The enclosed region has an area of zero.
Here, we have,
To find the area of the enclosed region between the two parabolas, we need to calculate the definite integral of the difference of the two functions over the interval where they intersect.
The two parabolas are given by:
y=6x²
y=x² + 9
To find the intersection points, we set the two equations equal to each other:
6x² =x² + 9
Now, let's solve for
6x² =x² + 9
=> 6x² - x² = 9
=> 5x² = 9
=> x² = 9/5
=> x = ±√9/5
Since we are looking for the area between the two curves, we only need to consider the positive x value:
x = √9/5
=> x = 3/√5
Now, the definite integral to find the area between the curves is given by:
[tex]A = \int\limits^b_a {(y_2 - y_1)} \, dx[/tex]
where [tex]y_2[/tex] and [tex]y_1[/tex] are the equations of the two curves, and a and b are the x-coordinates of the intersection points.
In this case,
[tex]y_2 = x^2+9 \\ y_1 = 6x^2[/tex]
So, the area A is:
A = [tex]\int\limits^\frac{3}{\sqrt{5} } _0 {(9-5x^2)} \, dx[/tex]
Now, integrate with respect to x:
[tex]A = [9x - \frac{5x^3}{3} ]^{\frac{3}{\sqrt{5} }}_0[/tex]
solving we get,
A = 0
The enclosed region has an area of zero.
This result may seem counterintuitive, but it means that the two parabolas intersect at a point and do not enclose any finite area between them.
Instead, they share a single point in the coordinate plane.
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The area of the fenced field, enclosed by the parabolas y=6x² and y=x²+9, can be found by integrating the difference of the two functions from their points of intersection, which are x=-3 and x=3.
Explanation:The subject of this question is Mathematics, more specifically it's an application of calculus. To find the area enclosed by two curves, we need to integrate the difference of the two functions from where they intersect. In this case, the two parabolas y=6x² and y=x²+9. To find the points of intersection X₁ and X₂, we make the two equations equal and solve for x, which gives us x= -3 and x=3.
Then we take the integral of (6x² - (x² + 9)) from -3 to 3. The result of this integral will give us the area between these two parabolas, which represents the area of the fenced field.
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A company makes wax candles in the shape of a cylinder. Each candle has a radius of 2 inches and a height of 7 inches. How much wax will the company need to make 210 candles?
Answer:
Volume of wax for 210 candles=18463.2 cubic inches
Step-by-step explanation:
Each wax candle has a cylindrical shape with
Radius of candle, r =2Height of candle, h =7Number of candles to be made=210
Volume of one wax candle =π × [tex]r^{2}[/tex] × h
=π ×[tex]2^{2}[/tex] × 7
=3.14×4×7
=87.92 cubic inches
Volume of 210 wax candles=210×87.92
=18,463.2 cubic inches
Answer:
18,471.6
Step-by-step explanation:
Earth orbits the sun at an average speed of 29.79 kilometers per second. Find how long it take, to the nearest hundredth of a second, for earth to travel 500 kilometers
Answer:
16.78 seconds
Step-by-step explanation:
speed = Distance Traveled / time
thus speed =29.79 Km/sec
time =distance Traveled/speed (from above formula)
time taken=500 km ÷ 29.79 Km/sec
∴time taken=16.78 seconds
The function f(x)=lnx is transformed into the equation f(x)=ln(9.2x). Select from the drop-down menus to correctly identify the parameter and the effect the parameter has on the parent function. The function f(x)=ln(9.2x) is a of the parent function by a factor of _________.
The function f(x)=ln(9.2x) is a horizontal compression of the parent function by a factor of 1/9.2
Step-by-step explanation:
The multiplication of a function by a number compresses or stretches the function vertically while to compress or stretch the function horizontally, the input variable is multiplied with a number.
i.e.
[tex]For\ f(x) => g = f(bx)[/tex]
where b is a constant.
Now
If b>0 then the function is compressed horizontally
The given function is:
[tex]f(x) = ln\ x\\Transformed\ to\\f(x) = ln\ (9.2x)[/tex]
As the variable in function is multiplied with a number greater than zero, the function will stretch horizontally.
The function f(x)=ln(9.2x) is a horizontal compression of the parent function by a factor of 1/9.2
Keywords: Transformation
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Answer:
The function f(x)=In (9.2x) is a horizontal compression of the parent function by a factor of 5/46
Step-by-step explanation:
I just took this test and that was the correct answer :( good luck everyone
John can jog twice as fast as he can walk. He was able to jog the first mile to his grandmas house but then he got tired and walked the remaining 4 miles. If the total trip took 0.75 hours, then what was his average jogging speed
Answer:
12 mph
Step-by-step explanation:
The relationship between jogging speed and walking speed means the time it takes to walk 4 miles is the same as the time it takes to jog 8 miles. Then the total travel time (0.75 h) is the time it would take to jog 1+8 = 9 miles. The jogging speed is ...
(9 mi)(.75 h) = 12 mi/h . . . average jogging speed
__
Check
1 mile will take (1 mi)/(12 mi/h) = 1/12 h to jog.
4 miles will take (4 mi)/(6 mi/h) = 4/6 = 8/12 h to walk.
The total travel time is (1/12 +8/12) h = 9/12 h = 3/4 h. (answer checks OK)
_____
Comment on the problem
Olympic race-walking speed is on the order of 7.7 mi/h, so John's walking speed of 6 mi/h should be considered quite a bit faster than normal. The fastest marathon ever run is on the order of a bit more than 12 mi/h, so John's jogging speed is also quite a bit faster than normal. No wonder he got tired.
The mean annual incomes of certified welders are normally distributed with the mean of $50,000 and a population standard deviation of $2,000. The ship building association wishes to find out whether their welders earn more or less than $50,000 annually. A sample of 100 welders is taken and the mean annual income of the sample is $50,350. If the level of significance is 0.10, what conclusion should be drawn?
A. Do not reject the null hypothesis as the test statistic is less than the critical value of z.
B. Do not reject the null hypothesis as the test statistic is less than the critical value of t.
C. Reject the null hypothesis as the test statistic is greater than the critical value of t.
D. Reject the null hypothesis as the test statistic is greater than the critical value of z.
Answer:
D. Reject the null hypothesis as the test statistic is greater than the critical value of z.
Step-by-step explanation:
[tex]H_{0}:[/tex] welders earn $50,000 annually
[tex]H_{a}:[/tex] welders' income does not equal $50,000 annually
Sample size 100>30, therefore we need to calculate z-values of sample mean and significance.
z-critical at 0.10 significance is 1.65
z-score of sample mean (test statistic) can be calculated as follows:
[tex]\frac{X-M}{\frac{s}{\sqrt{N} } }[/tex] where
X is the mean annual income of the sample ($50,350)M is the mean annual income assumed under null hypothesis ($50,000)s is the population standard deviation ($2,000)N is the sample size (100)Then z=[tex]\frac{50,350-50,000}{\frac{2,000}{\sqrt{100} } }[/tex] =
1.75.
Since test statistic is bigger than z-critical, (1.75>1.65), we reject the null hypothesis.
To answer the question, a z-test was performed comparing the sample mean income of welders to the population mean. The test statistic (1.75) was found to be greater than the critical z-value (±1.645) for a 0.10 alpha level, therefore we reject the null hypothesis.
Explanation:The question is asking us to conduct a hypothesis test to determine whether shipbuilders' welders earn more or less than the population mean income of certified welders, which is $50,000. In statistics, we normally use a z-test for such a comparison when we know the population standard deviation. Given that the level of significance (alpha) is 0.10, we must find the critical z-value that corresponds to this alpha level, calculate the test statistic for the sample mean of $50,350, then compare our test statistic with the critical value to make our decision.
Since we have a large sample size (n=100) and the population standard deviation is known, a z-test is appropriate. The test statistic is calculated using the formula:
z = (X_bar- μ) / (σ/√n)
Where X_bar is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Plugging in the values:
z = ($50,350 - $50,000) / ($2,000/√100) = $350 / $200 = 1.75
To determine whether to reject the null hypothesis, we must compare the test statistic to the critical value of z for a significance level of 0.10. For a two-tailed test (since we want to know if it's more or less, not just more), the critical z-values are approximately ±1.645. Since our test statistic of 1.75 is greater than 1.645, we reject the null hypothesis, implying that there is enough evidence to suggest the mean annual income of the sample of welders is different from $50,000. However, as the question does not specify the direction of the alternative hypothesis (whether we were testing for higher or lower earnings, specifically), we cannot conclude that welders earn more than $50,000 without further information.
A part of a line consisting of two endpoints and all points between them
Answer:
segment
Step-by-step explanation:
We know that a line has no end points.
If we take a part from the line then it is called line segment.
The line segment has starting point and end point.
A part of a line consisting of two endpoints and all the points between them is called segment.
Therefore, the answer is segment
Line segment
LineA line segment is a section of a line that is defined by two distinct end points and includes all points on the line between them. So, a part of a line made up of two ends and all points in between is known as a line segment.A line is a collection of points that extends in two opposite directions and is infinitely thin and long.A line is a one-dimensional figure with no thickness that extends in both directions indefinitely.Find out more information about line here:
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A clock was showing the time accurately on Wednesday at 4pm. On the following Saturday, at 2pm, the clock was running late by 35 seconds. On average, how many seconds did the clock skip every 30 minutes?
Answer: The clock skip 0.25 seconds every 30 minutes.
Step-by-step explanation:
Since we have given that
Clock was correct on Wednesday at 4 pm.
At 2 pm , on saturday, the clock was running late by 35 seconds.
From 4 pm wednesday to 4 pm thursday = 24 hours
From 4 pm thursday to 4 pm friday = 24 hours
From 4 pm friday to 2 pm saturday = 22 hours
So, total hours = 24+24+22 = 70 hours
We need to find the number of seconds that the clock skip every 30 minutes.
So, it becomes
[tex]\dfrac{70}{0.5}=\dfrac{35}{T}\\\\70T=35\times 0.5\\\\T=\dfrac{35\times 0.5}{70}\\\\T=0.25[/tex]
Hence, the clock skip 0.25 seconds every 30 minutes.
The clock was 35 seconds late over a period of 70 hours, which equals 140 intervals of 30 minutes. Therefore, the clock skipped an average of 0.25 seconds every 30 minutes.
To determine the average number of seconds the clock skipped every 30 minutes, we first need to calculate the total time difference and then divide by the number of 30-minute intervals.
The clock was showing the correct time on Wednesday at 4 pm.
It was 35 seconds late by Saturday at 2 pm.
Time elapsed from Wednesday 4 pm to Saturday 2 pm is 2 days and 22 hours, which equals (2×24 + 22) hours = 70 hours.
Converting 70 hours into minutes: 70×60 = 4200 minutes.
Number of 30-minute intervals in this period: 4200 / 30 = 140 intervals.
Average seconds skipped per 30-minute interval: 35 seconds / 140 intervals = 0.25 seconds.
Thus, the clock skipped an average of 0.25 seconds every 30 minutes.
slader the internal revenue service claims it takes an average of 3.7 hours to complete a 1040 tax form, assuming th4e time to complete the form is normally distributed witha standard devait of the 30 minutes:
a. What percent of people would you expect to complete the form in less than 5 hours?
b. What time interval would you expect to include the middle 50 % of the tax filers?
Answer:
0.9953, 3.3629<x<4.0371
Step-by-step explanation:
Given that slader the internal revenue service claims it takes an average of 3.7 hours to complete a 1040 tax form, assuming th4e time to complete the form is normally distributed witha standard devait of the 30 minutes:
If X represents the time to complete then
X is N(3.7, 0.5) (we convert into uniform units in hours)
a) percent of people would you expect to complete the form in less than 5 hours
=[tex]100*P(x<5)\\= 0.9953[/tex]
b) P(b<x<c) = 0.50
we find that here
c = 4.0371 and
b = 3.3629
Interval would be
[tex](3.3629, 4.0371)[/tex]
Q3:
A company sells bikes for $120 each. They pay a monthly rent of $1,800 for their store and each bike costs them $60 in materials. Write the revenue and cost functions and find the break-even point by graphing.
Let the number of bikes = x and total money = y.
Set up two equations:
The first equation is the cost function, which would be rent plus 60 times the number of bikes:
y = 1800 + 60x
The second equation would be revenue, where total money would be equal to 120 times the number of bikes sold:
y = 120x
Now you can graph both equations by setting the equations to equal each other. The point on the graph, where the loib=ne crosses the X axis is the break even point
Graph
120x = 1800 +60x
See attached picture for the graph and you can see the break even point is 30 bikes.
You can check by replacing X with 30 to see if the equations equal each other:
1800 + 60(30) = 1800 +1800 = 3600
120(30) = 3600
Terry and Callie do word processing. For a certain prospectus Callie can prepare it two hours faster than Terry can. If they work together they can do the entire prospectus in five hours. How long will it take each of them working alone to repair the prospectus? Round answers to the nearest 10th of an hour
Time taken by jerry alone is 10.1 hours
Time taken by callie alone is 8.1 hours
Solution:
Given:- For a certain prospectus Callie can prepare it two hours faster than Terry can
Let the time taken by Terry be "a" hours
So, the time taken by Callie will be (a-2) hours
Hence, the efficiency of Callie and Terry per hour is [tex]\frac{1}{a-2} \text { and } \frac{1}{a} \text { respectively }[/tex]
If they work together they can do the entire prospectus in five hours
[tex]\text {So, } \frac{1}{a-2}+\frac{1}{a}=\frac{1}{5}[/tex]
On cross-multiplication we get,
[tex]\frac{a+(a-2)}{(a-2) \times a}=\frac{1}{5}[/tex]
[tex]\frac{2 a-2}{(a-2) \times a}=\frac{1}{5}[/tex]
On cross multiplication ,we get
[tex]\begin{array}{l}{5 \times(2 a-2)=a \times(a-2)} \\\\ {10 a-10=a^{2}-2 a} \\\\ {a^{2}-2 a-10 a+10=0} \\\\ {a^{2}-12 a+10=0}\end{array}[/tex]
using quadratic formula:-
[tex]x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}[/tex]
[tex]x=\frac{12 \pm \sqrt{144-40}}{2}[/tex]
[tex]\begin{array}{l}{x=\frac{12 \pm \sqrt{144-40}}{2}} \\\\ {x=\frac{12 \pm \sqrt{104}}{2}} \\\\ {x=\frac{12 \pm 2 \sqrt{26}}{2}} \\\\ {x=6 \pm \sqrt{26}=6 \pm 5.1} \\\\ {x=10.1 \text { or } x=0.9}\end{array}[/tex]
If we take a = 0.9, then while calculating time taken by callie = a - 2 we will end up in negative value
Let us take a = 10.1
So time taken by jerry alone = a = 10.1 hours
Time taken by callie alone = a - 2 = 10.1 - 2 = 8.1 hours
Emma and Leah are both jewelry makers. Gemma made 106 beaded necklaces. Leah made 39 more necklaces than Gemma. Each necklace they make has exactly 104 beads on it. How many beads did both jewelers use altogether while making their necklaces?
Both jewelers used 26104 beads altogether while making necklaces.
Step-by-step explanation:
No. of necklaces made by Gemma = 106
Necklaces made by Leah = 106+39 = 145 necklaces
Total necklaces made = Gemma's + Leah's
Total necklaces made = [tex]106+145 = 251\ necklaces[/tex]
Beads used in 1 necklace = 104 beads
Beads used in 251 necklaces = 104*251 = 26104
Both jewelers used 26104 beads altogether while making necklaces.
Keywords: multiplication, addition
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A punch glass is in the shape of a hemisphere with a radius of 5 cm. If the punch is being poured into the glass so that the change in height of the punch is 1,5 cm/sec, at what rate is the exposed area of the punch changing when the height of the punch is 2 cm.
Answer:
28.27 cm/s
Step-by-step explanation:
Though Process:
The punch glass (call it bowl to have a shape in mind) is in the shape of a hemispherethe radius [tex]r=5cm[/tex] Punch is being poured into the bowlThe height at which the punch is increasing in the bowl is [tex]\frac{dh}{dt} = 1.5[/tex]the exposed area is a circle, (since the bowl is a hemisphere)the radius of this circle can be written as [tex]'a'[/tex]what is being asked is the rate of change of the exposed area when the height [tex]h = 2 cm[/tex] the rate of change of exposed area can be written as [tex]\frac{dA}{dt}[/tex]. since the exposed area is changing with respect to the height of punch. We can use the chain rule: [tex]\frac{dA}{dt} = \frac{dA}{dh} . \frac{dh}{dt}[/tex]and since [tex]A = \pi a^2[/tex] the chain rule above can simplified to [tex]\frac{da}{dt} = \frac{da}{dh} . \frac{dh}{dt}[/tex] -- we can call this Eq(1)Solution:
the area of the exposed circle is
[tex]A =\pi a^2 [/tex]
the rate of change of this area can be, (using chain rule)
[tex]\frac{dA}{dt} = 2 \pi a \frac{da}{dt}[/tex] we can call this Eq(2)
what we are really concerned about is how [tex]a[/tex] changes as the punch is being poured into the bowl i.e [tex]\frac{da}{dh}[/tex]
So we need another formula: Using the property of hemispheres and pythagoras theorem, we can use:
[tex]r = \frac{a^2 + h^2}{2h}[/tex]
and rearrage the formula so that a is the subject:
[tex]a^2 = 2rh - h^2[/tex]
now we can derivate a with respect to h to get [tex]\frac{da}{dh}[/tex]
[tex]2a \frac{da}{dh} = 2r - 2h[/tex]
simplify
[tex]\frac{da}{dh} = \frac{r-h}{a}[/tex]
we can put this in Eq(1) in place of [tex]\frac{da}{dh}[/tex]
[tex]\frac{da}{dt} = \frac{r-h}{a} . \frac{dh}{dt}[/tex]
and since we know [tex]\frac{dh}{dt} = 1.5[/tex]
[tex]\frac{da}{dt} = \frac{(r-h)(1.5)}{a} [/tex]
and now we use substitute this [tex]\frac{da}{dt}[/tex]. in Eq(2)
[tex]\frac{dA}{dt} = 2 \pi a \frac{(r-h)(1.5)}{a}[/tex]
simplify,
[tex]\frac{dA}{dt} = 3 \pi (r-h)[/tex]
This is the rate of change of area, this is being asked in the quesiton!
Finally, we can put our known values:
[tex]r = 5cm[/tex]
[tex]h = 2cm[/tex] from the question
[tex]\frac{dA}{dt} = 3 \pi (5-2)[/tex]
[tex]\frac{dA}{dt} = 9 \pi cm/s// or//\frac{dA}{dt} = 28.27 cm/s[/tex]
You want to buy a $230,000 home. You plan to pay 20% as a down payment, and take out a 30 year fixed loan for the rest. Round all answers to the nearest cent as needed.
Amount of down payment = $46000
Mortgage needs = $184000
Solution:
From the given,
Cost of the house = [tex]\$230000[/tex]
Percentage of down payment = [tex]20\%[/tex]
Number of years of fixed loan = 30
[tex]\text { Total down payment }=\text { cost of the house } \times \text { Percentage of down payment }[/tex]
[tex]\Rightarrow \frac{230000 \$\times 20}{100} \rightarrow 46000 \$[/tex]
[tex]\text {Mortgage needs}=\text { Total cost - Total down payment }[/tex]
[tex]\Rightarrow 230000 \$-46000 \$=184000 \$[/tex]
It can be concluded that the total down payment for the house and mortgage needs would be [tex]\$46000 \text{ and } \$184000[/tex]
1. Find the remainder if f(x) = 2x³ + 8x² – 5x + 5 is divided by x – 2.
Answer:
The answer to your question is 43
Step-by-step explanation:
2x³ + 8x² - 5x + 5 / x - 2
Process
1.- Use synthetic division
2 8 -5 5 2
4 24 38
2 12 19 43
Quotient 2x² + 12x + 19
Remainder 43
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Find the mean for the binomial distribution. Round to the nearest tenth.
n=1632; p=0.57
A) 939.9
B) 937.5
C) 922.7
D) 930.2
ALSO QUESTION IN PICTURE PLEASE
Answer:
The mean of a binomial distribution is given by mean = n x p where n = the number of items and p equals the probability of success. Here we have:
mean = 1632 x 0.57 = 930.2
Step-by-step explanation:
The mean for a binomial substitution = n x p
Mean = 1632 x 0.57 = 930.24
The answer would be D.
Picture:
Multiply P(x) by X, then add those together:
0 x 0.42 = 0
1 x 0.12 = 0.12
2 x 0.34 = 0.68
3 x 0.05 = 0.15
4 x 0.07 = 0.28
Mean = 0 + 0.12 + 0.68 + 0.15 + 0.28 = 1.23
For a group of graduating college seniors, a researcher records each student’s rank in his/her high school graduating class and the student’s rank in the college graduating class. Which correlation should be used to measure the relationship between these two variables?
Answer:
Spearman's correlation
Step-by-step explanation:
A researcher records each student’s rank in his/her high school graduating class and the student’s rank in the college graduating class.
The correlation that should be used to measure the relationship between these two variables is - Spearman's correlation
This correlation gives a statistical measure of similar relationship between paired data.
This is used to evaluate relationships involving ordinal variables.
The length of a rectangular driveway is four feet less than five times the width. The area is 672 feet squared. Find the width and length of the driveway
Answer: length of the drive way = 56 feet
Width of the driveway = 12 feet
Step-by-step explanation:
The rectangular driveway has two equal lengths and two equal widths. The area of the driveway is expressed as
length,l × width,w
The area is 672 feet squared. It means that
L×W = 672
The length of the rectangular driveway is four feet less than five times the width. It means that
L = 5W - 4
Substituting L = 5W - 4 into LW = 672
W(5W - 4) = 672
5W^2 - 4W - 672 = 0
5W^2 + 56W - 60W - 672 = 0
W(5W + 56) - 12(5W + 56) = 0
(W - 12)(5W + 56) = 0
W - 12 = 0 or 5W + 56 = 0
W = 12 or 5W = -56
W= 12 or W = - 56/5
The Width cannot be negative , so
W = 12
LW = 672
12L = 672
L = 672/12 = 56
What are the period and amplitude of the function?
The given graph displays a periodic function with a [tex]5[/tex]-unit period and a [tex]3[/tex]-unit amplitude. Therefore, option C is correct, accurately describing the function's characteristics based on the observed graph.
The given graph indicates a periodic function with repeating patterns. The period is the horizontal distance between two successive peaks or troughs. In this case, the graph repeats every [tex]5[/tex] units horizontally, so the period is indeed [tex]5[/tex]. The amplitude is the vertical distance from the midline to the peak or trough.
Here, the vertical distance is [tex]3[/tex] units, confirming the amplitude as [tex]3[/tex].
Therefore, according to the graph, option C is correct with a period of [tex]5[/tex] and an amplitude of [tex]3[/tex], aligning with the observed characteristics of the function's periodicity and vertical range.
The angle measurements in the diagram are represented by the following expressions.
Solve for x and then find the measure of ∠B.
Answer: 70
Step-by-step explanation:
8x+6=4x+38
4x=32
x=8
<B=4x+38=4*8+38=70
Answer:
x = 8
∠B = 70°
Step-by-step explanation:
∠A = ∠B through alternate exterior angles.
∠A = ∠B
8x + 6 = 4x + 38
8x - 4x + 6 = 38
4x + 6 = 38
4x = 38 - 6
4x = 32
x = 32 ÷ 4
x = 8
∠B = 4x + 38
4(8) + 38
32 + 38
= 70
Osmin had a gross pay of 624.86 last week. She earns 12.85 per hour plus a 3% commission on all sales. She knows she worked 40hrs last week but can't remember her total sales. What were her total sales ?
Her total sales was $3695.33
Step-by-step explanation:
Osmin had a gross pay of $624.86 last week
She earns $12.85 per hourShe earns 3% commission on all salesShe knows she worked 40 hrs last weekShe wants to know what was her total sales
Assume that her total sales is $x
∵ Her gross pay = $624.86
∵ She earns 12.85 per hour plus a 3% commission on all sales
∵ She worked for 40 hrs last week
∵ Her total sales was $x
- Put all of these in an equation represents her gross pay
∵ Her gross pay = her rate per hour × the number of hours + 3% of x
∴ 624.86 = 12.85 (40) + 3% (x)
∴ 624.86 = 514 + [tex]\frac{3}{100}[/tex] x
∴ 624.86 = 514 + 0.03 x
- Subtract 514 from both sides
∴ 110.86 = 0.03 x
- Divide both sides by 0.03
∴ 3695.33 = x
∴ Her total sales = $3695.33
Her total sales was $3695.33
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Alicia draws an equilateral triangle and then rotates it about its center. Through which angle measures can she rotate the equilateral triangle to map it onto itself?
a. 60°
b. 90°
c. 120°
d. 180°
e. 240°
f. 300°
Answer:
The answers should be c. 120° and e. 240°
Step-by-step explanation:
Consider the provided information.
Equilateral triangle has 3 equal sides.
Now we need to rotate the equilateral triangle so that the equilateral triangle to map it onto itself.
For this we need to rotate each of those sides to an adjacent side. (Shown in figure)
This can be happen 3 times as there are 3 sides,
A circle has 360° and [tex]\dfrac{1}{3}\times 360^{\circ}=120^{\circ}[/tex].
Thus, a 120° rotation map it onto itself.
Any other angle multiple of 120° will do the same.
Hence, the answers should be c. 120° and e. 240°
Trevor is making payments on a car that cost $26,555 he makes 36 equal payments if he runs to equal payments up to the nearest whole dollar about how much will he overpay after 36 months
Answer:
The over payment amount after 36 months is $ 0.37
Step-by-step explanation:
Given as :
The cost of the Car = $ 26,555
The number of times payment done for 36 month = 36
So, The the cost of car for 36 equal payment = [tex]\dfrac{\textrm Total cost of car}{\textrm number of times payment done }[/tex]
i.e The the cost of car for 36 equal payment = [tex]\frac{26,555}{36}[/tex]
∴ The the cost of car for 36 equal payment = $ 737.6389
Now rounding this value to nearest whole dollar = $ 738
Note - A) If after decimal , number are above 4 then round it to 1 above digit
B) If after decimal , number 4 or less then simply remove all number after decimal
So, The over payment after 36 months = $ 738 - $ 737.63 = $ 0.37
Hence The over payment amount after 36 months is $ 0.37 Answer
It takes Carl 45 minutes to drive to work using two roads. She drives 32 mph on a small road for 1/2 hour. Then she drives 56 mph on a small road for 1/4 hour. How far does she travel for work?
Answer:
The Total distance she travel fro work is 30 miles .
Step-by-step explanation:
Given as :
the total time taken to cover distance = 45 minutes
Let The total distance cover = D miles
The distance cover at the speed of 32 mph = [tex]D_1[/tex] miles
The time taken to cover [tex]D_1[/tex] miles distance = [tex]\frac{1}{2}[/tex] hour
Distance = Speed × Time
∴ [tex]D_1[/tex] = 32 mph × [tex]\frac{1}{2}[/tex] h
or, [tex]D_1[/tex] = 16 miles
Again ,
The distance cover at the speed of 56 mph = [tex]D_2[/tex] miles
The time taken to cover [tex]D_2[/tex] miles distance = [tex]\frac{1}{4}[/tex] hour
∴ [tex]D_2[/tex] = 56 mph × [tex]\frac{1}{4}[/tex] h
or, [tex]D_2[/tex] = 14 miles
So , The total distance she travel for work = [tex]D_1[/tex] + [tex]D_2[/tex]
Or, The total distance she travel for work = 16 miles + 14 miles = 30 miles
Hence The Total distance she travel fro work is 30 miles . Answer
Mr. Lynch buys some turkey slices, wheat rolls, and cheese for $45. The ratio of the amount of money he spends on cheese to the amount of he spends on turkey slices is 2:3
Given a ratio of cheese to turkey slices is 2:3, Mr. Lynch spent $18 on cheese and $27 on turkey slices out of a total of $45.
Explanation:Mr. Lynch is faced with a real-world example of the concept of ratios. In this case, the ratio of cheese to turkey slices is 2:3, meaning for every 2 parts of cheese, he's spending on 3 parts of turkey slices. The total amount spent is $45. We can solve this ratio problem to identify the individual costs for cheese and turkey slices by using a simple mathematical method.
Firstly, let's understand how a ratio works. It's a way to compare amounts of different things. Given the ratio 2:3 (cheese to turkey), add the ratio numbers together to get the total parts, i.e., 2 + 3 = 5. These 5 parts represent the total amount of $45 spent.
Next, we need to find the value of one part. To do this, divide the total amount spent by the total parts: $45/5 parts gives us 9: this determines that each part is worth $9.
Finally, to find the amounts spent on cheese and turkey slices, multiply the number of parts each item has in the ratio by the value of one part. So, for cheese, it's 2 parts x $9 = $18, and for turkey slices, it's 3 parts x $9 = $27.
In conclusion, Mr. Lynch spent $18 on cheese and $27 on turkey slices.
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A rectangle is inscribed in an equilateral triangle so that one side of the rectangle lies on the base of the triangle. Find the maximum area the rectangle can have when the triangle has side length 14 inches.
Answer:
A(max) = 42.43 in²
Dimensions:
a = 7 in
b = 6,06 in
Step-by-step explanation: See annex
Equilateral triangle side L = 14 in, internal angles all equal to 60°
Let A area of rectangle A = a*b
side b tan∠60° = √3 tan∠60° = b/x b = √3 * x
side a a = L - 2x a = 14 - 2x
A(x) = a*b A(x) = ( 14 - 2x ) * √3 * x
A(x) = 14*√3*x - 2√3 * x²
Taking derivatives both sides of the equation
A´(x) = 14√3 - 4√3*x
A´(x) = 0 ⇒ 14√3 - 4√3*x = 0 ⇒ 14 - 4x = 0 x = 14/4
x = 3,5 in
Then
a = 14 - 2x a = 14 - 7 a = 7 in
b = √3*3,5 b = *√3 *3,5 b = 6,06 in
A(max) = 7 *6,06
A(max) = 42.43 in²
What is the binomial expansion of (2x – 3)^5?
A) (2x)^ 5 – 15(2x)^ 4 + 90(2x)^ 3 – 270(2x)^ 2 + 405(2x) – 243
B) (2x)^ 5 + 15(2x)^ 4 – 90(2x)^ 3 + 270(2x)^ 2 – 405(2x) + 243
C) (2x)^ 5 + 15(2x)^ 4 + 90(2x)^ 3 + 270(2x)^ 2 + 405(2x) + 243
D) 2(x)^ 5 – 30(x)^ 4 + 180(x)^ 3 – 540(2x)^ 2 + 810(x) – 243
Answer:
C
Step-by-step explanation:
(2x + 3)^5 = C(5,0)2x^5*3^0 +
C(5,1)2x^4*3^1 + C(5,2)2x^3*3^2 + C(5,3)2x^2*3^3 + C(5,4)2x^1*3^4 + C(5,5)2x^0*3^5
Recall that
C(n,r) = n! / (n-r)! r!
C(5,0) = 1
C(5,1) = 5
C(5,2) = 10
C(5,3) = 10
C(5,4) = 5
C(5,5) = 1
= 1(2x^5)1 + 5(2x^4)3 + 10(2x^3)3^2 + 10(2x^2)3^3 + 5(2x^1)3^4 + 1(2x^0)3^5
= 2x^5 + 15(2x^4) + 90(2x^3) + 270(2x^2) + 405(2x) +243
= 32x^5 + 15(16x^4) + 90(8x^3) + 270(4x^2) + 810x + 243
= 32x^5 + 240x^4 + 720x^3 + 1080x^2 + 810x + 243
Answer:
the answer is C
Step-by-step explanation:
help me solve this problem!!
Answer:
initial size: 75doubling time: 7.51 minutesafter 115 minutes: about 3,056,900reaches 11,000: 54.03 minutesStep-by-step explanation:
For given points (t1, y1), (t2, y2), I like to write the exponential function as ...
y(t) = y1·(y2/y1)^((t-t1)/(t2-t1))
This can be converted to other forms (such as a·b^t or a·e^(kt)) fairly easily, but those tend not to reproduce the given numbers exactly as this form does.
Using (15, 300) and (35, 1900) as our data values, the exponential function can be written as ...
y(t) = 300·(19/3)^((t-15)/20)
__
a) The initial size of the culture is the value of y(0).
y(0) = 300·(19/3)^(-15/20) ≈ 75.144
y(0) ≈ 75 . . . initial population
__
b) The doubling period will be the value of t that satisfies ...
(19/3)^(t/20) = 2
Taking logarithms, we have ...
(t/20)·log(19/3) = log(2)
t = 20·log(2)/log(19/3) ≈ 7.5104 . . . . minutes
The doubling time is about 7.51 minutes.
__
c) Evaluating the formula for t=115, we have ...
y(115) = 300·(19/3)^(100/20) ≈ 3056912.346
The count after 115 minutes will be about 3,056,900.
__
d) Solving y(t) = 11,000, we have ...
11000 = 300·(19/3)^((t-15)/20)
11000/300 = (19/3)^((t-15)/20)
log(110/3) = (t-15)/20·log(19/3)
t = 20·log(110/3)/log(19/3) + 15 ≈ 54.027
It will take about 54.03 minutes for the count to reach 11,000.
_____
I find a graphing calculator to be a nice tool for solving problems like this.
a store sells two different brands of lemonade mix. for brand a 1/2 cup if mix makes a pitcher. for brand b 1/4 cup of mix makes a pitcher. the container for brand a contains 4 more cups of mix than the container for brand b. both containers make the same number of pitchers of lemonade. how many pitchers of lemonade can each container make?
Answer:
The number of pitchers produced by each container = 16 .
Step-by-step explanation:
Given,
Brand A requires [tex]\frac{1}{2}[/tex] cup of a mix for a pitcherBrand B requires [tex]\frac{1}{4}[/tex] cup of a mix for a pitcherBoth containers produce the same number of pitchers2 Containers :Brand A : contains four more cups of mix than Brand BBrand B : contains [tex]x[/tex] cups of mix⇒∴ The number of cups of mix in brand A = [tex]x+4[/tex];
Number of pitchers = [tex]\frac{TOTAL.NO.OF.MIX}{NO.OF.MIX.FOR.ONE }[/tex]Number of pitchers produced by the containers :
Brand A : [tex]=\frac{x+4}{\frac{1}{2} } \\=2*(x+4)\\=2x+8[/tex]Brand B : [tex]=\frac{x}{\frac{1}{4} }\\=4*x\\=4x[/tex]Since both are equal:
⇒[tex]2x+8 = 4x\\8=2x\\x=4[/tex]
Thus the number of cups of mix in Brand B = [tex]x=4[/tex];
The number of pitchers produced by each container :
= [tex]\frac{4}{\frac{1}{4} } \\= 4*4\\=16[/tex]
∴The number of pitchers produced by each container = 16.
Each container can make 16 pitchers.
What is a Fraction?In mathematics, a fraction is used to denote a portion or component of the whole. It stands for the proportionate pieces of the whole.
As per the given data:
For making a pitcher of brand, A 1/2 cup of a mix is required.
For making a pitcher of brand, B 1/4 cup of a mix is required.
For brand A, 4 more cups than brand B .
Let's assume the number of cups for brand B as x
∴ Number of cups for brand A = x + 4
Total number of pitchers = Total number of cups / cups for one pitcher
For brand A number of pitchers:
= [tex]\frac{x + 4}{\frac12}[/tex] = 2(x + 4)
For brand B number of pitchers:
= [tex]\frac{x}{\frac14}[/tex] = 4x
The number of pitchers will be same for both brand A and B
∴ 2(x + 4) = 4x
= 2x + 8 = 4x
x = 4
The number of pitchers = [tex]\frac{4}{\frac14}[/tex] = 16
Hence, each container can make 16 pitchers.
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Solve the linear programming problem. Minimize and maximize Upper P equals negative 20 x plus 30 y Subject to 2 x plus 3 y greater than or equals 30 2 x plus y less than or equals 26 negative 2 x plus 3 y less than or equals 30 x comma y greater than or equals 0
Answer:
Maximum = 540 at (6,14)
Minimum = 300 at (0,10) or (12,2).
Step-by-step explanation:
The given linear programming problem is
Minimize and maximize: P = 20x + 30y
Subject to constraint,
[tex]2x+3y\ge 30[/tex] .... (1)
[tex]2x+y\le 26[/tex] .... (2)
[tex]-2x+3y\le 30[/tex] .... (3)
[tex]x,y\geq 0[/tex]
The related equation of given inequalities are
[tex]2x+3y=30[/tex]
[tex]2x+y=26[/tex]
[tex]-2x+3y=30[/tex]
Table of values are:
For inequality (1).
x y
0 10
15 0
For inequality (2).
x y
0 26
13 0
For inequality (3).
x y
0 10
15 0
Pot these ordered pairs on a coordinate plane and connect them draw the corresponding related line.
Check each inequality by (0,0).
[tex]2(0)+3(0)\ge 30\Rightarrow 0\ge 30[/tex] False
[tex]2(0)+(0)\le 26\Rightarrow 0\le 26[/tex] True
[tex]-2(0)+3(0)\le 30\Rightarrow 0\le 30[/tex] True
It means (0,0) is included in the shaded region of inequality (2) and (3), and (0,0) is not included in the shaded region of inequality (1).
From the below graph it is clear that the vertices of feasible region are (0,10), (6,14) and (12,2).
Calculate the values of objective function on vertices of feasible region.
Point P = 20x + 30y
(0,10) P = 20(0) + 30(10) = 300
(6,14) P = 20(6) + 30(14) = 540
(12,2) P = 20(12) + 30(2) = 300
It means objective function is maximum at (6,14) and minimum at (0,10) or (12,2).
During a manufacturing process, a metal part in a machine is exposed to varying temperature conditions. The manufacturer of the machine recommends that the temperature of the machine part remain below 141°F. The temperature T in degrees Fahrenheit x minutes after the machine is put into operation is modeled by
T = 0.005x² + 0.45x + 125.
Will the temperature of the part ever reach or exceed 141F? Use the discriminant of a quadratic equation to decide.
A. yes
B. no
Answer:
Yes, it will reach or exceed 141 degree F
Step-by-step explanation:
Given equation that shows the temperature T in degrees Fahrenheit x minutes after the machine is put into operation is,
[tex]T = 0.005x^2 + 0.45x + 125[/tex]
Suppose T = 141°F,
[tex]\implies 141 = 0.005x^2 + 0.45x + 125[/tex]
[tex]\implies 0.005x^2 + 0.45x + 125 - 141 =0[/tex]
[tex]\implies 0.005x^2 + 0.45x - 16=0[/tex]
Since, a quadratic equation [tex]ax^2 + bx + c =0[/tex] has,
Real roots,
If Discriminant, [tex]D = b^2 - 4ac \geq 0[/tex]
Imaginary roots,
If D < 0,
Since, [tex]0.45^2 - 4\times 0.005\times -16 = 0.2025 + 32 > 0[/tex]
Thus, roots of -0.005x² + 0.45x + 125 are real.
Hence, the temperature can reach or exceed 141 degree F.
The temperature of the part will exceed 141°F during the manufacturing process.
Explanation:To determine if the temperature of the part will ever reach or exceed 141°F, we need to find the value of x when the temperature T equals 141°F. We can do this by setting the equation T = 0.005x² + 0.45x + 125 equal to 141 and solving for x using the quadratic formula.
The quadratic formula is given by x = (-b ± √(b² - 4ac))/(2a), where a, b, and c are the coefficients of the quadratic equation. In this case, a = 0.005, b = 0.45, and c = 125 - 141 = -16.
Calculating the discriminant, which is the value inside the square root in the quadratic formula, we get b² - 4ac = 0.45² - 4(0.005)(-16) = 0.2025 + 0.32 = 0.5225. Since the discriminant is positive, the quadratic equation has two real and distinct solutions, which means the temperature of the part will exceed 141°F at some point during the manufacturing process.
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