Answer:
C) 6 square inches.
Step-by-step explanation:
The length of the logo = 3 inches
The width of the logo = 2 inches.
It is rectangle shape.
The area of a rectangle = length × width
So, the space needed to work on the logo = 3 × 2 = 6 square inches.
The answer is C) 6 square inches.
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
In a research study conducted to determine if arrests were related to the socioeconomic class of the offender, the chi square critical score was 9.488 and the chi square test statistic was 12 2 We can conclude that:
A. The variables are dependent
B. The variables are independent
C. The probability of getting these results by random chance alone is 5
D. Being in a certain socioeconomic class causes arrests
Suppose that textbook weights are normally distributed. You measure 28 textbooks' weights, and find they have a mean weight of 76 ounces. Assume the population standard deviation is 12.3 ounces. Based on this, construct a 95% confidence interval for the true population mean textbook weight. Round answers to 2 decimal places.
Answer:
Step-by-step explanation:
We want to find 95% confidence interval for the mean of the weight of of textbooks.
Number of samples. n = 28 textbooks weight
Mean, u =76 ounces
Standard deviation, s = 12.3 ounces
For a confidence level of 95%, the corresponding z value is 1.96. This is determined from the normal distribution table.
We will apply the formula
Confidence interval
= mean +/- z ×standard deviation/√n
It becomes
76 +/- 1.96 × 12.3/√28
= 76 +/- 1.96 × 2.3113
= 76 +/- 4.53
The lower end of the confidence interval is 76 - 4.53 =71.47
The upper end of the confidence interval is 76 + 4.53 = 80.53
Therefore, with 95% confidence interval, the mean textbook weight is between 71.47 ounces and 80.53 ounces
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
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.
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:
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]
The ___________ of a lens or mirror is a rotational symmetry axis of the surfaces.
Answer:
Optical axis
Step-by-step explanation:
Optical axis is the rotational symmetry axis of the surfaces.
A line with a certain degree of rotational symmetry is called as the optical axis in an optical system.
It is the straight line that passes through the geometric center of the lens and joins two curvature centers of its surfaces.
It is also called as the principal axis.
99 POINTS BRAINLIEST!!! No fake answers!
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
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
Evaluate (x + y)^0 for x = -3 and y = 5.
Answer:
The answer is 1.
Answer:
1
Step-by-step explanation:
any variable of power 0 equal to 1
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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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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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
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.
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 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.
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
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).
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]
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]
URGENT!!!
Find the equation x^2 + y^2 + Dx + Ey + F = 0
of the circle that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle.
(0, 0), (8, 8), (16, 0)
Answer:
D= -16
E= 0
F= 0
Step-by-step explanation:
The given equation is [tex]x^{2} + y^{2} + Dx + Ey + F = 0[/tex]
It is also given that the circle passes through (0,0) (16,0) and (8,8).
Inserting (0,0) in the equation, it gives
[tex]0 + 0 + 0 + 0 + F = 0[/tex]
This gives F = 0 .
Now inserting (16,0) , it gives
[tex]16^{2} + 0^{2} + D(16) + E(0) + 0 = 0[/tex]
[tex]D(16) = -256[/tex]
[tex]D = \frac{-256}{16}[/tex]
D = -16
Now inserting (8,8) , it gives
[tex]8^{2} + 8^{2} + (-16)(8) + (E)(8) + 0 = 0[/tex]
[tex]-16 + E = -16[/tex]
E = 0
Thus the equation of circle is
[tex]x^{2} + y^{2} + (-16)x = 0[/tex]
We can draw the following graph and thus verify that points (0,0) (8,8) and (16,0) lie on graph.
The equation of the circle passing through points (0, 0), (8, 8), and (16, 0) is x^2 + y^2 - 16x - 16y = 0. Solving the system of equations derived from substituting the given points into the circle equation confirms these coefficients for D and E.
Explanation:To find the equation of the circle that passes through the points (0, 0), (8, 8), and (16, 0), we can use the standard form of a circle's equation:
x
2
+
y
2
+ D
x
+ E
y
+ F = 0
Because the circle passes through the origin (0,0), we know that F = 0. With the remaining points (8, 8) and (16, 0), we can substitute these coordinates into the equation to form a system of equations.
Using point (8,8), the equation becomes:
64 + 64 + 8D + 8E + F = 0
Using point (16,0), the equation becomes:
256 + 16D + F = 0
Since F = 0, the system of equations is:
8D + 8E + 128 = 016D + 256 = 0Solving these equations, we get:
D = -16E = -16The equation of the circle is therefore:
x2 + y2 - 16x - 16y = 0To verify the result, plotting the points and the graph of the circle on a graphing utility should show that the points lie on the circumference of the circle.
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.
One leg of a right triangle is 4 mm shorter than the longer leg in the hypotenuse is 4 mm longer than the longer leg find the links of the sides of the triangle
Answer:
Step-by-step explanation:
The right triangle has three sides which can be called legs. The legs are; shorter leg. Longer leg and hypotenuse
Let the longer leg be x
One leg of a right triangle is 4 mm shorter than the longer leg. This means
The shorter leg = x - 4
the hypotenuse is 4 mm longer than the longer leg. This means
The hypotenuse = x + 4
So the legs of the triangle are
Shorter leg or side = x-4
Longer leg or side = x
Hypotenuse = x + 4
The lengths of the sides of the triangle are 12 mm, 16 mm, and 20 mm.
Explanation:Let's use variables to represent the lengths of the sides:
Shorter leg: x mmLonger leg: x + 4 mmHypotenuse: x + 8 mmAccording to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse:
a² + b² = c²
Plugging in the values, we have:
x² + (x + 4)² = (x + 8)²
Expanding and simplifying, we get:
x² + x² + 8x + 16 = x² + 16x + 64
Combining like terms, we get:
x² - 8x - 48 = 0
Factoring the quadratic equation, we find:
(x - 12)(x + 4) = 0
Therefore, x = 12 or x = -4. We discard the negative value, so the lengths of the sides of the triangle are:
Shorter leg: 12 mmLonger leg: 16 mmHypotenuse: 20 mmQ3:
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
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°
Which percent is equivalent to 3/4 ?
A) 25%
B) 50%
C) 60%
D) 75%
Answer:
d is the correct answer
Answer:75
Step-by-step explanation:
Simplify the function f(x) = 1/3 (81) 3x/4 Then determine the key aspects of the function.
Answer:
[tex]f(x)=3^{3x-1}[/tex].
The domain of the function is the set of all real number and the range is [tex](0,\infty)[/tex]
Step-by-step explanation:
Given:
The function is given as:
[tex]f(x)=\frac{1}{3}(81)^{\frac{3x}{4}}[/tex]
Using the rule of the exponents, [tex]a^{mn}=(a^m)^n[/tex],
[tex]f(x)=\frac{1}{3}((81)^{\frac{1}{4}})^{(3x)}\\f(x)=\frac{1}{3}(\sqrt[4]{81} )^{3x}\\f(x)=\frac{1}{3}(3)^{3x}\\f(x)=\frac{3^{3x}}{3^1}[/tex]
Using the rule of the exponents,[tex]\frac{a^m}{a^n}=a^{m-n}[/tex],
[tex]f(x)=3^{3x-1}[/tex]
Therefore, the simplified form of the given function is:
[tex]f(x)=3^{3x-1}[/tex]
Key aspects:
The given function is an exponential function with a constant base 3.
Domain is the set of all possible values of [tex]x[/tex] for which the function is defined.
The domain of an exponential function is a set of all real values.
The range of an exponential function is always greater than zero.
Therefore, the domain of this function is also all real values and the range is from 0 to infinity.
Domain: [tex]x \epsilon (-\infty,\infty)[/tex]
Range: [tex]y\epsilon (0,\infty)[/tex]
Answer: 1/3 27 all real numbers y>0
Step-by-step explanation:
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.
Learn more about temperature calculation here:https://brainly.com/question/36032928
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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: