps!!!
The value of the x is 11 and the measure of the angle B is 82 degrees.
What are vertically opposite angles?It is defined as the angles when two lines intersect each other and at the intersecting point, some pair of angles are formed which we call vertically opposite angles, as the name describes that they have vertically opposite angles.
We have:
The angle measurements in the diagram are represented by the following expressions:
Angle A = 8x - 6
Angle B = 4x + 38
By using the vertically opposite angle property in angle B
As we know the alternate exterior angle are equal
8x - 6 = 4x + 38
4x = 44
x =11
Measure of angle B = 4(11) + 38 = 82 degrees
Thus, the value of the x is 11 and the measure of the angle B is 82 degrees.
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A group of 18 patrons each owe $15 at a restaurant. What is the total amount owed by all 18 customers?
Answer:
$270
Step-by-step explanation:
18x15=$270
HOPE THAT HELPED:)
A restaurant owner wants to determine the effectiveness of his servers. the owner places a survey regarding the servers' effectiveness with randomly selected customer bills. what is the sample?
The sample is the randomly selected customers.
Sampling:
Suppose I want to estimate the proportion of New York state residents who are Buffalo Bills fans. So i ask, lets say, 1000 randomly selected New York state residents whether they are Buffalo Bills fans, and expand this to the entire population of New York State residents.
Sampling is a process used in statistical analysis in which a predetermined number of observations are taken from a larger population. The methodology used to sample from a larger population depends on the type of analysis being performed, but it may include simple random sampling or systematic sampling.
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The difference in length of a spring on a pogo stick from its non-compressed length when a teenager is jumping on it after θ seconds can be described by the function f of theta equals 2 times cosine theta plus radical 3.
Part A: Determine all values where the pogo stick's spring will be equal to its non-compressed length.
Part B: If the angle was doubled, that is θ became 2θ, what are the solutions in the interval [0, 2π)? How do these compare to the original function?
Part C: A toddler is jumping on another pogo stick whose length of their spring can be represented by the function g of theta equals 1 minus sine squared theta plus radical 3 period At what times are the springs from the original pogo stick and the toddler's pogo stick lengths equal?
The given function for the difference in length is presented as follows;
[tex]f( \theta) = 2 \cdot cos( \theta) + \sqrt{3} [/tex]
When the pogo stick will be equal to its non compressed length, the difference is zero, therefore;
[tex]f( \theta) = 2 \cdot cos( \theta) + \sqrt{3} = 0[/tex]
[tex] 2 \cdot cos( \theta) = - \sqrt{3} [/tex]
[tex]\theta= arccos \left( \frac{ - \sqrt{3} }{2} \right) [/tex]
Which gives;
[tex] \theta = \frac{12 \cdot \pi \cdot n1 + 5\cdot \pi}{6} [/tex]
[tex] \theta = -\frac{12 \cdot \pi \cdot n1 + 5\cdot \pi}{6} [/tex]
Part B; If the angle was doubled, we have;
[tex]f( \theta) = 2 \cdot cos(2 \cdot \theta) + \sqrt{3} = 0[/tex]
Therefore;
[tex] 2 \cdot cos(2 \cdot \theta) = - \sqrt{3} [/tex]
Which gives;
[tex] \theta = \frac{12 \cdot \pi \cdot n1 + 5\cdot \pi}{12} [/tex]
[tex] \theta = -\frac{12 \cdot \pi \cdot n1 + 5\cdot \pi}{12} [/tex]
Between 0 and 2•π, we have;
[tex] \theta = \frac{5\cdot \pi}{12} [/tex]
[tex] \theta = \frac{ 17\cdot \pi}{12} [/tex]
Part C;
The toddler's pogo stick is presented as follows;
[tex]g( \theta) = 1- son^2( \theta) + \sqrt{3} [/tex]
Integrating the original function between 0 and theta gives;
[tex]2 \cdot sin( \theta) + \sqrt{3}\cdot \theta [/tex]
The original length =
Therefore, when the lengths are equal, we have;
[tex]1- son^2( \theta) + \sqrt{3} = 2 \cdot sin( \theta) + \sqrt{3}\cdot \theta [/tex]