Answer:
Does not exceed 2% in both cases.
Step-by-step explanation:
Given that a manufacturer of interocular lenses will qualify a new grinding machine if there is evidence that the percentage of polished lenses that contain surface defects does not exceed 2%.
Sample proportion = [tex]\frac{6}{250} \\=0.024[/tex]
Create hypothesses as
[tex]H_0: p = 0.02\\H_a : p >0.02[/tex]
(Right tailed test at 5% significance level)
P difference = 0.004
Std error = 0.0089
test statistic Z = p diff/std error = 0.4518
p value = 0.326
Since p >alpha, we accept nullhypothesis
b) For confidence interval 97% we have
Margin of error = 2.17* std error = 0.0192
Confidence interval
= [tex](0.024-0.0191, 0.024+0.0191)\\= (0.0049, 0.0431)\\[/tex]
Since 2% = 0.02 lies within this interval we accept null hypothesis.
Does not exceed 2%
Since
Final answer:
To determine whether the new grinding machine can be qualified, we need to test the hypothesis that the percentage of polished lenses with surface defects does not exceed 2%.
Explanation:
To determine whether the new grinding machine can be qualified, we need to test the hypothesis that the percentage of polished lenses with surface defects does not exceed 2%. The null hypothesis (H₀) is that the proportion of defective lenses is equal to or less than 2%, while the alternative hypothesis (Ha) is that the proportion of defective lenses is greater than 2%. We can use a one-sample proportion test to analyze the data.
(a) The hypotheses are:
H₀: p ≤ 0.02 (proportion of defective lenses)
Ha: p > 0.02 (proportion of defective lenses)
With a significance level of α = 0.05, we can calculate the p-value from the sample data. Using a normal approximation to the binomial distribution, we find that the p-value is 0.0165.
(b) The question in part (a) can also be answered using a confidence interval. We can calculate a confidence interval for the proportion of defective lenses and see if it includes or excludes the value of 0.02. If the confidence interval includes 0.02, it suggests that the machine can be qualified. If the confidence interval does not include 0.02, it suggests that the machine cannot be qualified.
The base and sides of a container is made of wood panels. The container does not have a lid. The base and sides are rectangular. The width of the container is x cm . The length is double the width. The volume of the container is 54cm3 . Determine the minimum surface area that this container will have.
Answer:
Minimum surface area =[tex]70.77 cm^2[/tex]
Step-by-step explanation:
We are given that
Width of container=x cm
Length of container=2x cm
Volume of container=[tex]54 cm^3[/tex]
We have to find the minimum surface areas that this container will have.
Volume of container=[tex]l\times b\times h[/tex]
[tex]x\times 2x\times h=54[/tex]
[tex]2x^2h=54[/tex]
[tex]h=\frac{54}{2x^2}=\frac{27}{x^2}[/tex]
Surface area of container=[tex]2(b+l)h+lb[/tex]
Because the container does not have lid
Surface area of container=[tex]S=2(2x+x)\times \frac{27}{x^2}+2x\times x[/tex]
[tex]S=\frac{162}{x}+2x^2[/tex]
Differentiate w.r.t x
[tex]\frac{dS}{dx}=-\frac{162}{x^2}+4x[/tex]
[tex]\frac{dx^n}{dx}=nx^{n-1}[/tex]
Substitute [tex]\frac{dS}{dx}=0[/tex]
[tex]-\frac{162}{x^2}+4x=0[/tex]
[tex]4x=\frac{162}{x^2}[/tex]
[tex]x^3=\frac{162}{4}=40.5[/tex]
[tex]x^3=40.5[/tex]
[tex]x=(40.5)^{\frac{1}{3}}[/tex]
[tex]x=3.4[/tex]
Again differentiate w.r.t x
[tex]\frac{d^2S}{dx^2}=\frac{324}{x^3}+4[/tex]
Substitute x=3.4
[tex]\frac{d^2S}{dx^2}=\frac{324}{(3.4)^3}+4=12.24>0[/tex]
Hence, function is minimum at x=3.4
Substitute x=3.4
Then, we get
Minimum surface area =[tex]\frac{162}{(3.4)}+2(3.4)^2=70.77 cm^2[/tex]
Please help! Question above
Answer:
see below
Step-by-step explanation:
The next step in copying the angle is to copy the width of it (the length of its chord) to the new location. First, you have to set the compass to that chord length, GF.
Solve for (g).
−3+5+6g=11−3g
g= ?
Answer:
g=1
Step-by-step explanation:
Given equation is \[−3+5+6g=11−3g\]
Simplifying, \[2+6g=11−3g\]
Bringing all the terms containing g to the left hand side and the numeric terms to the right hand side,
\[6g + 3g=11−2\]
=> \[(6 + 3)g=9\]
In other words, \[g=\frac{9}{9}\]
Or, g = 1
The given equation is satisfied for g=1.
Validation:
Left hand side of the equation: -3+5+6 = 8
Right hand side of the equation: 11-3 = 8
A recent study reported that the prevalence of hyperlipidemia (defined as total cholesterol over 200) is 30% in children 2-6 year of age. If 12 children are analyzed:
a.What is the probability that at least 3 are hyperlipidemic?
Answer:
The probability is 0.74719
Step-by-step explanation:
Let's start defining the random variable X.
X : ''Number of children with hyperlipidemia out of 12 children''
X can be modeled as a binomial random variable.
X ~ Bi (n,p)
Where n is the sample size and p is the ''success probability''.
We defining as a success to find a child that has hyperlipidemia.
The probability function for X is :
[tex]P(X=x)=(nCx).p^{x}.(1-p)^{n-x}[/tex]
Where nCx is the combinatorial number define as :
[tex]nCx=\frac{n!}{x!(n-x)!}[/tex]
We are looking for [tex]P(X\geq 3)[/tex]
[tex]P(X\geq 3)=1-P(X\leq 2)[/tex]
[tex]P(X\geq 3)=1-[P(X=0)+P(X=1)+P(X=2)][/tex]
[tex]P(X\geq 3)=1-[(12C0)0.3^{0}0.7^{12}+(12C1)0.3^{1}0.7^{11}+(12C2)0.3^{2}0.7^{10}][/tex]
[tex]P(X\geq 3)=1-(0.7^{12}+0.07118+0.16779)=1-0.25281=0.74719[/tex]
There is a probability of 0.74719 that at least 3 children are hyperlipidemic.
To find the probability that at least 3 out of the 12 children are hyperlipidemic, we can use the binomial probability formula. The probability is 36.21% or 0.3621.
Explanation:To find the probability that at least 3 out of the 12 children are hyperlipidemic, we need to use the binomial probability formula. The probability of success is 30% or 0.3 (since 30% of children are hyperlipidemic), and the probability of failure is 1 - 0.3 = 0.7.
The formula for the binomial probability is P(X >= k) = 1 - P(X < k), where X follows a binomial distribution with n trials (12 children in this case) and probability of success p (0.3).
To find P(X < k), we need to calculate the probabilities for X = 0, 1, and 2 children being hyperlipidemic and then sum them up.
P(X = 0) = [tex](0.7)^{12[/tex] = 0.0687P(X = 1) = 12C1 * [tex](0.3)^1 * (0.7)^{{11[/tex] = 0.2332P(X = 2) = 12C2 * [tex](0.3)^2 * (0.7)^{10[/tex] = 0.3361Summing up these probabilities, we get P(X < 3) = 0.0687 + 0.2332 + 0.3361 = 0.6379
Finally, the probability of at least 3 children being hyperlipidemic is P(X >= 3) = 1 - P(X < 3) = 1 - 0.6379 = 0.3621 or 36.21%.
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The owner of a store sells raisins for $3.20 per pound and nuts for $2.40 per pound. He decides to mix the raisins and nuts and sell 50 lb of the mixture for $2.72 per pound. What quantities of raisins and nuts should he use
Answer:
Step-by-step explanation:
First we have to find the worth of 50 Ib of both mixtures, so we multiply 50 by 2.72 since the owner wants to sell the mixture at $2.72 per pound
50 Multiplied by $2.72 equals $135
Then we divide that amount by 2 since we are considering two types of products, raisins and nuts
$135 divided by 2 equals $67.5
So we find the amount of raisins that is worth $67.5 and we know a pound of raisins cost $3.2
We divide $67.5 by $3.2 which will give 21.09 Ib (Amount for the raisins)
We divide $67.5 by $2.4 which will give 28.13 Ib (Amount for the nuts)
Final answer:
The store owner should mix 20 lb of raisins with 30 lb of nuts to make a 50 lb mixture that can be sold for $2.72 per pound. This is determined by solving a system of linear equations.
Explanation:
The owner of the store wants to mix raisins and nuts to sell a 50 lb mixture for $2.72 per pound. To find out the quantities of raisins and nuts he should use, we can set up a system of equations. Let R represent the amount of raisins in pounds and N represent the amount of nuts in pounds. The two equations area:
R + N = 50 (the total weight of the mixture)3.20R + 2.40N = 50 * 2.72 (the total cost of the mixture)We can solve this system using the substitution or elimination method. Assuming we choose elimination, we can multiply the first equation by 2.40 to make the coefficient of N the same in both equations:
2.40R + 2.40N = 120 (multiplying the first equation by 2.40)3.20R + 2.40N = 136 (the second equation)Subtracting these equations gives us:
0.80R = 16Dividing both sides by 0.80 gives us the amount of raisins:
R = 20 lbUsing the first equation (R + N = 50), we can find the amount of nuts:
N = 50 - RN = 50 - 20N = 30 lbSo the owner should mix 20 lb of raisins with 30 lb of nuts to make the mixture.
The grocery store sold 1346 cans of tomato soup in January. Have as many cans were sold in July. How many cans of tomato soup were sold in January and July?
Answer:
2,692 cans of tomato soup were sold in January and July
Step-by-step explanation:
The number of cans of soup sold in January = 1346
Store sold same number of soup cans in July.
⇒The number of cans of soup sold in July = 1346
So, the total soup cans sold in January and July
= Sum of soup cans sold in both months
= 1346 + 1346
=2,692
Hence, 2,692 cans of tomato soup were sold in January and July.
A statistic instructor randomly selected four bags of oranges, each bag labeled 10 pounds, and weighed the bags.They weighed 9.3, 9.7, 9.2, and 9.7 pounds. assume that the distribution of the weights is normal. Find a 95% confidence interval for the mean weight of all bags of oranges.
We are 95% confident the population mean is between ____ and ____?
The 95% confidence interval for the mean weight of all bags of oranges is calculated using the sample mean and the sample standard deviation. After calculation, we are 95% confident the mean weight is between 9.193 and 9.757 pounds.
Explanation:To find the 95% confidence interval for the mean weight of all bags of oranges, we first need to calculate the sample mean and sample standard deviation. The sample mean, [tex]\overline{x}[/tex], is the sum of all sample weights divided by the number of samples. In this case, it is (9.3+9.7+9.2+9.7)/4 = 9.475 pounds.
The sample standard deviation, s, is the square root of the sum of the squared differences between each sample weight and the sample mean, divided by the number of samples minus one. The s value here is approximately 0.253 pounds.
For a 95% confidence interval with a sample size of 4, the z-score is 2.776 (obtained from a standard z-table). The margin of error is the z-score multiplied by the standard deviation divided by the square root of the sample size. This is approximately 0.282 pounds. So, the 95% confidence interval is (9.475-0.282, 9.475+0.282) = (9.193, 9.757) pounds.
So we can say that we are 95% confident that the mean weight of all bags of oranges is between 9.193 and 9.757 pounds.
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please help me solve the screenshot below.
Answer:
The arrangement of the given equation in the slope - intercept form are
[tex]y=-x+4[/tex][tex]y=2x-5[/tex]Step-by-step explanation:
Given:
x + y = 4 and
y - 2x = -5
Slope - intercept form :
[tex]y=mx+c[/tex]
Where,
m is the slope of the line.
c is the y-intercept.
When two points are given say ( x1 , y1 ) and ( x2 , y2) we can remove slope by
Slope,
[tex]m=\frac{y_{2}- y_{1}}{x_{2}- x_{1}}[/tex]
Intercepts: Where the line cut X axis called X- intercept and where cut Y axis is called Y- intercept.
So, the Slope -intercept form of
x + y = 4 is [tex]y=-x+4[/tex]
and
y - 2x = -5 is [tex]y=2x-5[/tex]
How do you do this question?
Answer:
C
Step-by-step explanation:
f"(x) < 0, which means the function is concave down at all values of x.
For any such function, within the domain of a ≤ x ≤ b, the secant line S(x) is below the curve of f(x), and the tangent line T(x) is above the curve of f(x).
Here's an example:
desmos.com/calculator/fyektbi9yl
At the beginning of the year, a sporting goods store had $250,000 worth of inventory. The store’s buyers purchased an additional $115,000 worth of inventory during the year. At year’s end, the value of the inventory was $185,000. What was the store’s cost of goods sold?
Answer:
180000
Step-by-step explanation:
A sprinkler sprays water over a distance of 40 feet and rotates through an angle of 80 degrees. find the area watered by the sprinkler.
A: 1117.01 ft^2
B: 558.51 ft^2
C: 111.70 ft^2
D: 55.85 ft^2
PLEASE HELP I WILL GIVE BRAINLIEST!! TEST GRADE AND TEST IS TIMED!
Answer:
[tex]1117.01 \mathrm{ft}^{2} \text { is the watered are by the sprinkler. }[/tex]
Option: A
Step-by-step explanation:
A sprinkler sprays water over a distance of (r) = 40 feet
Rotates through an angle of (θ) = 80°
80° convert to radians
[tex]\text { Radians }=80^{\circ} \times\left(\frac{\pi}{180}\right)[/tex]
[tex]\text { Radians }=80^{\circ} \times 0.017453292[/tex]
θ in Radians = 1.396263402
We know that,
Area of sprinkler is [tex]\mathrm{A}=\frac{1}{2} \mathrm{r}^{2} \theta[/tex]
Substitute the given values,
[tex]A=\frac{1}{2} \times 40^{2} \times 1.396263402[/tex]
[tex]A=\frac{(1600 \times 1.396263402)}{2}[/tex]
[tex]\mathrm{A}=\frac{2234.021443}{2}[/tex]
[tex]\mathrm{A}=1117.01 \mathrm{ft}^{2}[/tex]
Area of sprinkler is [tex]1117.01 \mathrm{ft}^{2}[/tex]
A fruit stand sells two varieties of strawberries: standard and deluxe. A box of standard strawberries sells for $7, and a box of deluxe strawberries sells for $11. In one day the stand sold 110 boxes of strawberries for a total of $930. How many boxes of each type were sold?
Answer:
70 standard boxes, 40 deluxe boxes
Step-by-step explanation:
standard: x
deluxe: y
x + y = 110
7x + 11y = 930
Multiply the first equation by 7 and subtract.
7x + 7y = 770
-(7x + 11y = 930)
You get -4y = -160
y = 40
Substitute y into the first equation.
x + 40 = 110
x = 70
Suppose you draw a card, put it back in the deck, and draw another one. What is the probability that the first card is a two and the second is a three? (Enter your probability as a fraction.) P =
Answer:
0.006
Step-by-step explanation:
This is joint probability and thus the probability of the given event is the product of:
(probability of drawing a 2 on the first draw)*(prob. of dring a 3 on the second draw), or
4 4 16
----- * ------ = ------------ = 0.006
52 52 (52)^2
Answer:
1/169
Step-by-step explanation:
if you mean a deck as in the 52 cards then the answer is the following.
there are 4 cards in a deck with the number 2 on it. 2 of spades, diamonds, hearts and clubs. so the probability would be 4/42 because there is 4 out of the 52 cards there are. next there is also 4 cards with three on it, so 4/52 again.you multiply these two to get the answer(but before you do i would recommend to simplify it to 1/13) to get 1/169
A researcher conducts a study and finds that the outcome measure is normally distributed with a mean of 57, a median of 56, and a standard deviation of 6. Approximately 95% of the sample falls between which two values?
Answer:
56 and 57
Step-by-step explanation:
The numbers 56 and 57 are values that are very close it is better to estimate 56 and 57 then 6 and 56 and 57
The answer to the teacher's question is that approximately 95% of the sample lies within the range of 45 to 69. This conclusion is based on the features of the normal distribution
Explanation:The study's results use elements of statistics, with the distribution in question being a normal distribution. A characteristic of a normal distribution is that approximately 95% of measurements will fall within two standard deviations, both above and below the mean. Given that the mean in this case is 57 and the standard deviation is 6, we can calculate the values between which approximately 95% of the sample falls.
Doing the math, we find that: 57 - (2*6) = 45 and 57 + (2*6) = 69. Thus, approximately 95% of the sample lies between the values of 45 and 69.
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Two rainstorms occurred in one week. First storm 15 mL of rain fell per hour. Second storm 30 mL of rain fell per hour. The rain lasted for a total of 70 hours with a total of 1500 mL. What was the duration of each storm?
Answer:
The answer to your question is the first rain lasted 40 hours and the second rain lasted 30 hours.
Step-by-step explanation:
Data
1st 15 ml/h = x
2nd 30 ml/h = y
Total time = 70 h and 1500 ml
1.- Write 2 equations
First rain + second rain = 70 h
x + y = 70 (I)
15x + 30 y = 1500 (II)
2.- Solve equations by elimination
Multiply first equation by -15
-15x - 15y = - 1050
15x + 30y = 1500
0x + 15y = 450
y = 450 / 15
y = 30 h
x + 30 = 70
x = 70 - 30
x = 40 h
3.- Conclude
The first rain lasted 40 h and the second rain lasted 30 h.
Final answer:
The first rainstorm lasted for 40 hours and the second rainstorm lasted for 30 hours. A system of equations method was used to find the duration of each storm.
Explanation:
Calculating Duration of Rainstorms
Let's denote the duration of the first storm as x hours and the second storm as y hours. The total amount of rain for the first storm would then be 15 mL/hour × x hours and for the second storm 30 mL/hour × y hours. Given that the total duration of the rain is 70 hours, we can set up the following equation: x + y = 70. Additionally, since the total volume of rain is 1500 mL, we have 15x + 30y = 1500.
We can solve this system of equations by multiplying the first equation by 15 to eliminate variable y: 15x + 15y = 1050 and then subtracting this from the second equation: (15x + 30y) - (15x + 15y) = 1500 - 1050, which simplifies to 15y = 450. Thus, y = 30 hours. Finally, we substitute y into x + y = 70 to find x: x = 70 - 30, so x = 40 hours.
Hence, the first storm lasted for 40 hours and the second storm lasted for 30 hours.
A random sample of eighty-five students in Chicago city high schools takes a course designed to improve SAT scores. Based on these students, a 90% confidence interval for the mean improvement in SAT scores from this course for all Chicago city high school students is computed as (72.3, 91.4) points. The correct interpretation of this interval is__________.
A. that 90% of the students in the sample had their scores improve by between 72.3 and 91.4 points.B. that 90% of the students in the population should have their scores improve by between 72.3 and 91.4 points.C. Neither choice is correct.
Answer:
(B) The correct interpretation of this interval is that 90% of the students in the population should have their scores improve by between 72.3 and 91.4 points.
Step-by-step explanation:
Confidence interval is the range the true values fall in under a given confidence level.
Confidence level states the probability that a random chosen sample performs the surveyed characteristic in the range of confidence interval. Thus,
90% confidence interval means that there is 90% probability that the statistic (in this case SAT score improvement) of a member of the population falls in the confidence interval.
A hemispherical depression is cut out from one face of a cuboidal wooden block of edge 21cm such that the diameter of the hemisphere is equal to the edge of the cube. Determine the total surface area of the remaining block
Answer:
(2646 +110.25π) cm² ≈ 2992.4 cm²
Step-by-step explanation:
The area of a sphere is 4 times the area of a circle with the same radius. Hence the area of a hemisphere will be 2 times the area of that circle. This means carving a hemispherical depression in the face of the cube will add an area that is equal to the area of the circular hole.
Of course the total surface area of a cube is 6 times the area of one square face. The area of a circle is ...
A = πr² = π(d/2)² = (π/4)d²
The total surface area of the carved cube is ...
S = 6·(21 cm)² + (π/4)·(21 cm)² = (441 cm²)(6 +π/4)
S ≈ 2992.36 cm²
The total surface area of the remaining block is about 2992.4 cm².
Answer:
(2646 +110.25π) cm² ≈ 2992.4 cm²
Step-by-step explanation:
Is 3/13 closer to 1/2, 1 or 0
Answer: 3/13 is closer to 0 than 1/2 or 1.
- This is because 1/2 of 1/13 would be about 6.5/13.
- Since 3/13 is not close to 1/2, nor it will be to 1.
A rectangular deck is to be constructed using a rock wall as one side and fencing for the other three sides. There are 24 yards of fencing available. Determine the dimensions that would create the deck of maximum area. What is the maximum area? Enter only the maximum area. Do not include units in your answer.
The problem is solved by using calculus to find the maximum of the area function for a rectangle. The dimensions that yield maximum area with 24 yards of fencing are a length of 6 yards and a width of 12 yards, resulting in an area of 72 square yards.
Explanation:The problem involves maximizing the area of a rectangle with a fixed perimeter. Let's define the length of the rectangle as x and the width as y. Since you have 24 yards of fencing and you need to enclose three sides of the rectangle, the perimeter equation becomes 2x + y = 24, which can be rearranged as y = 24 - 2x.
The area of the rectangle can be represented by the equation A = xy, and substituting the y value from our perimeter equation, we get A = x(24 - 2x). To maximize this area, we take the derivative of A with respect to x and set it equal to zero, yielding the equation 24 - 4x = 0, or x = 6. Substituting x = 6 back into the y equation gives y = 12.
Therefore, the dimensions that maximize the area of the deck are a length of 6 yards and a width of 12 yards. Substituting these values back into the area equation gives a maximum area of 72 square yards.
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In survey conducted by Quinnipiac University from October 25-31, 2011, 47% of a sample of 2,294 registered voters approved of the job Barack Obama was doing as president.
What is the 99% confidence interval for the proportion of all registered voters who approved of the job Barack Obama was doing as president?
A) (0.460, 0.480)
B) (0.453, 0.487)
C) (0.450, 0.490)
D) (0.443, 0.497)
Answer:
D) (0.443, 0.497)
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The confidence interval for a proportion is given by this formula
[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
For the 99% confidence interval the value of [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2=0.005[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=2.58[/tex]
And replacing into the confidence interval formula we got:
[tex]0.47 - 2.58 \sqrt{\frac{0.47(1-0.47)}{2294}}=0.443[/tex]
[tex]0.47 + 2.58 \sqrt{\frac{0.47(1-0.47)}{2294}}=0.497[/tex]
And the 99% confidence interval would be given (0.443;0.497).
We are confident that about 44.3% to 49.7% of registered voters approved of the job Barack Obama was doing as president.
I reward with points!
1. Write an exponential function to represent the spread of Ben's social media post.
Step-by-step explanation:
Let [tex]f(x)[/tex] be the number of shares in [tex]x[/tex] days.
Let the function [tex]f(x)=ab^{x}[/tex].
It is given that each friend shares with three friends the next day.
So,[tex]f(x)=3\times f(x-1)[/tex]
substituting [tex]f(x)=ab^{x}[/tex]
[tex]ab^{x}=3\times ab^{x-1}[/tex]
So,[tex]b=3[/tex]
Given that at day [tex]0[/tex],there are [tex]2[/tex] shares.
So,[tex]f(0)=2[/tex]
[tex]a3^{0}=2[/tex]
[tex]a=2[/tex]
So,[tex]f(x)=2\times 3^{x}[/tex]
The value of China's exports of automobiles and parts (in billions of dollars) is approximately f ( x ) = 1.8208 e .3387 x , where x = 0 corresponds to 1998. In what year did/will the exports reach $12.3 billion?
The Chinese exports of automobiles and parts reached a value of approximately $12.3 billion around the year 2004.
Explanation:Let's first assume that the value of the function f(x) equals the targeted exports, $12.3 billion. Therefore, we have the equation 1.8208e⁽°³³⁵⁷ˣ⁾= 12.3. We can solve this equation for x, which represents the number of years since 1998.
First, divide both sides of the equation by 1.8208 to isolate e⁽°³³⁵⁷ˣ⁾. You will get e⁽°³³⁵⁷ˣ⁾ = 6.7475 approximately. To get rid of the base e, we take the natural logarithm (ln) of both sides. This gives us .3387x = ln(6.7475).
Divide this by .3387 to solve for x. The solution approximates to x = 5.9. This means the exports reach $12.3 billion approximately 6 years after 1998, which would be around the year 2004.
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The year with $12.3 billion in Chinese auto exports is found by setting the provided exponential function equal to it and solving for x, which represents years past 1998. We apply logarithms to solve the equation. The final value of x, when added to 1998, gives us the requested year.
Explanation:The goal is to find the year when the value of China's exports (represented by f(x)) reaches $12.3 billion. This can be achieved by setting f ( x ) = 1.8208 e .3387 x equal to $12.3 billion and solving for x using logarithmic properties.
Here are the steps:
Set f(x) = 12.3Therefore, 1.8208 e.3387 x = 12.3Divide both sides by 1.8208 to get e.3387 x = 12.3 / 1.8208Apply natural logarithm (ln) to both sides which results in .3387x = ln(12.3 / 1.8208)Finally, solve for x to find the year: x = ln(12.3 / 1.8208) / .3387Once you find the value of x, add this value to the base year 1998 to get the year when the exports reached $12.3 billion.
Note: You require a calculator with the capability to calculate natural logs to solve for x.
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Please help with this question show work please I need it today thankyou
Answer:
28. a) [tex]-7.3 + 7.4[/tex]
29. c) [tex]3n + 0.18[/tex]
Step-by-step explanation:
Question 28:
Given:
The equation is -9.6 + 9.5 + _________ = 0
In order to solve this, first let us simplify the terms -9.6 + 9.5.
-9.6 + 9.5 = -0.1
So, we are getting the sum of first 2 numbers as -0.1. Now, the result is 0. So, we know that, -a + a = 0. Therefore, the number that should come in the blanks should be -(-0.1) = 0.1
Now, among all the 4 choices, only choice (a) gives the sum as 0.1 as shown below:
-7.3 + 7.4 = 0.1
The options (b) and (c) has a result of 0 and option (d) sum is -0.1.
Therefore, the correct option is option (a).
Question 29:
Given:
The expression to simplify is given as:
[tex]2n+n+0.18[/tex]
Using commutative property of addition, we add the like terms together. Here, the like terms are [tex]2n\ and\ n[/tex]. So, we add them as shown below:
[tex]2n+1n=(2+1)n=3n[/tex]
Therefore, the given expression is simplified to [tex]3n + 0.18[/tex]. Hence, the correct option is option (c).
Candace's backyard has a flower garden that covers 40% of her backyard.If the total area of her backyard is 450 square feet, how many square feet is her flower garden? A40 B180 C270 D18,000
Answer: option B is the correct answer
Step-by-step explanation:
Candace's backyard has a flower garden that covers 40% of her backyard.
If the total area of her backyard is 450 square feet, it means that her garden covers 40% of 450 square feet.
To determine how many square feet is her flower garden, we will find the value of 40% of 450 square feet. It becomes
40/100 × 450= 0.4×460 = 180 square feet
Solve the equation by factoring. 2x2 = 28 - x
x = -3.5 or x = 4
x = 2 or x = -7
x = -2 or x = 7
x= -4 or x = 3.5
Answer:
2x^2 = 28 - x
-28+x -28+x
2x^2 + x - 28 = 0
ac=-56
m+p=-7,8
(2x-7)(2x+8)=0
2x-7+7=0+7
2x/2=7/2
x=3.5
2x+8-8=0-8
2x/2=-8/2
x= -4
Step-by-step explanation:
subtract from right to left to get the equation in a standard form = 0since there's no GCF and no two numbers that multiply to -28 and add to x, we can multiply a and c (-56), then find two numbers that multiply to ac and add to bx (-7,8). factor the equation into two binomials and set them equal to zero. the binomials must start with the square root of the first term and end with the square root of the second term.solve for x if the two binomials.x=-4 or 3.5
Final answer:
The quadratic equation 2x² = 28 - x is solved by rearranging it to 2x² + x - 28 = 0, factoring by grouping, and finding the solutions x = 4 and x = -3.5. The solutions are verified by substituting them back into the original equation.
Explanation:
To solve the equation by factoring, we first need to rearrange the equation to set it to zero. The original equation given is 2x² = 28 - x. Moving all terms to one side gives us 2x² + x - 28 = 0. We can solve this quadratic equation by factoring:
First, look for two numbers that multiply to give ac (in this case, 2 * -28 = -56) and add to give b (in this case, 1).
These two numbers are 7 and -8 since 7 * -8 = -56 and 7 + (-8) = -1.
Rewrite the middle term using these two numbers: 2x² + 7x - 8x - 28 = 0.
Factor by grouping: (2x² + 7x) - (8x + 28) = 0.
Factor out the common terms: x(2x + 7) - 4(2x + 7) = 0.
Factor out the common binomial: (x - 4)(2x + 7) = 0.
Solving each factor separately gives us: x = 4 and x = -7/2 or -3.5.
The solutions we find are x = 4 and x = -3.5. To check if these solutions are correct, we can substitute them back into the original equation and verify if they satisfy the equation, thereby confirming they are correct.
Use the algebraic procedure explained in section 8.9 in your book to find the derivative of f(x)=1/x. Use h for the small number. (Hint: Simplify f(x+h)-f(x) by finding a common denominator and combining the two fractions).
Answer:
By definition, the derivative of f(x) is
[tex]lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}[/tex]
Let's use the definition for [tex]f(x)=\frac{1}{x}[/tex]
[tex]lim_{h\rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\\lim_{h\rightarrow 0} \frac{\frac{x-(x+h)}{x(x+h)}}{h}=\\lim_{h\rightarrow 0} \frac{\frac{(-1)h}{x^2+xh}}{h}=\\lim_{h\rightarrow 0} \frac{(-1)h}{h(x^2+xh)}=\\lim_{h\rightarrow 0} \frac{-1}{x^2+xh)}=\frac{-1}{x^2+x*0}=\frac{-1}{x^2}[/tex]
Then, [tex]f'(x)=\frac{-1}{x^2}[/tex]
Suppose a spider moves along the edge of a circular web at a distance of 3 cm from the center.?
(a) If the spider begins on the far right side of the web and creeps counterclockwise until it reaches the top of the web, approximately how far does it travel?
Distance: ______units
(b) If the spider crawls along the edge of the web a distance of 1 cm, approximately what is the angle formed by the line segment from the center of the web to the spider's starting point and the line segment from the center of the web to the spider's finishing point?
Angle: _______degrees
Answer:
a) d = 4,712 cm
b) ∠ 108,52 ⁰
Step-by-step explanation:
The crcular path of the spider has radius = 3 cm
If the spider moves from the fa right side of the circle its start poin of the movement is P ( 3 , 0 ) (assuming the circle is at the center of the coordinate system. And the top of the web (going counterclockwise ) is the poin ( 0 , 3 )
So far the spider has traveled 1/4 of the circle
The lenght of the circle is
L = 2*π*r
The traveled distance for the spider (d) is
d = (1/4 )* 2*π*r ⇒ d = (1/2)*3,1416*3 ⇒ d = 4,712 cm
And now spider go ahead a new distance of approximately 1 more cm
Therefore the spider went a total of 5.712 cm
Now we know that
πrad = 180⁰ then 1 rad = 180/π ⇒ 1 rad = 57⁰
rad = lenght of arc/radius
so 5,712/3 = 1.904 rad
By rule of three
1 rad ⇒ 57⁰
1.904 rad ⇒ ? x
x = 57 * 1.904 ⇒ 108.52 ⁰
So the ∠ 108,52 ⁰
A potter forms a 100 cm3 piece of clay into a cylinder. As she rolls it, the length L,of the cylinder increases and the radius, r, decreases. If the length of the cylinder is increasing at 0.1 cm per second, find the rate at which the radius is changing when the radius is 5 cm.
Answer:
- 0.2 cm/s
Step-by-step explanation:
Volume of a cylinder = πr²l--------------------------------------------------------(1)
dV/dt =(dV /dr ) x (dr/dt) + (dV /dl ) x (dl/dt) ---------------------------------(2)
dV/dr = 2 πrl
dV/dl = πr²
dr/dt = Unknown
dl/dt = 0.1 cm/s
dV/dt = 0
From equation (1), the length of the cylinder can be calculated when r = 5cm
V = πr²L
100 = π (5)²L
L =100/25π
=4/π
To find the rate of radius change (dr/dt) we substitute known values into equation (2):
0 = 2 π (5) (4/π) x (dr/dt) + π(5)² x 0.1
0 = 40 (dr/dt) + 2.5π
40 (dr/dt) = -2.5π
dr/dt = -2.5π/40
= -0.1963 cm/s
≈ - 0.2 cm/s
The negative sign shows that the radius of the cylinder of constant volume decrease at a rate twice the length.
It takes 4 lawnmowers to cut 2 acres of grass in 2 hours. If the amount of time varies directly with the amount of grass and inversely with the number of lawnmowers, how many hours will it take 2 lawnmowers to cut 6 acres of grass?A. 3B. 4C. 8D. 12
Answer:
It take 12 hours for 2 lawnmowers to cut 6 acres of grass
Step-by-step explanation:
We are given that It takes 4 lawnmowers to cut 2 acres of grass in 2 hours.
Let [tex]T_1 , M_1 and W_1[/tex] denotes the time taken , No. of land mowers and Amount of grass respectively in case 1 .
[tex]T_1=2 \\W_1=2\\M_1=4[/tex]
Now The amount of time varies directly with the amount of grass and inversely with the number of lawnmowers
Let [tex]T_2 , M_2 and W_2[/tex] denotes the time taken , No. of land mowers and Amount of grass respectively i case 2
[tex]T_1=? \\W_1=6\\M_1=2[/tex]
So, [tex]\frac{M_1 \times T_1}{W_1}=\frac{M_2 \times T_2}{W_2}[/tex]
Substitute the values
[tex]\frac{4 \times 2}{2}=\frac{2 \times ?}{6}[/tex]
[tex]\frac{4 \times 2 \times 6}{2 \times 2}=?[/tex]
[tex]12=?[/tex]
Hence it take 12 hours for 2 lawnmowers to cut 6 acres of grass
Answer:
D. 12
Step-by-step explanation:
hope this helps :)
Which of the values satisfy the following inequality
|x-7.5|≤ 17
Select all that apply.
A: x= 20
B: x= -10
C: x= -9
D: x= 27
Answer:
A: x = 20
C: x = -9
Step-by-step explanation:
You can solve the inequality and compare that with the offered choices, or you can try the choices in the inequality to see if it is true. Either approach works, and they take about the same effort.
Solving it:
Unfold it ...
-17 ≤ x -7.5 ≤ 17
Add 7.5 ...
-9.5 ≤ x ≤ 24.5
The numbers 20 and -9 are in this range: answer choices A and C.
_____
Trying the choices:
A: |20 -7.5| = 12.5 ≤ 17 . . . . this works
B: |-10 -7.5| = 17.5 . . . doesn't work
C: |-9 -7.5| = 16.5 ≤ 17 . . . . .this works
D: |27-7.5| = 19.5 . . . doesn't work
The choices that work are answer choices A and C.