Final answer:
The antique purchased by Juan increases in value by 10% each year. The value of the antique after 2 years can be found by calculating the expression $200a, where a is a constant. The value of [tex]\( a \) is \( 1.21 \).[/tex]
Explanation:
To find the value of a, we need to represent the annual increase of 10% in terms of multiplication.
After the first year, the value of the antique increases by [tex]\( 10\% \) of its previous value, which is \( 0.10 \times 200 \) dollars.[/tex]
After the second year, the value of the antique increases by [tex]\( 10\% \) of its value at the end of the first year, which is \( 0.10 \times (200 + 0.10 \times 200) \) dollars.[/tex]
Generally, after \( n \) years, the value of the antique will be [tex]\( 200 \times (1 + 0.10)^n \) dollars.[/tex]
The expression given for the value of the antique 2 years after purchase is 200a , where a is a constant. This represents the value of the antique after 2 years.
Equating the expression to the value of the antique after 2 years, we have:
[tex]\[ 200a = 200 \times (1 + 0.10)^2 \][/tex]
Now, let's solve for \( a \):
[tex]\[ 200a = 200 \times (1.10)^2 \][/tex]
200a = 200 \times 1.21
200a = 242
Dividing both sides by 200:
[tex]\[ a = \frac{242}{200} \][/tex]
[tex]\[ a = 1.21 \][/tex]
Therefore, the value of [tex]\( a \) is \( 1.21 \).[/tex]
help me pass this course please somebody, anybody
Using the points (0, 0), (6, 0), and (0, 8) to form a triangle, find the length of the three sides of the triangle.
7, 8, 5
6, 8, 10
8, 6, 3
6, 8, 9
Answer:
6, 8, 10,
Step-by-step explanation:
you start at zero and go 6 units to the right, then from zero, you go 8 units up; this forms a right triangle with the smaller sides being 6 and 8. Then u can use the Pythagorean theorem to find the bigger side.
Why does math get so hard that you have an answer but you forget what your answer was because it was so so so so so so so so so so so so so so hard.Why is it hard?
Answer:
because if u write down the steps that u took to get the answer it wont be sosososososososo hard
Step-by-step explanation:
The measurements obtained for the interior dimensions of a rectangular box are 200 cm by 200 cm by 300 cm. If each of the three measurements has an error of at most 1 cm, which of the following is closest to the maximum possible difference, in cubic cm, between the actual capacity of the box and the capacity computed using these measurements?A. 100,000.B. 120,000.C. 160,000.D. 200,000.E. 320,000.
Answer:
C. 160,000
Step-by-step explanation:
Given that the measurements obtained for the interior dimensions of a rectangular box are 200 cm by 200 cm by 300 cm.
Also given that each of the three measurements has an error of at most 1 cm
COnsider the worst case where each dimension is increased by 1 cm
Then we measure the dimensions as 201 , 201 and 301
So volume would be measured as
[tex]201*201*301\\= 12160701[/tex] cubic cm
Actual volume of the box = [tex]200*200*300\\=12000000[/tex] cubic cm
Difference maximum possible = 160701 cubic cm
Out of the five options given option C is nearest to this value
So answer is
C. 160,000
Marcus is working at a local pizzeria where he makes $12.50 per hour and is also working at the university bookstore where he makes $9.50 per hour. He must make at least $300 per week to cover his expenses but cannot work more than 30 hours per week in order to attend classes. Write a system of inequalities that models this situation where p represents the hours he works at the pizzeria and b represents the hours he works at the bookstore.
Answer:
p+b[tex]\leq[/tex]30 .......(i)
12.50p+9.50b[tex]\geq[/tex]300 .......(ii)
Step-by-step explanation:
Marcus Hourly Rate at the local pizzeria = $12.50 per hour
Marcus Hourly Rate at the university bookstore = $9.50 per hour
Let the number of hours worked at the local pizzeria=p
Let the number of hours worked at the university bookstore=b
Since he cannot work more than 30 hours per week in order to attend classes, the total of the hours:
p+b[tex]\leq[/tex]30 .......(i)
If he earns $12.50 for p hours at the local pizzeria,
Income from local pizzeria=12.50p
If he earns $9.50 for b hours at the university bookstore,
Income from university bookstore=9.50b
He must make at least $300 per week, therefore his total income must not be less than $300
Total Income=Income from Pizzeria + Income from University bookstore
12.50p+9.50b[tex]\geq[/tex]300 .......(ii)
Therefore the system of inequalities that models this situation is given as:
p+b[tex]\leq[/tex]30 .......(i)
12.50p+9.50b[tex]\geq[/tex]300 .......(ii)
Triangle SRQ undergoes a rigid transformation that results in triangle VUT. 2 right triangles with identical side lengths and angle measures are shown. The second triangle is shifted up and to the right. Which statements are true regarding the transformation? Select two options. SQ corresponds to VU. AngleR corresponds to AngleU. UV corresponds to RS. AngleS corresponds to AngleT. QS corresponds to RS.
Answer:
The true statements are 2 and 3.
Step-by-step explanation:
Triangle SRQ undergoes a rigid transformation that results in triangle VUT
So, ΔSRQ ≅ ΔVUT
So, point S will map to point V, point R will map to point U and point Q will map to point T.
According to the previous, We will check the statements:
1) SQ corresponds to VU. Wrong because SQ corresponds to VT
2) ∠R corresponds to ∠U. True
3) UV corresponds to RS. True
4) ∠S corresponds to ∠T. Wrong because ∠S corresponds to ∠V
5) QS corresponds to RS. Wrong because QS corresponds to TV
So, The true statements are 2 and 3.
2) ∠R corresponds to ∠U
3) UV corresponds to RS.
Answer:
The true statements are 2 and 3.
Step-by-step explanation:
Triangle SRQ undergoes a rigid transformation that results in triangle VUT
So, ΔSRQ ≅ ΔVUT
So, point S will map to point V, point R will map to point U and point Q will map to point T.
According to the previous, We will check the statements:
1) SQ corresponds to VU. Wrong because SQ corresponds to VT
2) ∠R corresponds to ∠U. True
3) UV corresponds to RS. True
4) ∠S corresponds to ∠T. Wrong because ∠S corresponds to ∠V
5) QS corresponds to RS. Wrong because QS corresponds to TV
So, The true statements are 2 and 3.
2) ∠R corresponds to ∠U
3) UV corresponds to RS.Step-by-step explanation:
A researcher asks a sample of brothers and sisters to rate how positive their family environment was during childhood. In this study, the differences in ratings between each brother and sister pair were compared. The type of design described here is called a
Answer:
Matched sample design
Step-by-step explanation:
- A matched subject design uses separate experimental groups for each particular treatment, ( A sample of brothers and sisters - genders ).
- But relies upon matching every subject in one group with an equivalent in another. (The differences in ratings between each brother and sister pair were compared. )
- The idea behind this is that it reduces the chances of an influential variable skewing the results by negating it.
What is the length of the missing side FP? Round answer to the nearest tenth.
Answer:
Step-by-step explanation:
Considering the given triangle KFP, to determine FP, we would apply the sine rule. It is expressed as
a/SinA = b/SinB = c/SinC
Where a, b and c are the length of each side of the triangle and angle A, Angle B and angle C are the corresponding angles of the triangle. Likening it to the given triangle, the expression becomes
FP/SinK = FK/SinP = KP/SinF
Therefore
FP/Sin 49 = 66/Sin 85
Cross multiplying, it becomes
FPSin85 = 66Sin49
0.996FP = 45 × 0.7547
0.996FP = 33.9615
FP = 33.9615/0.996
FP = 34.1
The distribution of SAT scores of all college-bound seniors taking the SAT in 2014 was approximately normal with a mean of 149714971497 and standard deviation of 322322322. Let XXX represent the score of a randomly selected tester from this group. Find P(X>1800)P(X>1800)P, (, X, is greater than, 1800, ).
Answer:
P ( X > 1800) = 0.1734
Step-by-step explanation:
Given:-
- The mean, u = 1497
- The standard deviation, s.d = 322
Find:-
P(X>1800)
Solution:-
- We will denote a random variable X that follows a normal distribution for the SAT scores in 2014 with parameters mean (u) and standard deviation (s.d) as follows:
X ~ N ( 1497 , 322 )
- The following probability can be calculated by first computing the Z-score value:
P ( X < x ) = P ( X < Z )
Where,
Z = ( x - u ) / s.d
- P(X > 1800) have the corresponding Z-score value:
Z = ( 1800 - 1497 ) / 322
Z = 0.941
- Hence, using Z-table:
P ( X > 1800) = 1 - P ( Z < 0.9471 )
P ( X > 1800) = 1 - 0.8266
P ( X > 1800) = 0.1734
The probability that a randomly selected person scored above 1800 on the SAT is approximately 17.36%, after calculating the corresponding z-score and looking up the probability in the Standard Normal Distribution table.
Explanation:To find P(X>1800), we first need to calculate the z-score for an SAT score of 1800. The z-score is computed as:
z = (X - μ) / σ
Where X is the SAT score, μ is the mean, and σ is the standard deviation. Given μ = 1497 and σ = 322, we have:
z = (1800 - 1497) / 322 = 303 / 322 ≈ 0.941
Once we have the z-score, we can use the Standard Normal Distribution table to find P(Z > 0.941). We find that P(Z > 0.941) ≈ 0.1736. Thus, the probability that a randomly selected college-bound senior has an SAT score above 1800 is approximately 0.1736 or 17.36%.
Will mark the Brainliest !!!!! A team of 5 boys and 4 girls will be chosen from a group of 16 boys and 13 girls,
1. If Bob is one of the students, how likely is he to be picked?
2. If Jane is also a student, how likely is it that Jane and Bob will both be picked?
3. How likely is it that at least Jane or Bob will be picked?
4. How many different teams are possible?
Answer:
Step-by-step explanation:
The total number of permutations of boys and girls on the team are:
¹⁶P₅*¹³P₄
1.
Bob will be one of the fixed boys to be picked. Hence, actually 4 boys are to be picked from 15. The permutations of girls being picked remains the same.
Probability = Permutations with Bob as one of the boys / Total permutations
Probability = (¹⁵P₄*¹³P₄) / (¹⁶P₅*¹³P₄) = ¹⁵P₄ / ¹⁶P₅
Probability = [tex]\frac{15!}{(15-4)!}/\frac{16!}{(16-5)!}[/tex] = 15! / 16! = 1/16
2.
Now, Bob is one of the fixed boys and Jane is one of the fixed girls. Hence, actually 4 boys are to be picked from 15 and 3 girls are to be picked from 12.
Probability = Permutations with bob as one of the boys and jane as one of the girls / Total permutations
Probability = (¹⁵P₄*¹²P₃) / (¹⁶P₅*¹³P₄) = (1/16)*(1/13) = 1/208
3.
Now, the probability that at least Jane or Bob will be picked has been asked. This probability is a combination of three probabilities:
Probability = (Probability that only Bob will be picked) + (Probability that only Jane will be picked) + (Probability that both will be picked)
Probability = 1/16 + 1/13 + 1/208 = 0.123
4.
Total teams possible = ¹⁶P₅*¹³P₄ = 8994585600 teams are possible
A purchaser paid $1,539.13 for a computer system that originally cost $1,215.91. If the markup was 21% of the $1,539.13 selling price, then what is the percent markup based on cost?
Answer:
$1272.008264
Step-by-step explanation:
If the mark-up was 21%, then the final price is 121% of the original price. Simply divide $1,539.13 by 1.21 to get the original price of $1272.008264
The percent markup based on cost is calculated by finding the amount of the markup, dividing it by the original cost and multiplying by 100. In this case, the markup was 21% of the sale price, or $323.12. This correlates to a 26.56% markup based on the original cost ($1,215.91).
Explanation:The percent markup based on cost can be calculated by first determining the amount of the markup, then dividing the markup by the original cost of the item, and finally multiplying the result by 100 to express it as a percentage. According to the question, the markup is 21% of the $1,539.13 selling price. Therefore, to calculate the markup we multiply $1,539.13 by 0.21 which results in $323.12. This is the amount by which the original price was increased to get the selling price. To calculate the percent markup based on cost, we divide the markup amount ($323.12) by the original cost of the item ($1,215.91) and then multiply by 100. This gives us a percentage markup of approximately 26.56%. So, the percent markup based on cost is 26.56%.
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Dan and Carl share a 18-ounce box of cereal. By the end of the week, Dan has eaten 1 6 of the box, and Carl has eaten 2 3 of the box of cereal. How many ounces are left in the box?
Answer: 3 ounces of cereals was left in the box.
Step-by-step explanation:
Dan and Carl share a 18-ounce box of cereal. By the end of the week, Dan has eaten 1/6 of the box. This means that the amount of cereals that Dan ate is
1/6 × 18 = 3 ounces of cereals
Also, by the end of the week, Carl has eaten 2/3 of the box of cereal. This means that the amount of cereals that Carl ate is
2/3 × 18 = 12 ounces of cereals
The total amount of cereals that they ate is 12 + 3 = 15 ounces
Therefore, the amount of cereals left in the box is
18 - 15 = 3 ounces
Mr. Daniels is building a clubhouse for his children. He has decided that the floor will be square with an area of 64 square feet. Write this number using a power greater than 1 and a lesser base.
Step-by-step explanation:
Here, given:
The area of the square floor = 64 sq. ft
Now, as given the floor is in the shape of a square.
Let us assume the side length of the floor = k ft
Area of a square = (Side) x (Side)
= k x k = 64 sq ft
⇒ k ² = 64 = (8) ²
⇒ k = 8 ft
Hence, 64 sq ft = ( 8 ft) ²
Here, base = 8 and power = 2, which is greater than 1.
Which of the following is the solution to the quadratic equation x2 - 10x + 24 = 0?
x = -4, 6
x = 4, -6
x = 4, 6
x= -4, -6
Answer: the third option is the correct answer.
Step-by-step explanation:
The given quadratic equation is expressed as
x² - 10x + 24 = 0
We would apply the method of factorization by finding two numbers such that their sum or difference is -10x and their product is 24x^2. The two numbers are - 6x and - 4x. Therefore,
x² - 6x - 4x + 24 = 0
x(x - 6) - 4(x - 6) = 0
(x - 6)(x - 4) = 0
Therefore, the solutions to the equation are
x = 4 or x = 6
I am confused about the wording on this problem and also when I used the Pythagorean theorem it came out as wrong.
Answer:
Step-by-step explanation: they are asking “What is x + 3 + y”. So use Pythagorean’s theorem to get x (it should be 4) and then find y and I think u get square root of 13, then add 4 + 3 + square root 13
Sanjay bought 12 granola bars,which was 4 times as many granola bars as Lena bought.which equation shows the number of granola bars,b,that Lena bought?
Of the two production methods, a company wants to identify the method with the smaller population mean completion time. One sample of workers is selected and each worker first uses one method and then uses the other method. The sampling procedure being used to collect completion time data is based on
a.
matched samples.
b.
worker samples.
c.
pooled samples.
d.
independent samples.
Answer:
a) matched samples.
Step-by-step explanation:
Matched samples (also known as matched pairs, paired samples or dependent samples) are those samples which can be matched in pairs for one set of item and the sample data are not independent of each other. The pairs don’t have to be different people, it could be the same individuals at different time or tested on different activities for example
sampling the blood pressures of the same people before and after they receive a dosethe same people being measured when the group is given two different tests at different timesThe completion time data collection method used by the company, which involves each worker using both production methods, is based on matched samples.
Explanation:The current method used to collect completion time data involves each worker being required to use both production methods. This method is known as matched samples. It involves using the same subjects in two different conditions to measure the difference in outcomes. In this case, the matched samples design is being used to compare the completion times of the workers using both production methods. The design is a test of dependent means, classified as a matched pairs design. This design is useful in situations where the same subject is being tested in two different conditions. The matched pairs design allows for a more accurate comparison of the two conditions, as it eliminates the variability between different subjects.
PLEASE HELP
Find the range of the function f(n) = 5n −4 for the domain {−3, 0, 4}. List the values in order from least to greatest and use a comma to separate each value..
range: { }
The range of the function is [tex]\{-19,-4,16\}[/tex]
Explanation:
The function is [tex]f(n)=5n-4[/tex]
The domain of the function is [tex]\{-3,0,4\}[/tex]
We need to find the range of the function.
The range can be determined by substituting the values of domain in the function.
Thus, the range of the function when the domain is -3 is given by
[tex]f(-3)=5(-3)-4[/tex]
[tex]=-15-4[/tex]
[tex]=-19[/tex]
Thus, the range is -19 when [tex]n=-3[/tex]
The range of the function when the domain is 0 is given by
[tex]f(0)=5(0)-4[/tex]
[tex]=0-4[/tex]
[tex]=-4[/tex]
Thus, the range is -4 when [tex]n=0[/tex]
The range of the function when the domain is 4 is given by
[tex]f(4)=5(4)-4[/tex]
[tex]=20-4[/tex]
[tex]=16[/tex]
Thus, the range is 16 when [tex]n=4[/tex]
Thus, the range of the function is [tex]\{-19,-4,16\}[/tex] when their corresponding domain is [tex]\{-3,0,4\}[/tex]
Arranging the range in order from least to greatest is given by
[tex]\{-19,-4,16\}[/tex]
Hence, the range of the function is [tex]\{-19,-4,16\}[/tex]
Lauren coordinates a construction projects for a cement company. A government project requires constructing two rectangular concrete slabs of dimensions 24× 24× 1 feet. Additionally, the company sends a 20% surplus of concrete to ensure the job can be completed. If a cement truck can carry a maximum of 8 cubic yards of cement, what's the fewest number of trucks that Lauren should send? A)1 B)2 C)3 D)4 E)5lar
The fewest number of trucks Lauren should send is D) 4 trucks.
Step-by-step explanation:
Step 1:
The rectangular slab's dimensions are [tex]24 \times 24 \times 1[/tex] feet. Each truck can carry 8 cubic yards of cement.
First, we need to determine the volume of the slabs in yards. 1 foot = 0.333 yards. So 24 feet = [tex]24\times 0.3333[/tex] = 8 yards.
The volume of the slab = [tex]8 \times 8 \times 0.3333[/tex] = 21.3312 cubic yards.
Step 2:
The company sends a surplus of 20% to make sure the job can be completed. So the total cement sent is the required volume and an extra 20%.
The total cement sent = The required cement + 20%.
= 21.3312 + 20% = 25.597 cubic yards.
Step 3:
So to find the number of trucks needed, we divide the cement sent by the load each truck can carry. Each truck can carry 8 cubic yards of cement. So
The number of trucks needed = [tex]\frac{therequiredload}{load per truck} = \frac{25.597}{8} = 3.199625.[/tex]
If 3.199 trucks are needed, it means 4 trucks are needed which is option D.
A batch of 479 containers for frozen orange juice contains 3 that are defective. Two are selected, at random, without replacement from the batch. a) What is the probability that the second one selected is defective given that the first one was defective? Round your answer to five decimal places (e.g. 98.76543).
Answer:
0.00418
Step-by-step explanation:
The probability of the first one being defected is 3/479, as we have 3 defectives containers in a total of 479 containers.
If the first one is defected and removed, now we have 478 containers, with 2 being defective.
So the probability of the second one being picked being defective, given that the first one was defective, is 2/478 = 1/239 = 0.00418
________________shapes are radical alterations of visible reality simplifications, exaggerations, or transmutation that sometimes bear little resemblance to the original entities from which they were derived.1. geomatric2. organic3. contour4. abstract5. amorphous
Answer:4. Abstract.
Step-by-step explanation: Abstraction is a term used to describe a departure from reality in the expressions of image in art.
This kind of departure from accurate and actual representation can be slight,can be partial, or complete or total.
Abstract shapes are shapes used in depicting the virtual images of certain objects or people,it usually does not actually display reality or, it only shows the radical altering of the visual realities of different things.
Using the formula A=P(1+r)^t calculate the value of an initial investment of $4,500 after 10 years at 4% interest.
The solution is [tex]\$ 6661[/tex]
Explanation:
The initial investment is $4500
The time taken is 10 years.
The rate of interest is 4%
We shall determine the value of A using the formula [tex]A=P(1+r)^t[/tex]
where P is the initial investment,
r is the rate of interest and
t is time
Let us substitute the values [tex]P=4500[/tex] , [tex]t=10[/tex] and [tex]r=4 \%[/tex] in the formula [tex]A=P(1+r)^t[/tex]
Thus, we have,
[tex]A=4500(1+0.04)^{10}[/tex]
Adding the values within the bracket, we have,
[tex]A=4500(1.04)^{10}[/tex]
Simplifying, we get,
[tex]A=4500(1.4802)[/tex]
Multiplying, we have,
[tex]A=6661[/tex]
Thus, the value is $6661
Identify the sample chosen for the study. The number of hours a group of 12 children in Mrs. Smith's kindergarten class sleep in a day. Answer2 Points The 12 children selected in Mrs. Smith's kindergarten class. All children in Mrs. Smith's kindergarten class. The number of hours children sleep.
The sample in the study refers to the 12 children in Mrs. Smith's kindergarten class. A sample is a subset of people selected from a larger group for study purposes. In this research, the data being analyzed only pertains to these selected individuals.
Explanation:In this study, the sample chosen is the group of 12 children in Mrs. Smith's kindergarten class. This is because the data being collected and scrutinized is related to these particular individuals. In this context, 'sample' refers to the subset of people chosen from a larger group (the population) for research or study purposes. It is the set of individuals on which the study or experiment is conducted. In this case, the larger population could be considered all children in kindergarten, but the sample for the study is specifically the 12 children in Mrs. Smith's class, as the study only includes their sleep patterns.
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Use the given functions f and g to find f + g, f − g, fg, and f g . State the domain of each. (Enter your answer for the domain in interval notation.) f(x) = 3x + 6, g(x) = x + 2.
Find the domain of each problem.
f + g = Domain=
f-g= Domain=
(f)(g)= Domain=
f/g=Domain=
2.) Find (g ○ f)(x) and (f ○ g)(x) for the given functions f and g.
f(x) = 3/(x+5), g(x) = 3x − 6
(g ○ f)(x) =
(f ○ g)(x) =
3.) Use the method of completing the square to find the standard form of the quadratic function.
f(x) = x2 − 8x + 2
y =
4.) Use the vertex formula to determine the vertex of the graph of the function.
f(x) = x2 − 14x Write the function in standard form.
f(x) =
5.) Use the vertex formula to determine the vertex of the graph of the function.
f(x) = 3x2 − 10x + 1 Write the function in standard form.
f(x) =
Answer: The answers are stated below
Step-by-step explanation: Attached below is the explaination of the solution.
f + g =4x + 8 Domain - (-∞, ∞)
f - g = 2x + 4 Domain - (-∞, ∞)
(f)(g) = 3x² + 12x + 12 Domain - (-∞, ∞)
f/g = [tex]\frac{3x + 6}{x + 2}[/tex] Domain - [tex](-\infty, -2) \cup (-2, \infty)[/tex]
To solve the problems involving the functions f and g, we start by defining the functions:
Given functions:
[tex]f(x) = 3x + 6[/tex]
[tex]g(x) = x + 2[/tex]
1. Finding f + g, f - g, fg, and f/g:
f + g:
[tex]f + g = (3x + 6) + (x + 2) = 4x + 8[/tex]
Domain: All real numbers, [tex](-\infty, \infty)[/tex]
f - g:
[tex]f - g = (3x + 6) - (x + 2) = 2x + 4[/tex]
Domain: All real numbers, [tex](-\infty, \infty)[/tex]
fg:
[tex]fg = (3x + 6)(x + 2) = 3x^2 + 12x + 12[/tex]
Domain: All real numbers, [tex](-\infty, \infty)[/tex]
f/g:
[tex]f/g = \frac{3x + 6}{x + 2}[/tex]
However, g(x) cannot be zero: [tex]g(x) = 0[/tex] for [tex]x = -2[/tex].
Domain: All real numbers except [tex]x = -2[/tex] , which is [tex](-\infty, -2) \cup (-2, \infty)[/tex]
2. Finding (g ◦ f)(x) and (f ◦ g)(x):
(g ◦ f)(x):
[tex]g(f(x)) = g(3x + 6) = (3x + 6) + 2 = 3x + 8[/tex]
(f ◦ g)(x):
[tex]f(g(x)) = f(x + 2) = 3(x + 2) + 6 = 3x + 12[/tex]
3. Completing the square for f(x) = x² - 8x + 2:
First, take the coefficient of [tex]-8[/tex], halve it to get [tex]-4[/tex], and square it to get [tex]16[/tex].
Therefore:
[tex]f(x) = (x^{2} - 8x + 16) - 16 + 2 = (x - 4)^{2} - 14[/tex]
Now in standard form:
[tex]y = (x - 4)² - 14[/tex]
4. Vertex formula for f(x) = x² - 14x:
Vertex formula: [tex]x = -\frac{b}{2a}[/tex] where [tex]a=1, b=-14[/tex].
Therefore:
[tex]x = -\frac{-14}{2(1)} = 7[/tex]
Substituting [tex]x=7[/tex] back to find y:
[tex]f(7) = 7^{2} - 14(7) = 49 - 98 = -49[/tex]
Standard form:
[tex]f(x) = (x - 7)^{2} - 49[/tex]
5. Vertex for f(x) = 3x² - 10x + 1:
Vertex x-coordinate:
[tex]x = -\frac{-10}{2(3)} = \frac{10}{6} = \frac{5}{3}[/tex]
Substitute [tex]x=\frac{5}{3}[/tex] back to find y:
[tex]f(\frac{5}{3}) = 3(\frac{5}{3})^{2} - 10(\frac{5}{3}) + 1[/tex]
Calculate the value:
[tex]3(\frac{25}{9}) - \frac{50}{3} + 1 = \frac{75}{9} - \frac{150}{9} + \frac{9}{9} = -\frac{66}{9} = -\frac{22}{3}[/tex]
The standard form becomes:
[tex]f(x) = 3(x - \frac{5}{3})^{2} - \frac{22}{3}[/tex]
A jar contains six blue marbles and five red marbles. Suppose you choose a marble at random, and do not replace it. Then you choose a second marble. Find the probability of the following event. Both of the selected marbles are red.
The jar has 6+5=11 marbles.
We have to find the probability of the following event:
1.We pick a marble from a jar that has 11 marbles in total, 5 of them are red
2.We pick a marble from a jar that has now 10 marbles in total, 4 of them are red (because in the previous step we picked a red marble and did not put it back in the jar)
The probability of the first event is:
[tex]P_1=\frac{5}{11}[/tex]
The probability of the second event is:
[tex]P_2=\frac{4}{10}=\frac{2}{5}[/tex]
The probability of the both events to happen is:
[tex]P=P_1\cdot P_2=\frac{5}{11}\cdot \frac{2}{5}=\frac{2}{11}=0.1818[/tex]
student enrollment at a local school is concerning the community because the number of students has dropped to 504 which is a 20% decrease from the previous year. what was the student enrollment the previous year?
Answer:
The answer is 630 students
Step-by-step explanation:
For this case, we have 504 students and the student enrollment has decrease 20%, which mean 504 is the 80% of the total student enrollment of the previous year.
[tex]100 - 20 = 80[/tex]%
This is a direct proportion problem. As shown bellow:
504 -> 80%
x -> 100%
For solving this we use the Mathematical Rule of Three, a method of having three numbers to help calculate the unknown.
b -> c
x -> a
The algorithm for rule of three is the following:
[tex]x = \frac{a * b}{c} \\\\x= \frac{504*100}{80} =\frac{50400}{80}=630\\ \\x=630[/tex]