-2(x-3) = 5x+1
-2x +6 = 5x+1
subtract 5 x from each side
-7x+6 = 1
subtract 6 from each side
-7x = -5
divide both sides by -7
x = -5/-7 = 5/7
x = 5/7
The bookstore ordered 15 Sociology work books and 10 Economy 101 textbooks on August 1st to prepare for the fall semester. After realizing they miscounted, the bookstore order 1 more Sociology workbook and three more Economy 101 textbooks on August 15th. If August 1st the order was for $4300 and the August 15th order was $800 , find the cost of one Sociology workbook and one Economy 101 textbook?
The cost of one Sociology workbook is $140, and the cost of one Economy 101 textbook is $220, determined by solving two simultaneous equations based on the initial and additional orders placed by the bookstore.
Explanation:To solve for the cost of one Sociology workbook and one Economy 101 textbook, let's label the cost of one Sociology workbook as S and the cost of one Economy 101 textbook as E. Initially, the bookstore ordered 15 Sociology workbooks and 10 Economy 101 textbooks, spending a total of $4300. This gives us the equation: 15S + 10E = 4300. Later, the store ordered an additional 1 Sociology workbook and 3 Economy 101 textbooks for $800, leading to the equation: 1S + 3E = 800.
To solve these equations simultaneously, we can multiply the second equation by 15 to get 15S + 45E = 12000 and subtract the first equation from it: (15S + 45E) - (15S + 10E) = 12000 - 4300, simplifying to 35E = 7700. Dividing both sides by 35 gives us E = 220. Substituting E = 220 back into the first equation 15S + 10(220) = 4300 gives us 15S + 2200 = 4300, thus 15S = 2100 and S = 140.
Therefore, the cost of one Sociology workbook is $140 and the cost of one Economy 101 textbook is $220.
The sum of 7 consecutive odd numbers is 91. What is the sum of the two largest numbers in this set?
13-36x^2=-12 which value of x is a solution to he equation
Omar's Coffee Shop makes a blend that is a mixture of two types of coffee. Type A coffee costs Omar $4.85 per pound, and type B coffee costs $5.95 per pound. This month's blend used twice as many pounds of type B coffee as type A, for a total cost of $519.25 . How many pounds of type A coffee were used?
You roll a pair of fair dice until you roll “doubles” (i.e., both dice are the same). what is the expected number, e[n], of rolls?
The expected number of rolls, e[n], to get doubles on a pair of fair dice is 6. This is calculated by recognizing that getting doubles has a probability of 1/6, and the expectation for a geometric distribution is the inverse of the probability (1/p).
Explanation:To solve the problem of finding the expected number of rolls, e[n], to get doubles on a pair of fair dice, we first need to calculate the probability of rolling doubles. Since there are 6 faces on each die, there are a total of 6 x 6 = 36 possible outcomes when rolling two dice.
Out of these 36 possible rolls, there are 6 outcomes that result in doubles: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). This means the probability of getting doubles in one roll is 6/36, which simplifies to 1/6.
Because each roll is independent, we can model the scenario using a geometric distribution, where the expected value, or mean, is given by 1/p, where p is the success probability. Substituting p with 1/6, the expected number of rolls to get doubles would be 1/(1/6) = 6.
Therefore, the expected number of rolls needed to roll doubles is 6.
If we reject the null hypothesis, can we claim to have proved that the null hypothesis is false? why or why not?
A map is scaled so that 3 cm on the map is equal to 21 actual miles. if two cities on the map are 5 cm apart, what proportion would you use to solve the problem?
A cone-shaped paper drinking cup is to be made to hold 36 cm3 of water. find the height and radius of the cup that will use the smallest amount of paper. (round your answers to two decimal places.)
The formula for volume of cone is:
V = π r^2 h / 3
or
π r^2 h / 3 = 36 cm^3
Simplfying in terms of r:
r^2 = 108 / π h
To find for the smallest amount of paper that can create this cone, we call for the formula for the surface area of cone:
S = π r sqrt (h^2 + r^2)
S = π sqrt(108 / π h) * sqrt(h^2 + 108 / π h)
S = π sqrt(108 / π h) * sqrt[(π h^3 + 108) / π h]
Surface area = sqrt (108) * sqrt[(π h + 108 / h^2)]
Getting the 1st derivative dS / dh then equating to 0 to get the maxima value:
dS/dh = sqrt (108) ((π – 216 / h^3) * [(π h + 108/h^2)^-1/2]
Let dS/dh = 0 so,
π – 216 / h^3 = 0
h^3 = 216 / π
h = 4.10 cm
Calculating for r:
r^2 = 108 / π (4.10)
r = 2.90 cm
Answers:
h = 4.10 cm
r = 2.90 cm
The height of the cone is [tex]\boxed{4.10}[/tex] and the radius of the cone is [tex]\boxed{2.90}.[/tex]
Further explanation:
The volume of the cone is [tex]\boxed{V = \dfrac{1}{3}\left( {\pi {r^2}h} \right)}.[/tex]
The surface area of the cone is [tex]\boxed{S=\pi \times r\times l}[/tex]
Here l is the slant height of the cone.
The value of the slant height can be obtained as,
[tex]\boxed{l = \sqrt {{h^2} + {r^2}} }[/tex].
Given:
The volume of the cone shaped paper drinking cup is [tex]36{\text{ c}}{{\text{m}}^3}[/tex].
Explanation:
The volume of the cone shaped paper drinking cup [tex]36{\text{ c}}{{\text{m}}^3}[/tex].
[tex]\begin{aligned}V&=36\\\frac{1}{3}\left({\pi {r^2}h}\right) &= 36\\{r^2}&= \frac{{108}}{{\pi h}}\\\end{aligned}[/tex]
The surface area of the cone is,
[tex]\begin{aligned}S &= \pi\times\sqrt {\frac{{108}}{{\pi h}}}\times\sqrt {{h^2} + \frac{{108}}{{\pi h}}}\\&= \sqrt{108}\times\sqrt{\frac{{\pi {h^3} + 108}}{{\pi h}}}\\&=\sqrt {108}\times\sqrt {\pi h + \frac{{108}}{{{h^2}}}}\\\end{aligned}[/tex]
Differentiate above equation with respect to h.
[tex]\dfrac{{dS}}{{dh}}=\sqrt {108}\times \left( {\pi - \dfrac{{216}}{{{h^3}}}}\right)\times {\left( {\pi h + \dfrac{{108}}{{{h^2}}}}\right)^{ - \dfrac{1}{2}}}[/tex]
Substitute 0 for [tex]\dfrac{{dS}}{{dh}}[/tex].
[tex]\begin{aligned}\pi- \dfrac{{216}}{{{h^3}}}&= 0\\\dfrac{{216}}{{{h^3}}}&= \pi\\\dfrac{{216}}{{3.14}} &= {h^3}\\h &= 4.10\\\end{aligned}[/tex]
The radius of the cone can be obtained as,
[tex]\begin{aligned}{r^2}&=\frac{{108}}{{\pi \left({4.10} \right)}}\\{r^2}&= \frac{{108}}{{3.14 \times 4.10}}\\{r^2}&= 8.40\\r&= \sqrt {8.40}\\r &= 2.90\\\end{aligned}[/tex]
Hence, the height of the cone is [tex]\boxed{4.10}[/tex]and the radius of the cone is [tex]\boxed{2.90}[/tex].
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Mensuration
Keywords: cone shaped paper, drinking cup, volume, [tex]36{\text{ c}}{{\text{m}}^3}[/tex], height of cone, cup, smallest amount of paper, water.
Suppose you have a normally distributed set of data pertaining to a standardized test. the mean score is 1000 and the standard deviation is 200. what is the z-score of 900 point score? 1.5 1 0.5 â0.5
Find the area under the standard normal distribution curve to the left of z=-2.15 and to the right of z=1.62
Using the normal distribution table, which shows the percentage of the areas to the left a normal distribution, the area to the left z = - 2.15 and area to the right of z = 1.62 are 0.0158 and 0.0526 respectively.
1.)
The area under the normal distribution curve to the left of z = 2.15 can be expressed thus :
P(Z ≤ -2.15)
Using a normal distribution table ; the area to the left is
P(Z ≤ -2.15) = 0.0158
2.)
The area under the normal distribution curve to the right of z = 1.62 can be expressed thus :
P(Z ≥ 1.62) = 1 - P(Z ≤ 1.62)
Using a normal distribution table ; the area to the left is P(Z ≤ 1.62) = 0.94738
P(Z ≥ 1.62) = 1 - 0.94738 = 0.0526
Therefore, the area to the left z = - 2.15 and area to the right of z = 1.62 are 0.0158 and 0.0526 respectively.
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61702 67102 same or different?
Explain why rationalizing the denominator does not change the value of the original expression
Answer:
Because basically, you are multiplying by 1
Step-by-step explanation:
Let me explain this with an example. Rationalize the following expression:
[tex]\frac{5}{\sqrt{7} }[/tex]
In order to rationalize the denominator, the numerator and denominator of the fraction must be multiplied by the root of the denominator. So, what happen if you do that? Well first of all you aren't altering the expression because:
[tex]\frac{\sqrt{7} }{\sqrt{7} } =1[/tex]
Right? Because a certain quantity divided by itself is always equal to 1. So basically you are doing this because you want to rewrite the expression without altering its original value, it is the same when you do this:
[tex]9=3^2=3+3+3=\sqrt{81}[/tex]
Therefore, the only thing you do when you rationalize is remove radicals from the denominator of a fraction. Take a look:
[tex]\frac{5}{\sqrt{7} } *\frac{\sqrt{7} }{\sqrt{7} } =\frac{5*\sqrt{7} }{\sqrt{7}*\sqrt{7} } =\frac{5*\sqrt{7}}{(\sqrt{7} )^2} =\frac{5*\sqrt{7} }{7}[/tex]
You can check this new expression is equal to the original using a calculator:
[tex]\frac{5}{\sqrt{7} } \approx1.8898\\\\\frac{5*\sqrt{7} }{7} \approx1.8898[/tex]
If the APR of a savings account is 3.6% and interest is compounded monthly, what is the approximate APY of the account?
Answer:
3.66% ( approx )
Step-by-step explanation:
Since, the formula of annual percentage yield is,
[tex]APY = (1+\frac{r}{n})^n-1[/tex]
Where,
r = stated annual interest rate,
n = number of compounding periods,
Here, r = 3.6% = 0.036,
n = 12 ( ∵ 1 year = 12 months )
Hence, the annual percentage yield is,
[tex]APY=(1+\frac{0.036}{12})^{12}-1=1.03659 - 1 = 0.036599\approx 0.0366 = 3.66\%[/tex]
Car rental at Q.T. Rental is $22 per day plus an initial deposit of $36.Which expression shows how much it will cost to rent a car for d number of days?
Answer:
The expression is Price = 36 +22*d where d is each day that pass.
Step-by-step explanation:
The inicial 36 are the independant term since it's not related to how many days the car is rented. 22 is the slope since it depends on the days the car is rented.
A car uses 25L of petrol to travel 280km. What is the rate of petrol usage in kilometres per litre?
The rate of petrol per liter is 11.2 liter
What is unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Given:
In 25l car travels = 280km
For finding rate of petrol per liter we have to divide as
Rate of petrol = 280/25
Rate of petrol = 11.2 liter
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The rate of petrol usage for the car is 11.2 kilometers per litre, calculated by dividing the distance travelled, 280km, by the amount of petrol used, 25L.
The question asks to find the rate of petrol usage in kilometers per litre for a car that uses 25L of petrol to travel 280km. To calculate the fuel efficiency, you divide the distance travelled by the amount of petrol used.
Step-by-step calculation:
Distance travelled = 280 km
Amount of petrol used = 25L
Fuel efficiency (km/L) = Distance travelled / Amount of petrol used
Fuel efficiency = 280 km / 25L = 11.2 km/L
Therefore, the car's fuel efficiency is 11.2 kilometers per litre.
The brightness of a variable star adds a component to the simple harmonic motion we have studied in this lesson. Since the brightness is variable the vertical axis may no longer be equal to zero. Also included in the variance is a phase shift. In this case, the equation for this function would be: y = a cos w(t – c ) + b.
Suppose we have a variable star whose brightness alternately increases and decreases. For this star, the time between periods of maximum brightness is 6.5 days. The average brightness (or magnitude) of the star is 5.0 and its brightness varies by + 0.25 magnitude.
1. What is the amplitude of the function for this model?
2. What is the period?
3. What is w?
4. What is the vertical shift?
5. Is there a phase shift? If so, what is it?
6. What is the function
What is the midpoint of the line segment (-3,-2) and (1,4)
Answer:
(-1, 1)
Step-by-step explanation:
The midpoint (M) is found by averaging the coordinates of the end points:
M = ((-3, -2) +(1, 4))/2 = ((-3+1)/2, (-2+4)/2) = (-2/2, 2/2)
M = (-1, 1)
The midpoint of the line segment is (-1, 1).
Which value is a discontinuity of x^2+7x+1/x^2+2x-15? x=-1 x=-2 x=-5 x=-4
The value of x=-5 is a discontinuity of the function x^2+7x+1 / x^2+2x-15 since it makes the denominator of this function equal to zero.
Explanation:The subject of this question is the discontinuity of a rational function. In Mathematics, a function f(x) = (p(x))/(q(x)), where p(x) and q(x) are polynomials, is said to be discontinuous at a particular value of x if and only if q(x) = 0 at that value. From the equation in the question; x^2+7x+1/x^2+2x-15, we can determine its discontinuity by finding the values of x that would make the denominator equal to zero. This is done by solving the polynomial equation x^2+2x-15 = 0 for x. The solutions to this equation represent the values at which the function is discontinuous.
By applying the quadratic formula, (-b ± sqrt(b^2 -4ac))/(2a), where a = 1, b = 2, and c = -15, we get that x = -5, and 3. However, the values given in the question are x=-1, x=-2, x=-5, and x=-4. From these options, only x=-5 makes the denominator zero, thus, x = -5 is a point of discontinuity in the function x^2+7x+1 / x^2+2x-15.
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The area of one triangle is 150 square centimeters when it's height is 20 centimeters and it's base length is 15 centimeters. What is the area of a triangle having a height of 30 centimeters and a base length of 18 centimeters.
Carlos is putting money into a savings account. He starts with $750 in the savings account, and each week he adds $40 . Let S represent the total amount of money in the savings account (in dollars), and let W represent the number of weeks Carlos has been adding money. Write an equation relating S to W . Then use this equation to find the total amount of money in the savings account after 11 weeks.
Equation:
Total amount of money after 11 weeks:
Final answer:
The equation relating the total amount of money S in Carlos's savings to the number of weeks W he adds money is S = 750 + 40W. By substituting W with 11, we find that after 11 weeks, Carlos will have $1,190 in his savings account.
Explanation:
The question involves creating a linear equation to represent the relationship between the total amount of money S in Carlos's savings account and the number of weeks W he has been adding money. The equation can be written as S = 750 + 40W, where 750 represents the initial amount and 40 is the amount added each week. To find the total amount of money after 11 weeks, substitute W with 11:
S = 750 + 40(11) = 750 + 440 = 1190
Therefore, after 11 weeks, the total amount of money in the savings account would be $1,190.
The variable z is directly proportional to x and inversely proportional to y. When x is 12 and y is 18 z has the value 2 what is the value of z when x = 19 and y = 22
The daily mean temperature in a particular place is 83. how many cooling-degree days were accumulated?
Given the following triangle, if a = 12 and ∠B = 48°, find b to the nearest whole number.
Answer:
The value of b nearest whole number is, 13.
Step-by-step explanation:
We know an [tex]\angle B=48^{\circ}[/tex] and the side adjacent to it i.e, a=12.\
In a right triangle BCA ,
the tangent(tan) of an angle is the length of the opposite side divided by the length of the adjacent side.
i.e, [tex]\tan B=\frac{opposite}{Adjacent}=\frac{b}{a}[/tex]
Substitute the value of a=12 and [tex]\angle B=48^{\circ}[/tex] to solve for b in above expression:
[tex]\tan 48^{\circ}=\frac{b}{12}[/tex]
we have the value of [tex]\tan 48^{\circ}=1.110613[/tex]
then, [tex]1.20012724=\frac{b}{12}[/tex]
On simplify we get,
[tex]b=1.110613 \times 12=13.327356[/tex]
Therefore, the value of b nearest whole is, 13
Donna and Phyllis are reviewing for a math final. They have 150 pages to cover. Donna can review 37 pages an hour, but Phyllis can only go 2/3 as fast as Donna. About how long does it take each girl to finish?
Donna will finish reviewing in about 4.05 hours, while Phyllis will take approximately 6.08 hours.
Donna and Phyllis are reviewing for a math final and have 150 pages to cover. Let's calculate how long it takes each girl to finish reviewing:
Donna can review 37 pages an hour. To find the time she needs, we use the formula:
Time = Total Pages / Pages per Hour
Time = 150 / 37
Time ≈ 4.05 hours
Phyllis reviews at 2/3 of Donna's speed. So, her rate is:
Phyllis' rate = (2/3) × 37 ≈ 24.67 pages per hour
To find the time Phyllis needs, we use the same formula:
Time = Total Pages / Pages per Hour
Time = 150 / 24.67
Time ≈ 6.08 hours
In summary, Donna will finish reviewing in about 4.05 hours, while Phyllis will take approximately 6.08 hours.
Solve cos x +sqr root of 2 = -cos x for x over the interval 0,2pi
Assume that a procedure yields a binomial distribution with a trial repeated n times. use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. round to three decimal places. n = 14, x = 6 , p = 0.5
What standard deviation below the mean of normal young adults equals osteoporosis?
how do you write 20484163 in different forms
The indicated function y1(x) is a solution of the given differential equation. use reduction of order or formula (5) in section 4.2, y2 = y1(x) e−∫p(x) dx y 2 1 (x) dx (5) as instructed, to find a second solution y2(x). 9y'' − 12y' + 4y = 0; y1 = e2x/3
The indicated function y1(x) is a solution of the given differential equation.The general solution is [tex]y = c_1 e^{2x}- c_2e^{-6x}/8[/tex]
What is a differential equation?An equation containing derivatives of a variable with respect to some other variable quantity is called differential equations.
The derivatives might be of any order, some terms might contain the product of derivatives and the variable itself, or with derivatives themselves. They can also be for multiple variables.
Given differential equation is
y''-4y'+4y=0
and
[tex]y_1(x) = e^{2x}[/tex]
[tex]y_2(x) = y_1(x) \int\limits^a_b {e^{\int pdx} \, / y_1 ^2(x)dx[/tex]
The general form of equation
y''+P(x)y'+Q(x)y=0
Comparing both the equation
So, P(x)= - 4
[tex]y_2(x) = y_1(x) \int\limits^a_b {e^{\int pdx} \, / y_1 ^2(x)dx\\\\\\[/tex]
[tex]y_2(x) = e^{2x}\int e^{-4x} \, / e^{4x}dx[/tex]
[tex]y_2(x) = e^{2x}\int e^{-8x}dx\\\\y_2(x) = -e^{-6x}/8[/tex]
The general solution is
[tex]y = c_1 e^{2x}- c_2e^{-6x}/8[/tex]
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The area of a rectangle wall of a barn is 216 ft.² it's length is 6 feet longer than twice it's width. find the length and width of the wall of the barn