The explicit formula is given by the equation a(n) = 50n² + 25n + 25 and the recursive formula is given by a(n) = a(n-1) + 200, with a(1) = 100.
What is the finite differences method?A approach for identifying patterns in numerical sequences is the method of finite differences. It entails first identifying the differences between each word in the sequence, then repeating the procedure with the sequence of differences that results. If a pattern forms where the gaps between the gaps are always the same, the initial sequence exhibits a polynomial pattern. Both the recursive formula, which enables us to compute any term by utilising the previous term in the series, and the explicit formula for the sequence, which allows us to calculate any term in the sequence directly, can be found using the procedure.
Given the views on different days we have:
First finite difference: 300-100 = 200
Second finite difference: 1200-300 = 900
We observe that second finite difference is constant and hence follows a quadratic pattern.
Using the explicit formula we have:
a(n) = 50n² + 25n + 25
where, a(n) is the views on day n.
Using recursive formula:
a(n) = a(n-1) + 200, with a(1) = 100
Hence, the explicit formula is given by the equation a(n) = 50n² + 25n + 25 and the recursive formula is given by a(n) = a(n-1) + 200, with a(1) = 100.
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Find and simplify the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the following function. \[ \begin{array}{c} f(x)=6 \\ \frac{6(x+h-1)}{h} \\ x \end{array} \]
The simplified difference quotient for the function [tex]\(f(x)=6\) is \(\frac{x+h-1}{h/6}\).[/tex]
The difference quotient for the function \(f(x)=6\) is given by the formula \( \[tex]frac{f(x+h)-f(x)}{h}\)[/tex]. To simplify this difference quotient, substitute the given value of \(f(x)\) into the equation:
[tex]\( \frac{f(x+h)-f(x)}{h} = \frac{f(x+h)-6}{h} \).[/tex] Next, substitute the value of \(f(x)\) into the numerator:
[tex]\frac{6(x+h-1)}{h} \\[/tex] .
Finally, divide both sides of the equation by 6 to simplify the equation:
[tex]\( \frac{6(x+h-1)}{h} = \frac{x+h-1}{h/6} \)[/tex]
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A boy 5m tall observes a bird on top of a pole 20m high at an angle of elevation 30°, calculate the horizontal distance between the boy and the pole
Answer:
34.28 meters
Step-by-step explanation:
We can use trigonometry to solve this problem. Let's call the horizontal distance between the boy and the pole "d". Then we can draw a right triangle with the boy's height (5m) as one leg, the distance "d" as the other leg, and the hypotenuse being the distance from the boy's eyes to the top of the pole (which we don't know yet).
We can use the angle of elevation to find the length of this hypotenuse. The angle of elevation is the angle between the horizontal and the line of sight from the boy's eyes to the top of the pole. Since the boy is looking up at the bird, this angle is also the same as the angle between the hypotenuse and the vertical (i.e. the angle at the top of the triangle). So we have:
tan(30°) = opposite/adjacent
where "opposite" is the height of the pole (20m) and "adjacent" is the hypotenuse. Solving for "adjacent", we get:
adjacent = opposite/tan(30°) = 20/tan(30°)
We can simplify tan(30°) to 1/√3, so:
adjacent = 20/(1/√3) = 20√3
Now we can use the Pythagorean theorem to find the horizontal distance "d":
d^2 + 5^2 = (20√3)^2
Simplifying and solving for "d", we get:
d = √[(20√3)^2 - 5^2] = √(1200 - 25) = √1175
So the horizontal distance between the boy and the pole is approximately 34.28 meters (rounded to two decimal places
6) (8 pts) A hospital is interested in evaluating the percent of patients entering the emergency department who are admitted to the hospital. Data for randomly selected day was collected and out of 187 patients who entered the emergency department, 42 were admitted to the hospital. a) (6 pts) Calculate a 90% two-sided confidence interval for p, the percent of people entering the emergency department who are admitted to the hospital. b) (2 pts) In planning staffing to care for admitted patients, the hospital has assumed that 25% of people who enter the emergency department are admitted to the hospital. Based on your answer to part (a), is it reasonable for the hospital to use this assumption? Explain your answer using information from part (a)
90% people entering the emergency department is within the interval of [0.1559, 0.2933].
The confidence interval of the percent of patients entering the emergency department who are admitted to the hospital is [0.1763, 0.3137]. It is not reasonable for the hospital to assume that 25% of people who enter the emergency department are admitted to the hospital. Here's why.How to calculate a 90% two-sided confidence interval for p, the percent of people entering the emergency department who are admitted to the hospital:$$CI_p =\bigg(\hat{p}-Z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p}+Z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\bigg)$$ where $\hat{p} = \frac{x}{n}$, $\alpha = 0.10$, $Z_{\alpha/2} = 1.645$ (for a 90% confidence interval), and $n = 187$. The margin of error is given by $$ME = Z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$Plugging in the values, we get $$\hat{p} = \frac{42}{187} = 0.2246$$$$ME = 1.645 \cdot \sqrt{\frac{0.2246\cdot 0.7754}{187}} \approx 0.0687$$Therefore, the confidence interval for $p$ is $$CI_p = (0.2246-0.0687, 0.2246+0.0687) = (0.1559, 0.2933)$$The 90% two-sided confidence interval for the percent of people entering the emergency department who are admitted to the hospital is [0.1559, 0.2933].Since the interval doesn't include 0.25, the hospital should not use the assumption that 25% of people who enter the emergency department are admitted to the hospital. This is because the interval does not overlap with the value of 0.25. As a result, we are 90% confident that the true proportion of people who are admitted to the hospital after entering the emergency department is within the interval of [0.1559, 0.2933].
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1) The area of the shaded sector is 51.3 square feet. What is an estimate for the radius of the
circle? Round the answer to the nearest foot.
B
60°
D
120°
C
3.31 is an estimate for the radius of the circle.
What precisely is a circle?The circle fοrm is a clοsed twο-dimensiοnal shape because every pοint in the plane that makes up a circle is evenly separated frοm the "centre" οf the fοrm.
Each line tracing the circle cοntributes tο the fοrmatiοn οf the line οf reflectiοn symmetry. In additiοn, it rοtates arοund the center in a symmetrical manner frοm every perspective.
he radius of the circle, we need to use the formula for the area of a sector, which is:
Area of sector = (θ/360) x π x r
the radius of the circle "x" for now, and set up an equation using the given information
51.3 = (120/360) x π x
Simplifying this equation, we get:
= (51.3 x 360)/(120 x π)
x² ≈ 10.95
x ≈ √10.95
x ≈ 3.31
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PLEASE HELP ASAP!!! I'll MARK BRAINLIEST!!
Farmer Naxvip wanted to sell some hopts and totts at the market. She planned to charge $3 per hopt and $7 per tott. She expected to make at least $84. She expected to sell at most 24 units.
Write a system of statements, in standard form, modeling the relationships between amounts of hopts (x) and amount of totts (y).
Answer:
Let x be the number of hopts and y be the number of totts.
The first statement relates to the total amount of money expected to be made:
3x + 7y ≥ 84
This inequality states that the total revenue from selling hopts and totts should be at least $84.
The second statement relates to the total number of units expected to be sold:
x + y ≤ 24
This inequality states that the total number of hopts and totts sold should be at most 24 units. Therefore, the system of inequalities in standard form is:
3x + 7y ≥ 84
x + y ≤ 24
where x ≥ 0 and y ≥ 0 (since we cannot sell a negative number of hopts or totts).
Answer: Let "x" be the number of hopts that Farmer Naxvip plans to sell, and let "y" be the number of totts that she plans to sell. Then we can write the following system of inequalities to model the relationships between the amounts of hopts and totts:
3x + 7y >= 84
x + y <= 24
The first inequality represents the condition that Farmer Naxvip expects to make at least $84, while the second inequality represents the condition that she expects to sell at most 24 units.
Brainliest Appreciated! (:
7. A man wishes to invest $3500. He can buy savings bonds which pay simple
interest at the rate of 12% per annum or he can start a savings account which
pays compound interest at the same rate. Calculate the difference in the
amounts of the two investments at the end of the 3 years.
a)
1200 x 9 x2
=
Robert is currently 10 years old.Given that five years ago Sharon was twice as old as Robert was then, we can set up the following equation: y + 5 = 2(y)
We are given the information that Sharon is five years older than Robert, and five years ago Sharon was twice as old as Robert was then. This means we can create a system of equations to solve for Robert's age.
Let x = Robert's current age
Let y = Robert's age five years ago
Given that Sharon is five years older than Robert, we can set up the following equation:
x + 5 = Sharon's current age
Given that five years ago Sharon was twice as old as Robert was then, we can set up the following equation:
y + 5 = 2(y)
Solving the first equation for x, we get x = Sharon's current age - 5. Substituting this into the second equation, we get:
y + 5 = 2(Sharon's current age - 5)
Solving this equation for y, we get y = (Sharon's current age - 5)/2.
Since Sharon is five years older than Robert, Sharon's current age is x + 5. Substituting this into our equation for y, we get:
y = (x + 5 - 5)/2
Simplifying this equation, we get y = x/2. This means that Robert's age five years ago was half of his current age.
Since we know that Robert is currently 10 years old, Robert's age five years ago was 5. Therefore, Robert is currently 10 years old.
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Molly knows that 30% of the students at her school are boys and that there are 600 boys at her school. She wants to find the total number of students at the school, because she needs to order t-shirts for all the students.
What is the total number of students at Molly's school?
Answer:
Step-by-step explanation:
You ask yourself, "600 is 30% of how many students?". In equation form this looks like:
600 = .30x
Divide both sides by .30 and you'll get that the number of students is 2000
Frankie is flying to Seattle, Washington from Detroit, Michigan. He
leaves at 12 pm Eastern time. The flight is 5 hours. What time will it be in
Seattle (Pacific time) when he lands
Nancy has twice as many apples as jay. Jay has 3 more apples than Ava. Nancy has 22 apples. How many apples dose Ava have?
Answer:
8 apples
Step-by-step explanation:
Step-by-step explanation:
n = Nancy's apples.
j = Jay's apples.
a = Ava's apples.
n = 22
n = j × 2
j = n / 2 = 22/2 = 11
j = a + 3
a = j - 3 = 11 - 3 = 8
Ava has 8 apples.
James had 20 minutes to do a three-problem quiz. He spent 8 1 4 minutes on question A and 3 4 5 minutes on question B. How much time did he have left for question C?
Answer:
8.41
Step-by-step explanation:
8.14+3.45=11,59
20-11,59=8.41
For isosceles trapezoid LNOP, m\angle N=84m∠N=84, m\angle O=\left(4y-4\right)m∠O=(4y−4), now find the value of y, m\angle Lm∠L and m\angle Pm∠P
When we simplify the equation, we obtain: 2y + 20 = 56 2y = 36 y = 18 Hence, mL = 140 – 2y = 104°, mP = 104°, and mN = mO = (180° – mL – 84° – (4y–4)°)/2 = 76°.
The base angles are equivalent because the LNOP is an isosceles trapezoid. As a result, mP = mL. Since we now know that a quadrilateral's total angles equal 360°, we can say. 360° = mL + mN + mO + mP Inputting the values provided yields: m∠L + 84° + (4y-4)° + m∠L = 360° When we simplify the equation, we obtain: 2m∠L + 4y + 80 = 360 2m∠L = 280 - 4y m∠L = 140 - 2y The non-parallel sides of LNOP are congruent since it is an isosceles trapezoid. As a result, mN = mO. We are aware of: 180° - mL = mN + mO Inputting the values provided yields: 84° + (4y-4)° = 180° - (140-2y)° When we simplify the equation, we obtain: 2y + 20 = 56 2y = 36 y = 18 Hence, mL = 140 – 2y = 104°, mP = 104°, and mN = mO = (180° – mL – 84° – (4y–4)°)/2 = 76°.
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factor 36abc + 54d????
What is the remainder when 5x3 + 2x2 - 7 is divided by x + 9?
Answer:
The remainder when 5x3 + 2x2 - 7 is divided by x + 9 is -692.
Explanation:
We can use long division to divide 5x^3 + 2x^2 - 7 by x + 9:
-5x^2 + 43x - 385
x + 9 | 5x^3 + 2x^2 + 0x - 7
5x^3 + 45x^2
--------------
-43x^2 + 0x
-43x^2 - 387x
--------------
387x - 7
Therefore, the remainder when 5x^3 + 2x^2 - 7 is divided by x + 9 is 387x - 7.
Hope this helps, sorry if this is wrong! :]
Let W1 and W2 be independent geometric random variables with parameters p1 and p2. Find: a) P(W1=W2); b) P(W1W2); d) the distribution of min(W1,W2); e) the distribution of max(W1,W2).
a) The probability of that happening is P(W1 = 1, W2 = 2) = P(W1 = 2, W2 = 1) = p1p2.
b) P(W1 = 1)P(W2 > 1) + P(W1 = 1)P(W2 > 2) + P(W1 = 2)P(W2 > 3) + ...= p1(1 - p2) + p1(1 - p2)p2 + p1p2(1 - p2) + ...= p1(1 - p2)(1 +p2 + p22 + ...) = p1(1 - p2)/(1 - p2)2= p1/(1 - p2)2
c) The probability that their minimum value is k is P(min(W1, W2) = k) = P(W1 = k, W2 > k) + P(W1 > k, W2 = k) = p1(1 - p2)k-1p2 + p2(1 - p1)k-1p1
d) The probability that their maximum value is k is P(max(W1, W2) = k) = P(W1 = k, W2 = k) + P(W1 = k, W2 > k) + P(W1 > k, W2 = k) = p1p2k-1 + p1(1 - p2)k-1 + p2(1 - p1)k-1
a) P(W1 = W2)The probability that W1 = W2 is 0. If W1 and W2 have different values, then W1 is equal to either 1 or 2. The probability of that happening is P(W1 = 1, W2 = 2) = P(W1 = 2, W2 = 1) = p1p2. The probability of that happening is P(W1 = 1, W2 = 2) = P(W1 = 2, W2 = 1) = p1p2.b) P(W1 < W2)The probability of W1 being less than W2 is P(W1 = 1, W2 = 2) + P(W1 = 1, W2 = 3) + P(W1 = 2, W2 = 3) + ... This may be written as P(W1 = 1)P(W2 > 1) + P(W1 = 1)P(W2 > 2) + P(W1 = 2)P(W2 > 3) +...= p1(1 - p2) + p1(1 - p2)p2 + p1p2(1 - p2) + ...= p1(1 - p2)(1 + p2 + p22 + ...) = p1(1 - p2)/(1 - p2)2= p1/(1 - p2)2
d) Distribution of min(W1, W2)If W1 and W2 are independent geometric random variables with parameters p1 and p2, respectively, then the probability that their minimum value is k is P(min(W1, W2) = k) = P(W1 = k, W2 > k) + P(W1 > k, W2 = k) = p1(1 - p2)k-1p2 + p2(1 - p1)k-1p1e) Distribution of max(W1, W2)If W1 and W2 are independent geometric random variables with parameters p1 and p2, respectively, then the probability that their maximum value is k is P(max(W1, W2) = k) = P(W1 = k, W2 = k) + P(W1 = k, W2 > k) + P(W1 > k, W2 = k) = p1p2k-1 + p1(1 - p2)k-1 + p2(1 - p1)k-1
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Spiral Review Solve for x
.
A linear pair of angles is shown. The left side is two x. The right side measures fifty degrees.
Enter the correct answer in the box.
Solution: x=
value of variable x is 65 degree.
define straight lineA straight line is a geometric object that extends infinitely in both directions and has a constant slope or gradient. It is the shortest distance between two points, and it can be described by an equation in the form of y = mx + c, where m is the slope or gradient of the line and c is the y-intercept.
Define supplementary angleThe term "supplementary angles" refers to two angles whose sum is equal to 180 degrees. In other words, if angle A and angle B are supplementary angles, then:
∠A +∠B = 180 degrees
To find value of x
A straight line's angle total is 180°.
2x+50°=180°
2x=130°
x=65
Hence, value of variable x is 65 degree.
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can I get help with this question?
Answer:
A'(-2, 0)B'(0, 1)C'(1, -1)Step-by-step explanation:
You want the coordinates of triangle ABC after dilation by a factor of 1/2.
DilationWhen the center of dilation is the origin, each of the coordinates is multiplied by the dilation factor:
(x, y) ⇒ (x/2, y/2)
A(-4, 0) ⇒ A'(-2, 0)
B(0, 2) ⇒ B'(0, 1)
C(2, -2) ⇒ C'(1, -1)
i need help on these
Answer:
a) x = 20
b) x = 9
Step-by-step explanation:
Mean:a) [tex]\boxed{\bf Mean =\dfrac{sum \ of \ all \ the \ data}{Number \ of \ data}}[/tex]
[tex]\dfrac{16 +x + 3 + 14 +57}{5}=22\\[/tex]
x + 90 = 22*5
x + 90 = 110
x = 110 - 90
x = 20
b)
[tex]\bf \dfrac{4 + x + 5 +6 + 1}{5} = 5[/tex]
x + 16 = 5 * 5
x + 16 = 25
x = 25 - 16
x = 9
Alex is a single taxpayer with $80,000 in taxable income. His investment income consists of $500 of qualified dividends and short-term capital gains of $2,000. Use the tables to complete the statement
Due to Alex's salary falling inside the 22% tax bracket, his short-term capital gains would be subject to the same rate of taxation as his income.
What is short-term capital?A profit realized from the sale of a capital asset, such as a piece of personal or investment property, that has been possessed for one year or less is referred to as a short-term gain.
These profits are classified as ordinary income subject to tax at your personal income tax rate. Gain earned by selling assets that are held for a year or less are called short-term capital gains.
Alex is a single taxpayer who has taxable income of $80,000.
His investment income is made up of $2,000 in short-term capital gains and $500 in qualifying dividends.
As a result of his tax rate income falling between 38,601 and 425,800, which is below 15%, his qualifying dividends would be subject to a 15% tax.
Hence, we must deduct 15% of 500 and 22% of 2000 before combining them.
15% x 500
= 15 x 5
= 75
22% x 2000
= 22 x 20
= 440.
The total is = 75 + 440 = 515.
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a particle moves along the x-axis so that at any time t>0, its velocity is given by v(t)=4-6t^2. if the particle is at a position x=7 at t=1 time, what is the position of the particle at time t=2?
Answer:
-11.
Step-by-step explanation:
We know that the velocity function v(t) is the derivative of the position function x(t).
So, we can integrate v(t) to find x(t) up to a constant of integration:
∫v(t) dt = ∫(4 - 6t^2) dt = 4t - 2t^3 + C
where C is the constant of integration.
We can find the value of C by using the initial condition that the particle is at position x=7 at t=1:
x(1) = 4(1) - 2(1)^3 + C = 7
C = 5
So, the position function is:
x(t) = 4t - 2t^3 + 5
To find the position of the particle at time t=2, we can substitute t=2 into the position function:
x(2) = 4(2) - 2(2)^3 + 5 = -11
Therefore, the position of the particle at time t=2 is -11.
NEED HELP ASAP WILL MARK BRAINLIST !!
Todd and Eric went to the book store. Eric spent $15 less than three times the
amount that Todd spent. If the shoppers spent a total of $197 in books, How much
did Eric spend?
$34
$53
$65
$71
$29
$59
$48
If the shoppers spent a total of $197 in books, the amount that Eric spend is $144.
How much did Eric spend?Let's assume that Todd spent x dollars in the bookstore.
According to the problem, Eric spent $15 less than three times the amount that Todd spent, which can be written as:
3x - 15
The total amount spent by both shoppers is $197, so we can set up the equation:
x + (3x - 15) = 197
Simplifying and solving for x, we get:
4x - 15 = 197
4x = 212
x = 53
Therefore, Todd spent $53 in the bookstore.
To find how much Eric spent, we can substitute Todd's value into the expression we derived for Eric's spending:
3x - 15 = 3(53) - 15 = 144
So Eric spent $144 in the bookstore.
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Use the following statement to answer parts a) and b). Five hundred raffle tickets are sold for $3 each. One prize of $200 is to be awarded. Winners do not have their ticket costs of $3 refunded to them. Raul purchases one ticket.
a) Determine his expected value.
b) Determine the fair price of a ticket.
11. 3#12
Therefore, the fair price of one raffle ticket is $2.60, which is slightly less than the amount Raul paid ($3).
a) The fair price of one raffle ticket is $3. This is because 500 tickets were sold at this price and one prize of $200 is to be awarded. Therefore, the 500 tickets collected add up to $1,500, while the prize to be awarded is $200. The net amount to be divided among all the ticket holders is $1,300.
b) The fair price of one raffle ticket is $2.60. This is calculated by dividing the total prize money of $200 by the total number of tickets sold (500). Therefore, 500 tickets multiplied by $2.60 gives a total prize money of $1,300 which is equal to the total ticket sales of $1,500 less the prize money of $200.
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Alexander went to the store to buy some walnuts. The price per pound of the walnuts is $8 per pound and he has a coupon for $1 off the final amount. With the coupon, how much would Alexander have to pay to buy 2 pounds of walnuts? Also, write an expression for the cost to buy pp pounds of walnuts, assuming at least one pound is purchased.
Alexander would have to pay $15 to buy 2 pounds of walnuts with the coupon. The expression is cost = ($8/pound) x pp - $1.
What is a cost function?The functional connection between cost and output is referred to as the cost function. It examines the cost behaviour at various output levels under the assumption of constant technology. An essential factor in determining how well a machine learning model performs for a certain dataset is the cost function. It determines and expresses as a single real number the difference between the projected value and expected value.
Given that, the price per pound of walnuts is $8.
2 pounds x $8/pound = $16
Alexander would get $1 off the final amount.
Thus,
$16 - $1 = $15
So Alexander would have to pay $15 to buy 2 pounds of walnuts with the coupon.
The expression for the cost can be written as:
cost = ($8/pound) x pp - $1
Hence, Alexander would have to pay $15 to buy 2 pounds of walnuts with the coupon. The expression is cost = ($8/pound) x pp - $1.
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Estimate a 20% tip on a dinner bill of $169. 86 by first rounding the bill amount to the nearest ten dollars
Answer:
Tip is $34
Round up bill amount $170
Step-by-step explanation:
20/ 100 x 170 = $34
Tip is $34
BILL plus tip is $204
What is the value expression
2( X + 4) - (y * 8)
when x= 1/8
and y= 3/16
a 11/4
b 65/2
c 21/4
d 27/4
Answer:
d. 27/4, or 6.75
Step-by-step explanation:
[tex]2( \frac{1}{8} + 4) - ( \frac{3}{16} \times 8)[/tex]
[tex]2( \frac{33}{8} ) - \frac{3}{2} [/tex]
[tex] \frac{33}{4} - \frac{3}{2} = \frac{33}{4} - \frac{6}{4} = \frac{27}{4} = 6.75[/tex]
How are the products of -3(1) and - 3(-1) the same? How are they different?
Answer:
Step-by-step explanation:
45342
Find the area of the sector whose radius and central angle are 42cm and 60° respectively.
Answer:
Step-by-step explanation:
[tex]D=\frac{\theta}{360} \times2\times\pi \times r[/tex]
[tex]D=\frac{60}{360} \times2\times\pi \times 42[/tex]
[tex]=\frac{1}{6} \times84\times\pi[/tex]
[tex]=12\pi cm^2[/tex]
Justin is joining a gym the gym is offering a discount on the fee to join and on the monthly rate the discounted price in dollars the gym charges can be represented by the equation Y equals 10 X +5
Part A what are the slope and the Y intercept of the equation what do the slope and y-intercept each represent in this situation ?.
Part B the regular price in dollar the gym charges can be represented by the equation Y equals 15 X +20. How much money in dollars does Justin save the first month by joining the gym at the discounted price rather than the regular price
Part C Justin create a system of equation based on the equation from part a and the equation from part B, the solution to the system of equation is (-3, -25) Why is the point (-3, ,-25) not possible solution in this solution ?
Part A:
The equation Y = 10X + 5 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Therefore, in this equation, the slope is 10 and the y-intercept is 5.
The slope represents the rate of change in the monthly rate of the gym membership. For every one unit increase in the number of months, the monthly rate will increase by $10. The y-intercept represents the initial cost of joining the gym, which is $5.
Part B:
To find out how much money Justin saves the first month by joining the gym at the discounted price, we need to calculate the difference between the regular price and the discounted price for the first month.
The regular price for the first month can be found by plugging in X = 1 into the equation Y = 15X + 20, which gives Y = 35.
The discounted price for the first month can be found by plugging in X = 1 into the equation Y = 10X + 5, which gives Y = 15.
Therefore, Justin saves $20 (35 - 15) the first month by joining the gym at the discounted price rather than the regular price.
Part C:
The system of equations is:
Y = 10X + 5 (discounted price)
Y = 15X + 20 (regular price)
The solution to the system of equations is (-3, -25), which means that if X = -3, then Y = -25 is a solution to both equations. However, this solution is not possible in this situation because X represents the number of months, which cannot be negative. Therefore, the point (-3, -25) is not a valid solution.
Isabella's ice cream parlor uses waffle cones that have a diameter of 5 in. and a height of 6 in. what is the volume of ice cream that completely fills one cone to the top? enter your answer as a decimal in the box. use 3.14 for pi.
The volume of ice cream that fills one waffle cone to the top is 13.09 cubic inches.
The waffle cone has the shape of a circular cone. The volume of a cone is given by the formula [tex]V = (1/3) * \pi * r^2 * h[/tex], where r is the radius of the circular base and h is the height of the cone.
The diameter of the cone is given as 5 inches, so the radius is half of the diameter, or 2.5 inches. The height is given as 6 inches. Substituting these values into the formula, we get:
V = [tex](1/3) * 3.14 * (2.5 inches)^2 * 6 inches[/tex]
V = 13.09 cubic inches
Therefore, the volume of ice cream that completely fills one waffle cone to the top is 13.09 cubic inches (rounded to two decimal places).
To find the volume of ice cream that fills the waffle cone completely to the top, we first note that the waffle cone has the shape of a circular cone. We use the formula for the volume of a cone, which is [tex]V = (1/3) * \pi * r^2 * h[/tex], where r is the radius of the circular base and h is the height of the cone.
We are given the diameter of the cone, which is 5 inches, and we find that the radius is half of the diameter, or 2.5 inches. We are also given the height of the cone, which is 6 inches. Substituting these values into the formula, we can calculate the volume of the cone as 13.09 cubic inches (rounded to two decimal places). Therefore, the volume of ice cream that completely fills one waffle cone to the top is 13.09 cubic inches.
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Without using a calculator, find the values of the integers a and b for which the solution of the equation (a) x√24 + √96 = √108 + x√12 is √a + b₁ (b), x√40 =x√5 + √10 is a + √№b 7
(a) The solution of the equation is √6 + 2.
(b) a = 10 and b = 2, and the solution of the equation is √10 + 7.
What is the solution of the equation?
((a) To solve this equation, we need to isolate the term with the variable x on one side and move all other terms to the other side.
Let's start by simplifying each term using the fact that;
√24 = √(4 × 6) = 2√6,
√96 = √(16 × 6) = 4√6,
√108 = √(36 × 3) = 6√3, and
√12 = √(4 × 3) = 2√3.
Then, we have:
x√24 + √96 = √108 + x√12
2x√6 + 4√6 = 6√3 + 2x√3
2(√6 + 2√3) = (x√3 + 2x√6)
2√6 + 4√3 = x(√3 + 2√6)
Now, we can equate the coefficients of √6 and √3 on both sides to get a system of equations:
2 = x
4 = 2x
Solving this system, we find that x = 2 and therefore a = 6 and b₁ = 2.
So, the solution of the equation is √6 + 2.
(b) To solve this equation, we also need to isolate the term with the variable x on one side and move all other terms to the other side.
Let's start by squaring both sides of the equation to eliminate the square roots:
(x√40)² = (x√5 + √10)²
40x² = 5x² + 10 + 2x√50 + 10
35x² - 20 = 2x√50
Now, we can square both sides again to eliminate the remaining square root:
(35x² - 20)² = (2x√50)²
1225x⁴ - 1400x² + 400 = 0
This is a quadratic equation in x². We can solve it using the quadratic formula:
x² = (1400 ± √(1400² - 4 × 1225 × 400)) / (2 × 1225)
x² = (1400 ± 200) / 245
x² = 2 or x² = 8/7
Since x² cannot be negative, we have x² = 2 and therefore x = √2.
Substituting this value of x back into the original equation, we have:
x√40 = x√5 + √10
√80 = √10 + √10
√80 = 2√10
So, a = 10 and b = 2, and the solution of the equation is √10 + 7.
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An open box (no lid) with a square base has a volume of 4 cubic feet. What dimensions will minimize the surface area?
If a square base has a volume of 4 cubic feet, the dimensions that minimize the surface area are: x = 2√2 and h = 4/x^2 = 1/2.
Let x be the side length of the square base and h be the height of the box. Since the volume is 4 cubic feet, we have:
V = x^2h = 4
Solving for h, we get:
h = 4/x^2
The surface area of the box, A, is given by:
A = x^2 + 4xh
Substituting h in terms of x, we get:
A = x^2 + 4x(4/x^2) = x^2 + 16/x
To minimize A, we take the derivative with respect to x and set it equal to zero:
dA/dx = 2x - 16/x^2 = 0
Solving for x, we get:
x = 2√2
To ensure that this is a minimum, we take the second derivative:
d^2A/dx^2 = 2 + 32/x^3
At x = 2√2, this is positive, indicating a minimum.
The box has a square base with side length 2√2 feet and height 1/2 feet.
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