a rectangle has its base on x-axis and its upper two vertices on the parabola y= 12 - x2. What is the largest area of rectangle?
A rectangle's greatest area is 32 square units.
Given:
A parabola's equation is y = 12 - x2, which is an even function
As a result, its rectangular shape is also even at the origin.
We are aware that the area of a rectangle is equal to its length times its width.
Here,
width = y, length = 2x
A = 2x, where (12 - [tex]x^{2}[/tex])
⇒ A = 24x - 2[tex]x^{3}[/tex]
Consider the A derivative with regard to x.
⇒ A' = 24 - 6 [tex]x^{2}[/tex]
When A' = 0, the area is the largest.
⇒ 24 - 6x2 = 0
⇒ [tex]x^{2}[/tex] = 4
⇒ x = 2
Replace x with its value in y = 12 - [tex]x^{2}[/tex].
⇒ y = 12 - 4
⇒ y = 8
Area = 2(2)(8) = 32
Hence, a rectangle's greatest area is 32 square units.
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A wall is supported by a brace 10 feet long, as shown in the diagram. If one end of the brace is placed 6 feet from the base of the wall, how many feet up the wall does the brace reach?
The height of the wall the brace reached is 8 feet.
What is the Pythagorean theorem?Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
We have,
Height of the wall the brace reached = x
Now,
10² = 6² + x²
100 = 36 + x²
x² = 100 - 36
x² = 64
x = 8
Thus,
The brace reached 8 feet of the wall.
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Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
an= 1-(0.6)^n lim n tends to infinite an=n^3/7n^3+1lim n tends to infinite an=an= 7+2n^2/n+7n^2 lim n tends to infinite an=n^3/5n+8 lim n tends to infinite an= an=e^6/n lim n tends to infinite an= Square root n+5/25n+5 lim n tends to infinite an=n^2 /square root n^3+9n lim n tends to infinite an=
The given series diverges.
Given:
aₙ = n²/√n³+7n
using series divergence test
If limₙ→∞ aₙ ≠0, then ∑ aₙ diverges .
limₙ→∞ (n²/√n³+7n) = limₙ→∞ (n²/√n³(1+7/n²)
= limₙ→∞ (n²/n³/²√1+7/n²)
= limₙ→∞ (√n/√1+7/n²)
= (∞/√1+0)
= ∞
Therefore the series diverges.
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write the sum as the logarithm of a single expression. assume that variables represent positive numbers.
The sum can be written in logarithms as 6z log 6
What is logarithm ?
The word "logarithm" comes from the combination of the Greek words "logos," which means "principle or concept," and "arithmos," which means "number." A logarithm is the power that must be increased to obtain a given number. It is shown by a number's log.
Use of property
log (ab) = log a + log b
log a^b = b log a
Given that,
log 6^8 + log 6^z
=log 6^8 + log 6^z
=log (6^8*6^z)
=log (6^6z)
=6z log 6
Therefore, the sum can be written in logarithms as 6z log 6
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when you graduated from high school you had a choice between coming to jmu and working at walmart where you would earn $37,870 per year. suppose tuition at jmu is $20,000 per year, textbook expenses are $2,699 per year, and room and board expenses are $15,000 per year. assume that when you graduate you will get a job that pays you $75,000 per year.
Therefore , the best choice is you to join jmu and after that you will able to earn the same amount you spend on the graduation in 2 year.
What is equation ?Equation: A statement that two variable or integer expressions are equal. In essence, equations are questions, and the motivation for the development of mathematics has been the systematic search for the answers to these questions.
Here,
Given:
You would earn $37,870 per year at walmart
Tuition at jmu is $20,000 per year, textbook expenses are $2,699 per year, and room and board expenses are $15,000 per year.
and you will get a $75,000 per year after graduation.
So the total expense is = 20,000 + 15000 + 2699 =37,699
Therefore , the total expense at jmu = $37,699
and total 4 year graduation course fees = 37,699 * 4=1,50,796
Thus, total fees you have to pay = $1,50,796
and after this you will get a job of $75,000 per year.
you will able to earn the same amount you spend on the graduation in 2 year.
Therefore , the best choice is you to join jmu and after that you will able to earn the same amount you spend on the graduation in 2 year.
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Rick is looking for an expression equal to 64 + 88. In that expression, the greatest common factor of the two numbers in the parentheses must be 1. Which expression should he choose?
The expression that Rick choose is 99 + 53.
What is parentheses?
The well-known () symbols, often known as "round brackets," are used in pairs to group things together or designate the order of operations in an equation. You will frequently need to employ brackets in math when forming or resolving equations. They support number grouping and operation definition.
Given expression is 64 + 88
The simplest form of the expression 64 + 88 is 152.
152 = 99 + 53.
152 = 151 + 1
152 = 149 + 3
152 = 139 + 13
152 = 137 + 15
152 = 131 + 21
152 = 127 + 25
152 = 113 + 39
152 = 109 + 43
152 = 107 + 45
152 = 103 + 49
152 = 101 + 51
etc.
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Given the functions defined by f(x)=3x2−2 and g(x)=x2−2, calculate the following: (f∘g)(−2)
Answer:
10
Step-by-step explanation:
f(x) = 3x^2 - 2
g(x) = x^2 - 2
fog(x) = 3(x^2-2)^2 - 2 = 3x^4 + 12 - 12x^2 - 2 = 3x^4 - 12x^2 + 10
fog(-2) = 3(-2)^4 -12(-2)^2 + 10
fog(-2) = 48-48+10 = 10
The owner of a football team claims that the average attendance for games is 67,800, he therefore justified in moving the team to a city with a larger stadium.
express the null hypothesis in symbolic form- use the correct symbol for indicated parameter
The null hypothesis is H0: μ ≤ 68,800 for the given case of The owner of a football team claiming that the average attendance for games is 67,800.
Since we know that a null hypothesis (H0) shows that no significant variation exists between variables in other words it can be said that a single variable is much not different than its mean. since from the above case we were given the information that The owner of a football team claims that the average attendance for games is 67,800. So, the null hypothesis the that the average attendance at games is less than or equal to 68,800.
H0: μ ≤ 68,800
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22 is what percent of 145.1?
Answer:
15.161
Step-by-step explanation:
Answer:
31.922
Step-by-step explanation:
Vinnie's Mini-Golf hosts children's birthday parties. Parents can reserve the party room for kids to eat pizza and cake after playing golf. The total price for a party with c children is 8.5c + 35 dollars. Mrs. Kim scheduled her son's party for 8 children.
What price will Mrs. Kim pay for her son's party?
Mrs Kim will pay 103 dollars for her son's party.
What is a substitution method?
The algebraic approach to solving simultaneous linear equations is known as the substitution method. The value of one variable from one equation is substituted in the second equation in this procedure, as the name implies.
Given, the total price for the party with c children = 8.5c + 35 dollars
For 8 children, c = 8 then,
Total price = 8.5*8 + 35 = 103 dollars
Hence, Mrs Kim will pay 103 dollars for her son's party.
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Three CDs cost 52.17 If each CD costs the same, how much did one CD cost?
If the cost of 3 CDs is 52.17 then the cost of 1 CD is 17.723.
What do you mean by cost?
Cost is the sum of money needed to produce, market, and sell goods and services or to purchase assets. A cost is added to expense when something is sold or consumed.
Let the cost of 1 CD be x
Then the cost of three CDs will be equal to 3x,
And the cost of three CDs is equal to 53.17.
Equating the costs,
3x = 53.17
x = 53.17/3
x = 17.723
Therefore, the cost of 1 CD is 17.723.
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let g be a group and h be a subgroup such that the index is finite. aha-1 prove that is a finite set
The assertion that there exists a normal subgroup N of G such that N is of finite index in G and [N ⊂ H] is true and can be proved if "G" is a group and "H" is a subgroup of finite index.
Given,
The statement, "G" is a group and "H" is a subgroup of finite index.
We have to prove that there exists a normal subgroup N of G such that N is of finite index in G and [N ⊂ H].
Group: A group is a finite or infinite collection of elements and a binary operation (referred to as the group operation) in discrete mathematics that together meet the four essential qualities of closure, associativity, identity, and inverse.
Subgroup: If a subgroup "H" also satisfies the four essential criteria of closure, associativity, identity element, and inverse, it is a subset of a group, let's say "G" [denoted by (H ≤ G)].
Here,
By left multiplication, the group G affects the collection of left co-sets (G/H).
Hence it induces the permutation representation [ρ:G→Sₙ], where
[n = |G : H|].
(A permutation representation is a group homomorphism, it should be noted.)
Now, let (N = kerρ) be the kernel of the homomorphism (ρ). Then [N◃G].
The quotient group G/N is isomorphic to a subgroup of Sn according to the first isomorphism theorem.
In particular, G/N is a finite group, hence the index [G:N] is finite.
Finally, we show that [N ⊂ H].
For any (x ∈ N) = [ker (ρ)], we have [x(gH) = gH] for any [g ∈ G].
In particular we have (xH = H),
Hence (x ∈ H).
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On the first day of vacation, you read one-quarter of a novel. On the second day, you read half of the remaining pages. On the third day, you read the last 126 pages of the novel.
(a) How many pages does the novel have?
(b) How many pages did you ready by the end of the second day?
Answer:
A. 336
B. 210
Step-by-step explanation:
To solve, we need to first multiply 126 by 2, which gives us 252. This is how many pages we had left after the first day. Then, since we read one-quarter on the first day, 252 is three-quarters, or 75% of the book. To find what number 252 is 75% of, we multiply both sides by 100 and divide by 75. This gives us 336, and the total amount. Next, we need to find how many pages we read on the first day. To do this, we find 25% of 336, which is 84. Then add 84 (how much we read on the first day) to 126 (how much we read on the second day), which gives us 210.
Answer:
(a) 336 pages
(b) 210 pages
=======================================================
Work Shown:
Part (a)
x = number of pages
x/4 = amount read on day 1
3x/4 = amount left to read after day 1 is over
(1/2)*(3x/4) = 3x/8 = amount read on day 2
(x/4) + (3x/8) = (2x/8)+(3x/8) = 5x/8 = amount read on days 1 and 2
x - (5x/8) = 8x/8 - (5x/8) = 3x/8 = amount left after day 2 is over
3x/8 = 126
3x = 8*126
3x = 1008
x = 1008/3
x = 336
There are 336 pages total
-------------------
Check:
336 pages total.
You read one-quarter of that on day one, so you read 336/4 = 84 pages on the first day.
There are 336-84 = 252 pages left (note that 3/4 * 336 = 252)
Then on day two, you read half of what's left, so you read 252/2 = 126 pages. The other half (126) is the last 126 pages mentioned for day 3, so the answer to part (a) is confirmed.
-------------------
There are at least three ways we can solve part (b)
Add up the number of pages read on days 1 and 2 to get 84+126 = 210 (refer to the "check" section above).In the first section we found that 5x/8 represents how many pages have been read by the end of day 2. Plug in x = 336 to find that 5*336/8 = 210Since we know there are 336 pages total, and 126 is what needs to be read on the third day to finish up the book, this must mean 336-126 = 210 pages have been read so far for days 1 and 2 combined.Either way, 210 pages have been read by the end of the 2nd day.
The diagram shows two different buttons and their weights.
8 grams
A
12 grams
B
The buttons are sold in bags.
Each bag contains 144 grams of buttons.
a) How many of button A are needed to fill one bag?
b) How many of button B are needed to fill one bag?
A box of buttons contains the same number of each type of button.
It contains 220g of buttons.
c)
How many of each type of button are in the box?
The number of buttons A will be equal to 11. And the number of buttons B will be equal to 11.
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
Given that container of buttons contains a similar number of each kind of button it contains 220g of buttons.
The weight of the button is given as follows;
Button A=8 grams
Button B = 12 Grams
Consider x be the number of each button. Then the equation is;
8x + 12x = 220
20x = 220
x = 11
The number of buttons A is 11. And the number of buttons B is 11.
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The probability of event A is 0.28, and the
probability of event B is 0.74. What is the
probability of both occurring if A and B are
independent events?
The joint probability for events A and B is:
P(A and B) = 0.2072
How to find the probability of two events happening at the same time?We know that if two events A and B, each with correspondent probabilities
P(A) and P(B) are independet, then the joint probability (this is the probability that both events happen at the same time) is given by the product between the individual probabilities.
In this particular case we know that:
P(A) = 0.28
P(B) = 0.74
Then the joint probability of A and B is:
P(A and B) = P(A)*P(B) =0.28*0.74
P(A and B) = 0.2072
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A credit card account had a balance of $939 on May 5. A purchase of $216 was made on May 17, and a payment of $100 was made on May 20. The interest on the average daily balance is 1.4%
per month.
Find the average daily balance on the June 5 bill.
The average daily balance on the June 5 bill is $1,037.32, including the finance charge.
What is the average daily balance?The average daily balance is the total balance divided by the number of days in the billing cycle.
May has a 31-day billing cycle.
The average daily balance determines the finance charge, which is a product of the monthly interest rate (MPR) and the ending average daily balance.
Daily Balance
May 5 Credit card balance $939 $11,260 ($939 x 12)
May 17 Purchase 216 $4,620 ($1,155 x 4)
May 20 Payment -100 $15,825 ($1,055 x 15)
Total daily balances $31,705
Average daily balance $1,023 ($31,705 ÷ 31)
Interest rate per month = 1.4%
Finance charge for the month = $14.32 ($1,023 x 1.4%)
June 5 balance = $1,037.32 ($1,023 + $14.32)
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Q2 Convert the following readings to the unit indicated:
a) 0.05 m=______cm
b)50 gm= ______kg
c)25 mm=______km
a) 1 m = 100 cm.
0.05 m = 0.05×100
= 5 cm
b) 1000 gm = 1kg
50 = 50/1000
= 0.05 kg
c) 1000000 mm = 1 km
25 = 25/1000000
= 0.0000025 km
Hope understands.
when graphing x is greater than or equal to 1, where does the shading occur
The price of a double-dip ice cream cone is increasing at the rate of 18e0.08 cents per year, wheret is measured in years and t = 0 corresponds to 2000. Find the total change in price between t years 2000 and 2007. (Round your answer two decimal places.)
Between the years 2000 and 2007, there was a total price change of 24.79.
What is unitary method ?Area is the total amount of space that an object's shape or a flat (2-D) surface occupy.
Create a square on paper by using a pencil. Tw dimensions make it up. A shape's area on paper is the space it takes up.
Imagine that your square is made up of smaller unit squares.
The area of a figure is equal to the number of unit squares required to completely cover the surface area of a particular 2-D shape. Square cms, square feet, square inches, square meters, etc. are a few common units for measuring area.
To get the area of the square figures presented below, draw unit squares with 1-centimeter sides. Therefore, the shape will be measured.
Hence, Between the years 2000 and 2007, there was a total price change of 24.79.
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Solve the equation 2y+5=y-7
Answer: -12
Step-by-step explanation:
Graph triangle ABC with vertices
A( − 4, − 3), B(0, – 2),
and C(2, 2) and and its
image after a 90° clockwise
rotation about point C.
Check the picture below.
NEED HELP WITH THESES PLEASE
Answer:
angle congruence just watch some videos about it explaining that
it takes a boat 3 hours to travel 48 miles downstream and 4 hours to travel 40 miles upstream. What is the speed of the boat in still water?
The speed of the boat in still water is 3 miles per hour
Given that,
A boat travels 40 miles upstream in 4 hours and 48 miles downstream in 3 hours.
To find : The speed of the boat in still water
We use the distance formula. D = r * t
48/3 = 16 mph | Trip 1 |
40/4= 10mph | Trip 2 |
16-10 = 6 mph difference traveling upstream or downstream. Thus the speed of the current is 3 miles per hour.
Therefore, The speed of the boat in still water is 3 miles per hour.
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Find the total income produced by a continuous income stream in the first 15 years if the rate of flow is given by the following function, where t is time in years.
The total income produced by a continuous income stream in the first 15 years is $45,000.
What is continuous income stream?
We can see the money flowing into the account continuously rather than in a lot of discrete payments if the deposits are made often enough and the intervals between deposits are short compared to the account's overall lifespan. This scenario will be referred to as a constant revenue stream.
Given function is
f(t) = 3000
Now integrate both side of the function and the lower limit is t = 0 and upper limit is t = 15.
[tex]= \int^{15}_{0}3000 dt[/tex]
[tex]=3000[t]_0^{15}[/tex]
= 3000 [ 15 - 0]
=45000
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Determine which is the graph of the given function. f(x)= {2x+5 if x <0},f(x) = {5 if 0 <=x<=10}, f(x) = {2x-10 if x >10}
The graph of the piece-wise function in this problem is given by the image shown at the end of the answer.
How to graph the piece-wise function?A piece-wise function is a function that has different definitions, based on the input values (values of x) of the function.
The function in this problem has three definitions, given as follows:
Left of x = 0.Between x = 0 and x = 10. (closed).Right of x = 10.The behavior for each definition is given as follows:
Left of x = 0: increasing with a slope of 2 and intercept of 5.Between x = 0 and x = 10: constant at y = 5.Right of x = 10: increasing with slope of 2 and intercept of -10.Then, considering these three definitions, the graph of the function is presented at the end of the answer.
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The accompanying data, or the length in centimeters and girths (in centimeters) of 12 harbor seals. Find the equation of the regression line. Then construct a scatterplot of the data and draw the regression line. Then use the regression equation to predict the value of Y for each of the given X values, if meaningful, if the X value is not meaningful to predict the value of y, explain why not.
The equation of the regression line is defined as follows:
y = 0.71x + 8.27.
The estimates are given as follows:
a) x = 140 cm: 107.67 cm.
b) x = 172 cm: 130.39 cm.
c) x = 164 cm: 124.71 cm.
d) x = 158 cm: 120.45 cm.
How to obtain the linear regression equation?The linear regression equation is obtained inserting the points of the data-set into a linear regression calculator.
From the table given in this problem, the points are given as follows:
(138, 106), (168, 130), (151, 116), (144, 105), (160, 124), (159, 119), (125, 103), (137, 103), (154, 120), (147, 110), (146, 108), (147, 110).
Inserting these points into a calculator, the equation is given as follows:
y = 0.71x + 8.27.
Then the estimates are obtained replacing x into the equation and obtaining the value of y, for example, for x = 140 cm:
y = 0.71(140) + 8.27 = 107.67 cm.
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A projectile is fired straight up from ground level. After t seconds, its height above the ground is s feet, where s= - 161? + 112t. For what time period is the projectile at least 160 feet above the ground?
The time period the projectile at least 160 feet above the ground is 2 sec to 5sec.
What is time period?
The duration of time that a behavior lasts or a condition persists. Depending on the type of activity or situation under consideration, it may be measured in seconds or in millions of years.
Given function that represents height of the projectile above the ground is s feet is s = - 16t² + 112t.
To find the time period is the projectile at least 160 feet above the ground, make an inequality:
- 16t² + 112t ≥ 160
Subtract 160 from both sides:
- 16t² + 112t - 160 ≥ 0
Divide both sides by 16:
- t² + 7t - 10 ≥ 0
- t² + 2t + 5t - 10 ≥ 0
-t(t -2) + 5(t - 2) ≥ 0
(t - 2)( -t+ 5) ≥ 0
Either, t - 2 ≥ 0 → t ≥ 2
or, -t + 5 ≥ 0 → t ≤ 5
2 ≤ t ≤ 5
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Burton Bush wants to retire in Arizona when he is 80 years of age. Burton, who is now 55, believes he will need $400,000 to retire comfortably. To date, he has set aside no retirement money. If he gets an interest rate of 6% compounded annually, what will he have to invest today (to the nearest dollar) to reach his goal?
Multiple Choice
$160,000
$91,243
$96,500
$93,199
Burton Bush has to invest $93,199 to get $400,000 at the end. Hence, option D is correct.
What is compound interest?Interest that is added to a loan and deposit sum is known as compound interest. In our everyday lives, it is the notion that is employed the most frequently. Compound interest is calculated as a sum using the interest and principal accrued over time. Compound interest versus simple interest differ primarily in this way.
As per the given information in the question,
Rate, r = 6%
Number of years, n = 25
Present value, PV = ?
Final value, FV = $400,000
Use the equation of compound interest:
[tex]PV = FV (1 + r)^-^n[/tex]
Put values from above in the given equation,
PV = 400,000 (1 + 0.06)⁻²⁵
PV = 400,000 × 0.23
PV = $93,199.
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Find the exact value of sec 30^0 - sin 45^0 + sin 90^0
The exact value of sec 30°- sin 45° + sin 90° is 1.45.
What is the trigonometric ratio?
The trigonometric functions in mathematics are real functions that connect the right-angled triangle's angle to the ratios of its two side lengths. They are extensively employed in all fields of geometry-related study, including geodesy, solid mechanics, celestial mechanics, and many others.
Here, we have
sec 30°- sin 45° + sin 90°
we have to find the exact value of the given expression,
sec 30° = 2/√3
sin 45° = 1/√2
sin 90° = 1
Put all values in the given expression, and we get
sec 30°- sin 45° + sin 90° = 2/√3 - 1/√2 + 1
sec 30°- sin 45° + sin 90° = 1.15 - 0.70 + 1
sec 30°- sin 45° + sin 90° = 1.45
Hence, the exact value of sec 30°- sin 45° + sin 90° is 1.45.
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Factor: 5x2 – 15x – 20.
Answer:
=5x2-(20-5)x-20
=5x2-20x+5x-20
=5x(x-4)+5(x-4)
=(x-4)(5x+5)
Answer:-5(2+3x)
Step-by-step explanation:
