Donna the trainer has two solo workout plans that she offers her clients: Plan A and Plan B. Each client does either one or the other (not both). On Friday there were 7 clients who did Plan A and 9 who did Plan B. On Saturday there were 5 clients who did Plan A and 3 who did Plan B. Donna trained her Friday clients for a total of 12 hours and her Saturday clients for a total of 6 hours.
How long does each of the workout plans last?
How high is a 40-foot ramp if it is propped at a 30 degree angle?
What is the surface area of the square pyramid?
(Figure is not drawn to scale.)
Eloise started to solve a radical equation in this way: Square root of negative 2x plus 1 − 3 = x Square root of negative 2x plus 1 − 3 + 3 = x + 3 Square root of negative 2x plus 1 = x + 3 Square root of negative 2x plus 1 − 1 = x + 3 − 1 Square root of negative 2 x = x + 2 (Square root of negative 2 x)2 = (x − 4)2 −2x = x2 − 8x + 16 −2x + 2x = x2 + 8x + 16 + 2x 0 = x2 + 10x + 16 0 = (x + 2)(x + 8) x + 2 = 0 x + 8 = 0 x + 2 − 2 = 0 − 2 x + 8 − 8 = 0 − 8 x = −2 x = −8 Both solutions are extraneous because they don't satisfy the original equation. What error did Eloise make?
Please see attached image for the choices and proper formatting of the problem.
What she did:
[tex] \sqrt{-2x + 1 } - 1 = x + 3 - 1 \sqrt{-2x} = x + 2[/tex]
What she should have done:
[tex]( \sqrt{-2x + 1} ^{2} = (x + 3)^2 -2x + 1 = (x + 3)^2 -2x + 1 - 1 = (x + 3)^2 - 1 -2x = (x +3)^2 - 1[/tex]
The error that Eloise made in solving the radical equation was that she subtracted 1 before squaring both sides.
To add, the equation where at least one variable expression is fixed inside a radical, usually a square root is called a radical equation.
Answer:
We need to subtract 1 after squaring both sides
Step-by-step explanation:
Step1: [tex]\sqrt{-2x+1} -3 = x \\[/tex]
adding 3 both sides
[tex]\sqrt{-2x+1} =3 + x \\[/tex]
we need to square here first and then we need to subtract
and this is the error which Eloise make in solving it .
Susie divided a9-pound bag of apples into 5 equal piled. How many pounds of apples are in each piles?
divide the total weight by the number of piles
9/5 = 1.8 pounds per pile
The area of a square is 2500 cm .what is the side legth of the painting?
How many cubic feet of dirt are there in a hole that is 3’ deep x 3’ wide x 3’ long? 0?
the volume of the hole would be 3*3*3 = 27 cubic feet.
however since it is a hole, there is nothing in it.
PLEASE HELP.
Widget wonders produces widgets. They have found that the cost, c(x), of making x widgets is a quadratic function in terms of x.
The company also discovered that it cost $23 to produce 2 widgets, $103 to produce 4 widgets, and $631 to produce 10 widgets.
Find the total cost of producing 6 widgets.
Answer:
The total cost of producing 6 widgets is $231.
Step-by-step explanation:
Given : Widget wonders produces widgets. They have found that the cost, c(x), of making x widgets is a quadratic function in terms of x.
To find : The total cost of producing 6 widgets.
Solution :
Cost is given by [tex]c (x) = ax^2 + bx + d[/tex]
Cost $23 to produce 2 widgets,[tex]c(2) = a(2)^2 + b(2) + d[/tex]
[tex]23= 4a+2b+d[/tex] .........[1]
Cost $103 to produce 4 widgets,[tex]c(4) = a(4)^2 + b(4) + d[/tex]
[tex]103= 16a+4b+d[/tex] ............[2]
Cost $631 to produce 10 widgets.,[tex]c(10) = a(10)^2 + b(10) + d[/tex]
[tex]631= 100a+10b+d[/tex] ...........[3]
Now, we solve equation [1], [2] and [3]
Subtract equation [2]-[1] and [3]-[2]
[2]-[1] → [tex]12a+2b=80[/tex] ........[4]
[3]-[2] → [tex]84a+6b=528[/tex] .......[5]
Solving equation [4] and [5] by elimination method,
Multiply equation [4] by 3 and subtract from [5]
[tex]84a+6b-3(12a+2b)=528-3(80)[/tex]
[tex]84a+6b-36a-6b=528-240[/tex]
[tex]48a=288[/tex]
[tex]a=6[/tex]
Put in equation [4]
[tex]12(6)+2b=80[/tex]
[tex]72+2b=80[/tex]
[tex]2b=8[/tex]
[tex]b=4[/tex]
Substitute the value of a and b in [1] to get d
[tex]23= 4a+2b+d[/tex]
[tex]23= 4(6)+2(4)+d[/tex]
[tex]23= 24+8+d[/tex]
[tex]23= 32+d[/tex]
[tex]d=-9[/tex]
Substitute a=6,b=4,d=-9 in the cost equation,
The required equation form is [tex]c(x) = 6x^2 + 4x-9[/tex]
The total cost of producing 6 widgets.
Put x=6
[tex]c(6) = 6(6)^2 + 4(6)-9[/tex]
[tex]c(6) = 216+15[/tex]
[tex]c(6) =231[/tex]
Therefore, The total cost of producing 6 widgets is $231.
If a triangle has a height of 14 inches and a base of 9 inches what's the area
The product of a positive number and 14 more than seven times the number is 105. Find the two numbers. first number =______ and second number =_________
Answer:
can you explain it in more depth please
Step-by-step explanation:
The expression sin 50 cos 40 + cos 50 sin 40 is equivalent to
The expression sin 50 cos 40 + cos 50 sin 40 is equivalent to 1.
Explanation:The expression sin 50 cos 40 + cos 50 sin 40 can be simplified using the trigonometric identity:
sin(a + b) = sin a cos b + cos a sin b
By applying this identity, we can rewrite the expression as:
sin(50 + 40) = sin 90 = 1Therefore, the expression sin 50 cos 40 + cos 50 sin 40 is equivalent to 1.
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HELP
There are 20 alligators in the swamp. Each year, the number of alligators increases by 25%. There are 25 crocodiles in the swamp. Each year, 10 new crocodiles join the swamp.
Part A: Write functions to represent the number of alligators and crocodiles in the swamp throughout the years. (4 points)
alligators:
x = total number of alligators
n = number of years
x=20x1.25^n
crocodiles:
y = total number of crocodiles
n = number of years
y=25+10n
Tamara earns $8 an hour at her job working 25 hours per week. If 25% of her paycheck goes to taxes, what is Tamaras monthly cash flow? (Asssume this is her only source of income and that there are 4 pays per month)
using the graph of f(x)=log2x below approximate the value of y in the equation 2^2y=4
write the expression in factored form: m²-n²
HELP ROUND 1.75170179212 TO 6 DECIMAL PLACES
Answer:
The answer is 1.751702
Step-by-step explanation:
In this number there are 11 decimal places which can be found after the decimal point.
1.75170179212
As we want 6 decimal places, we need to look at the digit to the right of sixth decimal place. Therefore:
1.75170179212
So, in this case, the digit in the seventh place is a '7'.
If we want to round a number, there are two ways:
If the digit is less than 5, you leave the sixth decimal as it is.If the digit is equal or greater than 5, you add 1 up to the sixth decimal.As the seventh decimal is a '7' which is greater than '5'.
We add a 1 to the Sixth decimal place.
As the sixth decimal place was a '1'. When we round it, it will be a 2
1.751702
Identify the number that does not belong with the other three. Explain your reasoning.
A simple random sample of 1350 registered voters shows that 58% favor Candidate A over Candidate B. Which is the 99% confidence interval for the percent of the population of all registered voters who prefer Candidate A over Candidate B.
Final answer:
The 99% confidence interval for the percentage of registered voters who prefer Candidate A is between 54.6% and 61.4%.
Explanation:
To calculate the 99% confidence interval for the percentage of the population of registered voters who prefer Candidate A over Candidate B, you can use the formula for the confidence interval of a proportion:
The sample proportion (p') is given as 58%, which is 0.58 in decimal form. The size of the sample (n) is 1350.
To calculate the confidence interval, we also need the z-value for the 99% confidence level. The z-value for a 99% confidence level is approximately 2.576 (you can find this value in a standard z-table).
The formula for the margin of error (EBP) is:
EBP = z * sqrt((p'*(1-p'))/n)
Plugging the values into the formula:
EBP = 2.576 * sqrt((0.58*(1-0.58))/1350)
EBP ≈ 0.034
The confidence interval for the true binomial population proportion is given by:
(p' - EBP, p' + EBP)
Therefore, the 99% confidence interval is:
(0.58 - 0.034, 0.58 + 0.034)
(0.546, 0.614)
Interpreting this, we can say with 99% confidence that the true percentage of all registered voters who prefer Candidate A over Candidate B is between 54.6% and 61.4%.
Final answer:
To calculate the 99% confidence interval for the percent of registered voters who prefer Candidate A, we can use the formula CI = p ± Z * √(p*(1-p)/n). Plugging in the values, we find the 99% confidence interval is between 54.6% and 61.4%.
Explanation:
To calculate the 99% confidence interval for the percent of the population of all registered voters who prefer Candidate A over Candidate B, we can use the formula:
CI = p ± Z * √(p*(1-p)/n)
Where:
p = sample proportion
Z = Z-value for the desired confidence level
n = sample size
In this case, the sample proportion is 0.58, the sample size is 1350, and the Z-value for a 99% confidence level is approximately 2.58.
Plugging in these values, we get:
CI = 0.58 ± 2.58 * √(0.58*(1-0.58)/1350)
Simplifying the equation gives us the 99% confidence interval for the percent of registered voters who prefer Candidate A: between 0.546 and 0.614, or 54.6% and 61.4%.
Find the 6th term of the sequence with t1 = -4 and tn = 5tn-1
Which of the following statements are true? I. -(-6) = 6 and -(-4) > -4 III. 5 + 6 = 11 or 9 - 2 = 11 II. -(-4) < 4 or -10 > 10 - 10 IV. 17 > 2 or 6 < 9
The answer would be an option (D) 17 > 2 or 6 < 9 because 17 is greater than 2 and 6 is less than 9 is always true.
What is inequality?Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.
I. -(-6) = 6 and -(-4) > -4
Here -(-4) > -4 is incorrect
-(-4) = 4 is correct
These statements are not true
III. 5 + 6 = 11 or 9 - 2 = 11
Here 9 - 2 = 11 is incorrect
So 9 - 2 = 7 is correct
These statements are not true
II. -(-4) < 4 or -10 > 10 - 10
Here -(-4) < 4 is incorrect
So -(-4) = 4 is correct
These statements are not true
IV. 17 > 2 or 6 < 9
Here 17 is greater than 2 and 6 is less than 9 is always true.
Hence, the correct answer would be an option (D)
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what is the domain and range for the following function and its inverse f(x) = x2 – 2
Answer:
x2-2 domain an range are 0, -2
Step-by-step explanation:
A district manager rewards sales teams based on overall sales generated in a month. The data for earnings are shown in the table, where Low represents the lowest sales and High represents the highest sales generated by a single sales team member.Team Low High Range Mean Median IQR σTeam X 1970 2970 1200 2571.9 2684 426.3 313.8Team Y 250 375 125 315.8 311 59 37.8Team Z 950 1900 950 1529.9 1473 276 180.7Part A: If the manager wants to award the sales team that has the most consistent earnings among its team members, which team should it choose and why? Justify your answer mathematically. (5 points)Part B: If the manager wants to award the sales team with the highest average earnings, which team should it choose and why? Justify your answer mathematically. (5 points)
If the mean of a symmetric distribution is 82, which of these values is most likely to be the median of the distribution? A.92 B. 85 C.78 D.82
According to a recent survey, the salaries of assistant professors have a mean of $41,750 and a standard deviation of $7000 . assuming that the salaries of assistant professors follow a normal distribution, find the proportion of assistant professors who earn more than $48,000 . round your answer to at least four decimal places.
Using the z score formula and the z table, we find that approximately 18.67% of assistant professors earn more than $48,000 given a mean salary of $41,750 and a standard deviation of $7000.
Explanation:The question is asking for the proportion of assistant professors who earn more than $48,000 based on a normal distribution of salaries with a mean of $41,750 and a standard deviation of $7,000. To find this, we first need to calculate the z score, which is the number of standard deviations a data point is from the mean. The z score can be calculated using the formula z = (X - μ) / σ, where X is the data point, μ is the mean, and σ is the standard deviation.
Substituting the given values, we get z = (48,000 - 41,750) / 7,000 = 0.8929. Looking up this z value in the z table, we find that the proportion for z = 0.89 is 0.8133. This is the proportion of salaries that are below $48,000. Thus, to find the proportion of salaries above $48,000, we subtract this value from 1, giving us 1 - 0.8133 = 0.1867 or 18.67% when expressed as a percentage. So, approximately 18.67% of assistant professors earn more than $48,000.
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A ______ is an expression that can be written in the form of p/q where p and q are polynomials and q
The function h(x) = x2 + 6x + 7 represents a parabola.
Part A: Rewrite the function in vertex form by completing the square. Show your work. (6 points)
Part B: Determine the vertex and indicate whether it is a maximum or a minimum on the graph. How do you know? (2 points)
Part C: Determine the axis of symmetry for h(x). (2 points)
Part A.
The given equation is:
y = x^2 + 6x + 7
By completing the square:
y = (x^2 + 6x + 9) + 7 – 9
y = (x + 3)^2 – 2
y + 2 = (x + 3)^2
Part B.
The vertex form of a parabola is in the form:
y – k = 4p (x – h)^2
Where (h, k) is the vertex (x, y) of the parabola.
Therefore the vertex: (-3, -2)
Since 4p = 1, a positive number, therefore the parabola opens up which makes the vertex (-3, -2) the minima of the graph.
Part C.
The Axis of Symmetry is the x - coordinate of the vertex which is x = - 3
A 30 inch board is to be cut into three pieces? a 30 inch board is to be cut into three pieces so that the second piece is 4 times as long as the first piece and the third piece is 5 times as long as the first piece. if x represents the length of the first piece, find the length of all three pieces.
First, we construct an equation based on the problem: x+(4x)+(5x)=30. Solving this gives us x=3 inches. Thus, the first piece is 3 inches, the second piece is 12 inches and the third piece is 15 inches.
Explanation:In this problem, a 30 inch board is being divided into three pieces. If the first piece is designated as x, then the second piece would be 4 times as long which is 4x, and the third piece would be 5 times as long which is 5x. If all these pieces add up to the total length of the board which is 30 inches so we can construct the equation: x+(4x)+(5x)=30.
To solve, we combine like terms to get 10x = 30. Divide both sides by 10 to isolate x, so x=3 inches. Thus, the first piece is 3 inches, the second piece is 4 times the first piece which is 12 inches, and the third piece is 5 times the first piece which is 15 inches.
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what is 17pi/4 in decimal form nearest the thousandth
17 x PI = 53.40707511
divided by 4 = 13.35176878
rounded to nearest thousandth = 13.352
The population of greenville is currently 50,000 and declines at a rate of 1.2% every year. This models:
Exponential growth.
Exponential logarithm
Exponential decay
The quadratic function y = –10x2 + 160x – 430 models a store’s daily profit (y) for selling a T-shirt priced at x dollars. What equation do you need to solve to find the selling price or prices that would generate $50 in daily profit? What method would you use to solve the equation? Justify your choice.
a. The required equation is x² - 16x + 48 = 0
b. I would use factorisation to solve it and the selling prices that would generate a daily profit of $50 are $4 and $12 respectively.
a.
The required equation is x² - 16x + 48 = 0
The required equation
Since the quadratic function y = -10x² + 160x - 430 models a store’s daily profit (y) for selling a T-shirt priced at x dollars. Since we require a profit of $50, then y = 50.
So, y = -10x² + 160x - 430
-10x² + 160x - 430 = 50
-10x² + 160x - 430 - 50 = 0
-10x² + 160x - 480 = 0
Dividing through by -10, we have
x² - 16x + 48 = 0
So, the required equation is x² - 16x + 48 = 0
b.
I would use factorisation to solve it and the selling prices that would generate a daily profit of $50 are $4 and $12 respectively.
The methodTo determine the method you would use to solve the equation, you would need to determine the value of the discriminant.
DiscriminantFor a quadratic equation ax² + bx + c = 0, the discriminant is D = b² - 4ac
Since x² - 16x + 48 = 0 and its discriminant D = (-16)² - 4 × 48
= 256 - 192
= 48
= 64 > 0 and is a perfect square, so it is factorizable. The equation would have real and distinct roots,
So, x² - 16x + 48 = 0
x² - 4x - 12x + 48 = 0
x(x - 4) - 12(x - 4) = 0
(x - 4)(x - 12) = 0
x - 4 = 0 or x - 12 = 0
x = 4 or x = 12
I would use factorisation to solve it and the selling prices that would generate a daily profit of $50 are $4 and $12 respectively.
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