The correct symbol to compare the expressions is < (less than).
7 × (2/10) is equivalent to 1.4, which is less than 7. Therefore, 7 is greater than 1.4, and we can write 7 × (2/10) < 7 as the comparison between the expressions.
To compare the two expressions, we can analyze their values without actually multiplying them. The expressions are:
1. 7 × (2/10)
2. 7
Now let's simplify the first expression without multiplying:
7 × (2/10) = 7 × (1/5) (since 2 and 10 have a common factor of 2)
Now let's compare:
7 × (1/5) ? 7
Since we're multiplying 7 by a fraction that is less than 1 (1/5), the result will be smaller than 7. Therefore, the correct comparison symbol is "<":
7 × (1/5) < 7
The correct expression so formed is 7 × (2/10) < 7.
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Determine the equation of the circle graphed below.
The equation of the circle graphed is given as follows:
(x + 1)² + (y + 3)² = 36.
What is the equation of a circle?The equation of a circle of center [tex](x_0, y_0)[/tex] and radius r is given by:
[tex](x - x_0)^2 + (y - y_0)^2 = r^2[/tex]
The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle.
The coordinates of the center of the circle are given as follows:
(-1, -3).
The radius of the circle is given as follows:
r = 6 units.
Then the equation of the circle is given as follows:
(x + 1)² + (y + 3)² = 36.
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The following costs were for bikeway inc., a bicycle manufacturer that uses the high-low method:
output fixed costs variable costs total costs
950 $ 45,000 $ 95,000 $ 140,000
1,050 $ 45,000 $ 105,000 $ 150,000
1,100 $ 45,000 $ 110,000 $ 155,000
1,150 $ 45,000 $ 115,000 $ 160,000
at an output level of 1,000 bicycles, per unit total cost is calculated to be:
multiple choice
$139.13.
$145.00.
$121.50.
$126.09.
$100.00.
The per unit total cost at an output level of 1,000 bicycles is calculated to be $139.13.
To calculate the per unit total cost using the high-low method, follow these steps:
1. Identify the highest and lowest output levels (1,150 and 950 bicycles).
2. Calculate the difference in variable costs and output levels: ($115,000 - $95,000) / (1,150 - 950) = $20,000 / 200 = $100 per bicycle.
3. Calculate the variable cost for 1,000 bicycles: $100 x 1,000 = $100,000.
4. Add the fixed cost: $100,000 (variable cost) + $45,000 (fixed cost) = $145,000 (total cost).
5. Calculate the per unit total cost: $145,000 / 1,000 = $139.13 per bicycle.
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Imagine some colored blocks are laid out in a row: three red, two blue, three red, two blue and so on. If there are 65 colored blocks, how many would be red?
A. 52
B. 39
C. 26
D. 13
There are 39 red blocks out of 65 total blocks.
To solve this problem, we need to find the total number of blocks in each repeating pattern. The pattern is three red blocks followed by two blue blocks. So in each pattern, there are five blocks total.
To find the number of red blocks in 65 total blocks, we need to figure out how many times the pattern repeats. We can do this by dividing the total number of blocks (65) by the number of blocks in each pattern (5):
65 ÷ 5 = 13
So the pattern repeats 13 times.
In each pattern, there are three red blocks. So to find the total number of red blocks, we need to multiply the number of red blocks in each pattern (3) by the number of times the pattern repeats (13):
3 x 13 = 39
Therefore, there are 39 red blocks out of 65 total blocks.
The correct answer is option B.
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Use implicit differentiation to find the derivative of sin(y²)+x=eʸ
To find the derivative of sin(y²)+x=eʸ using implicit differentiation, we need to differentiate both sides of the equation with respect to x.
Starting with the left side, we use the chain rule and the derivative of sin(u), which is cos(u) times the derivative of u with respect to x:
d/dx(sin(y²)) = cos(y²) * d/dx(y²)
Using the power rule, we get:
d/dx(y²) = 2y * d/dx(y)
Putting it all together:
d/dx(sin(y²)) = 2y * cos(y²) * d/dx(y)
Now let's move on to the right side of the equation. The derivative of implicit function eʸ with respect to x is simply eʸ times the derivative of y with respect to x:
d/dx(eʸ) = eʸ * d/dx(y)
Putting it all together, we have:
2y * cos(y²) * d/dx(y) + 1 = eʸ * d/dx(y)
We can now solve for d/dx(y):
d/dx(y) = (1 - 2y * cos(y²)) / eʸ
Therefore, the derivative of sin(y²)+x=eʸ is:
d/dx(y) = (1 - 2y * cos(y²)) / eʸ.
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what is 10x10x10x10x10x10x10x10x103?
Answer:
1.03x 10^{10}
Step-by-step explanation:
No explanation, simple calculator calculation does the job.
Find the area of the shaded region:
Answer:
approximately 42.85 of whatever unit
Find the linearization of the function z = x =√y at the point (-2, 4). L(x, y)=
The linearization of the function z = x =√y at the point (-2, 4) is L(x, y) = 2 + (1/4)(y-4).
To find the linearization of the function z = x =√y at the point (-2, 4), we need to use the formula for the linearization:
[tex]L(x, y) = f(a, b) + f_x(a, b)(x-a) + f_y(a, b)(y-b)[/tex]
where f(a, b) is the value of the function at the point (a, b), f_x(a, b) is the partial derivative of f with respect to x evaluated at (a, b), f_y(a, b) is the partial derivative of f with respect to y evaluated at (a, b), and (x-a) and (y-b) are the distances from the point (a, b) to the point (x, y).
In this case, we have:
f(x, y) = √y
a = -2
b = 4
So, we need to find the partial derivatives f_x and f_y:
[tex]f_x(x, y) = 0f_y(x, y) = 1/(2√y)[/tex]
evaluated at (a, b):
f_x(-2, 4) = 0
f_y(-2, 4) = 1/(2√4) = 1/4
Now, we can plug in all the values into the linearization formula:
[tex]L(x, y) = f(-2, 4) + f_x(-2, 4)(x-(-2)) + f_y(-2, 4)(y-4)L(x, y) = √4 + 0(x+2) + (1/4)(y-4)L(x, y) = 2 + (1/4)(y-4)[/tex]
Therefore, the linearization of the function z = x =√y at the point (-2, 4) is L(x, y) = 2 + (1/4)(y-4).
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The main types of markets are called
answers)
(choose 1 of the 2 possible
1 point
O residential
customer
consumer
industrial
The main types of markets are:
Consumer markets: These are markets where individuals purchase goods or services for their personal use or consumption. Industrial markets: These are markets where businesses purchase goods or services for their own use in producing other goods or services.So, the correct answer is "consumer" and "industrial".
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Un medicamento tiene ciertos compuestos en las cantidades indicadas en la siguiente tabla:
Compuesto Cantidad en miligramos
A 0,6
B 0,402
C 0,08
D 0,46
Al ordenar la cantidad de compuesto que contiene dicho medicamento de menor a mayor, ¿Cuál es el orden correcto?
The correct order of the compounds in the medicine from least to greatest is: C, B, D, A.
To order the compounds in the medicine from least to greatest, we need to compare their amounts.
First, we can compare compounds A and C. Compound A has an amount of 0.6 milligrams, which is greater than the amount of compound C, which is only 0.08 milligrams. Therefore, we know that compound C is the smallest amount in the medicine.
Next, we can compare compounds B and D. Compound B has an amount of 0.402 milligrams, which is smaller than the amount of compound D, which is 0.46 milligrams. Therefore, we know that compound B is the second smallest amount in the medicine.
Finally, we can compare compounds A and D. Compound A has an amount of 0.6 milligrams, which is greater than the amount of compound D, which is only 0.46 milligrams. Therefore, we know that compound D is the third smallest amount in the medicine.
Therefore, the correct order of the compounds in the medicine from least to greatest is: C, B, D, A.
It's important to note that the amount of each compound in the medicine may have different effects on the patient's health, and that the order of the compounds from least to greatest may not necessarily reflect their importance or efficacy in treating a particular condition. The ordering of the compounds is simply a matter of comparing their relative amounts in the medicine.
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The triangle above has the following measures.
a=9cm
b=9√3cm
Use the 30-60-90 Thangle Theorem to find the
length of the hypotenuse Include correct units
Show all your work
Answer:
Step-by-step explanation:
The length of the hypotenuse is approximately 4.95 cm.
We have,
Since triangle ABC is a 45-45-90 triangle, we know that the measure of angle B is also 45 degrees.
Therefore, we can use the 45-45-90 Triangle Theorem, which states that in a 45-45-90 triangle,
the length of the hypotenuse is √2 times the length of either leg.
In this case,
We know that leg a = 3.5 cm, so we can find the length of the hypotenuse c using the formula:
c = a√2
Substituting the value of a, we get:
c = 3.5√2 ≈ 4.95 cm
Therefore,
The length of the hypotenuse is approximately 4.95 cm.
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complete question:
The triangle above has the following measures. mzC = 45° a = 3.5 cm Use the 45-45-90 Triangle Theorem to find the length of the hypotenuse. Include correct units. Show all your work.
Research on the major types of businesses in your province. Based from the data you have gathered, create 1 revenue problem involving quadratic functions.
The top industries are agriculture, mining, tourism, and manufacturing.
The quadratic equations are as given.A manufacturing company in my fiefdom produces and sells ceramic pots.
The company has fixed costs of$ 10,000 per month and variable costs of$ 5 per pot. The company's profit is given by the quadratic function R( x) = -0.2 x2 50x, where x is the number of pots produced and vended in a month.
What's the maximum profit that the company can induce in a month: To break this problem, we can use the formula for chancing the maximum value of a quadratic function, which is given by x = - b/ 2a. In this case, the measure of the x2 term is-0.2, and the measure of the x term is 50. Plugging these values into the formula, we get x = -50/( 2 *(-0.2)) = 125 Hence we obtain the quadratic equation.
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Projectile motion. the height, y metres, of a ball after it has been hit can be modelled by the equation y = -1/8x²+x+c, where x is the horizontal distance travelled by the object in metres and c is a constant. (i) given that the maximum height attained by the ball is 2.4 metres, find the value of c and the corresponding horizontal distance travelled by the ball.
The value of c is 0.4 and the corresponding horizontal distance travelled by the ball is 4 meters.
How to solve projectile equations?
We are given the projectile equation:
y = -1/8x² + x + c
To find the value of c, we can use the fact that the maximum height attained by the ball is 2.4 meters. The maximum height occurs at the vertex of the parabola, which is given by:
x = -b/2a
where a = -1/8 and b = 1.
Therefore,
x = -(1)/(2*(-1/8)) = 4
So, the corresponding horizontal distance travelled by the ball is 4 meters.
Now, we can use the maximum height attained by the ball to solve for c.
y = -1/8x² + x + c
Substituting x = 4 and y = 2.4, we get:
2.4 = -1/8(4)² + 4 + c
2.4 = -1/8(16) + 4 + c
2.4 = -2 + 4 + c
c = 0.4
Therefore, the value of c is 0.4.
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1. )Indicate the equation of the given line in standard form. Show all of your work for full credit.
The line containing the median of the trapezoid whose vertices are R(-1, 5) , S(1, 8), T(7, -2), and U(2, 0).
2. )Indicate the equation of the given line in standard form. Show all of your work for full credit.
The line containing the altitude to the hypotenuse of a right triangle whose vertices are P(-1, 1), Q(3, 5), and R(5, -5).
3. ) Indicate the equation of the given line in standard form. Show all of your work for full credit.
The line containing the diagonal, BD, of a square whose vertices are A(-3, 3), B(3, 3), C(3, -3), and D(-3, -3). Find two equations, one for each diagonal.
1) x+y=4. This is the equation of the line in standard form.
2) x+y=4. This is the equation of the line in standard form.
3) The equation of the other diagonal is x=0.
1) The median of a trapezoid connects the midpoints of the non-parallel sides. The midpoint of RT is ((-1+7)/2,(5-2)/2)=(3,1.5) and the midpoint of SU is ((1+2)/2,(8+0)/2)=(1.5,4). The line containing the median passes through these two points, so we can use them to find the equation of the line. The slope of the line is (4-1.5)/(1.5-3)=1.5/(-1.5)=-1. The midpoint formula for a line gives us (y-1.5)=-1(x-3), which simplifies to x+y=4. This is the equation of the line in standard form.
2) To find the altitude to the hypotenuse of a right triangle, we need to find the midpoint of the hypotenuse and the slope of the hypotenuse. The midpoint of PQ is ((-1+3)/2,(1+5)/2)=(1,3), and the midpoint of PR is ((-1+5)/2,(1-5)/2)=(2,-2). The slope of PQ is (5-1)/(3-(-1))=4/4=1, so the slope of the altitude is -1. We can use the point-slope form of a line to get y-3=-1(x-1), which simplifies to x+y=4. This is the equation of the line in standard form.
3) The diagonals of a square are perpendicular bisectors of each other, so we can find the equations of both diagonals using the midpoint and slope formulas. The midpoint of AC is ((3-3)/2,(3-3)/2)=(0,0), and the midpoint of BD is ((-3+3)/2,(3-3)/2)=(0,0). The slope of AC is (3-(-3))/(3-(-3))=6/6=1, so the slope of BD is -1. Using the point-slope form of a line, we can get y-0=-1(x-0), which simplifies to y=-x. This is the equation of one diagonal. To find the equation of the other diagonal, we use the midpoint of AB ((-3+3)/2,(3+3)/2)=(0,3) and the midpoint of CD ((3-3)/2,(-3-3)/2)=(0,-3). The slope of AB is (3-3)/(3-(-3))=0, so the slope of the other diagonal is undefined (since it's perpendicular to AB). The equation of the other diagonal is x=0.
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a fast food restaurant executive wishes to know how many fast food meals teenagers eat each week. they want to construct a 85% confidence interval with an error of no more than 0.06 . a consultant has informed them that a previous study found the mean to be 4.9 fast food meals per week and found the standard deviation to be 0.9 . what is the minimum sample size required to create the specified confidence interval? round your answer up to the next integer.
The minimum sample size which is need to to create the given confidence interval is equal to 467.
Sample size n
z = z-score for the desired confidence level
From attached table,
For 85% confidence level, which corresponds to a z-score of 1.44.
Maximum error or margin of error E = 0.06
Population standard deviation σ = 0.9
Minimum sample size required to construct a 85% confidence interval with an error of no more than 0.06,
Use the formula,
n = (z / E)^2 × σ^2
Plugging in the values, we get,
⇒ n = (1.44 / 0.06)^2 × 0.9^2
⇒ n = 466.56
Rounding up to the next integer, we get a minimum sample size of 467.
Therefore, the minimum sample size required to construct a 85% confidence interval with an error of no more than 0.06 is 467.
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Two wives and their husbands have tickets for a play. they have the first four seats on the left side of the center aisle. they will be arriving seperately from their jobs. so they agreee to take their seats from the inside to the aisle in whatever order they arrive. there is a propability of 2/3 that they will all have arrived by curtain time.
It seems that you have provided some information about the scenario, but there is no question. How may I assist you?
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Find the surface area of the figure below. 18 m 9sqrt(3) m 18
The surface area of the pyramid is 72 + 486√3 metres.
How to find the surface area of a pyramid?The pyramid above is an hexagonal pyramid. The surface area of the hexagonal pyramid can be found as follows:
surface area of a hexagonal pyramid = ph / 2 + B
where
B = base areap = perimeter of the baseheight of the pyramidTherefore,
Base area = 1 / 2pa
where
p = perimeter of the basea = apothemBase area = 1 / 2 × (18 × 6) × 9√3
Base area = 1 / 2 × 108 × 9√3
Base area = 54 × 9√3
Base area = 486√3 metres
surface area of a hexagonal pyramid = 108 × 18 / 2 + 486√3
Therefore,
surface area of a hexagonal pyramid = 972 + 486√3 metres
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Write an inequality that represents the cost of each cookie.
At Cindy's Sweet Treats, cookies are packaged in boxes of 8. Depending on the cookie flavor, the most a box can cost is $16
The inequality that represents the cost of each cookie is C ≤ $2, where C is the cost of each cookie.
An inequality is a mathematical expression that shows a relationship between two values that may not be equal. To represent the cost of each cookie using an inequality, we can first determine the cost per cookie by dividing the total cost of a box by the number of cookies in each box. In this case, that would be $16 divided by 8 cookies.
Let C represent the cost of an individual cookie. Since the most a box can cost is $16, the highest cost per cookie would be $16 / 8 = $2. To express this situation as an inequality, we can write:
C ≤ $2
This inequality indicates that the cost of each cookie (C) must be less than or equal to $2, ensuring that the total cost for a box of cookies does not exceed the maximum price of $16. By using this inequality, we can evaluate different cookie flavors and their respective costs to confirm that they meet Cindy's Sweet Treats' pricing requirements.
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The standard deviation of the scores on a skill evaluation test is 320 points with a mean of 1434 points. if 338 tests are sampled, what is the probability that the mean of the sample would differ from the population mean by less than 43 points? round your answer to four decimal places.
The probability that the mean of the sample would differ from the population mean by less than 43 points is approximately 0.7597 or 0.7600 (rounded to four decimal places).
Given that the standard deviation of the scores on a skill evaluation test is 320 points with a mean of 1434 points. And we have a sample of size n = 338.
We need to find the probability that the mean of the sample would differ from the population mean by less than 43 points.
The standard error of the mean is given by:
SE = σ/√n
where σ is the population standard deviation and n is the sample size.
Substituting the given values, we get:
SE = 320/√338
SE ≈ 17.398
To find the probability, we need to standardize the sample mean using the standard error as follows:
Z = (X - μ) / SE
where X is the sample mean, μ is the population mean, and SE is the standard error of the mean.
Substituting the given values, we get:
Z = (1434 - 1434) / 17.398
Z = 0
Since the mean difference is 0, we can find the probability of a difference less than 43 points by finding the probability that Z lies between -43/17.398 and 43/17.398.
Using a standard normal distribution table or calculator, we find that this probability is approximately 0.7597.
Therefore, the probability that the mean of the sample would differ from the population mean by less than 43 points is approximately 0.7597 or 0.7600 (rounded to four decimal places).
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120 people seated in the first 5 rows at a concert how many were between the ages of 11 and 17
A car-detailing service estimates that its daily cost of waxing a cars is C(q) = 0.06q²+37q + 360, If the service collects $65 for each car waxing, find the number of cars the service should wax daily in order to maximize profit
The service should wax approximately 233 cars daily to maximize profit.
To maximize profit, we need to find the number of cars (q) that would result in the highest profit. Profit function P(q) can be calculated as:
P(q) = Revenue - Cost
P(q) = 65q - (0.06q² + 37q + 360)
Now, to find the optimal value of q, we can calculate the derivative of the profit function with respect to q and set it to zero:
dP(q)/dq = 65 - (0.12q + 37)
0 = 65 - 0.12q - 37
Solve for q:
0.12q = 28
q = 28 / 0.12
q ≈ 233.33
Since the number of cars must be a whole number, we can round q to the nearest integer. Therefore, the service should wax approximately 233 cars daily to maximize profit.
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questions.
1) Choose the correct name for the set of numbers.
{..., -3, -2, 1, 0, 1, 2, 3, ...}
erc
The set of numbers is an example of the integers.
What is the best name for the set of numbers?The set of numbers is an example of the integers. Integers are whole numbers (positive or negative) and zero. They are often represented by the symbol "Z". In this set, we have all the whole numbers from negative infinity to positive infinity, including negative and positive 3, 2, 1, 0, and all the numbers in between. The use of ellipses indicates that the set goes on indefinitely in both directions. It is worth noting that 1 appears twice in the set, indicating that sets of integers may have repeated elements. Overall, the set of numbers shown is an infinite set of integers.
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How many integers between 100 and 300 have both 11 and 8 as factors?
176, 264. your welcome!
Every line segment used to make this hospital logo is 3 meters long. What is the total area of the logo in square meters?
1. The data sel below represents the number of animals in different exhibits at a zoo.
48, 86, 15, 27, 18, 52, 103
a. Write the data from least to greatest.
h. What is the minimum number of animals?
c. What is the maximum number of animals?
d. What is the median number of animals?
e. What is the median of the first half of the data? (first quartile)
f. What is the median of the second half of the data? (third quartile)
g. What is the interquartile range?
Answer:
a) 15, 18, 27, 48, 52, 86, 103
b) Minimum number = 15
c) Maximum number = 103
d) Median = 48
e) First quartile = 18
f) Third quartile = 86
g) Interquartile range = 68
Step-by-step explanation:
Part aTo write the data from least to greatest, arrange the numbers in ascending order:
15, 18, 27, 48, 52, 86, 103[tex]\hrulefill[/tex]
Part bThe minimum number in a set of data is the smallest value.
Therefore, the minimum number of animals is 15.
[tex]\hrulefill[/tex]
Part cThe maximum number in a set of data is the greatest value.
Therefore, the maximum number of animals is 103.
[tex]\hrulefill[/tex]
Part dThe median of a set of data is the middle value when all data values are placed in order of size.
[tex]\begin{array}{ccccccc}\sf 15, &\sf 18, &\sf 27, &\sf 48, &\sf 52, &\sf 86,& \sf 103\\ &&&\uparrow&&&\\&&&\sf median&&&\end{array}[/tex]
Therefore, the median is the fourth number, which is 48.
[tex]\hrulefill[/tex]
Part eThe lower quartile (Q₁) is the median of the data values to the left of the median.
[tex]\begin{array}{ccccccc}\sf 15, &\sf 18, &\sf 27, &\sf 48, &\sf 52, &\sf 86,& \sf 103\\ &\uparrow &&\uparrow&&&\\&\sf Q_1&&\sf median&&&\end{array}[/tex]
Therefore, the median of the first half of the data is 18.
[tex]\hrulefill[/tex]
Part fThe lower quartile (Q₃) is the median of the data values to the right of the median.
[tex]\begin{array}{ccccccc}\sf 15, &\sf 18, &\sf 27, &\sf 48, &\sf 52, &\sf 86,& \sf 103\\ &&&\uparrow &&\uparrow&\\&&&\sf median&&\sf Q_3&\end{array}[/tex]
Therefore, the median of the second half of the data is 86.
[tex]\hrulefill[/tex]
Part gThe interquartile range (IQR) is the difference between the third quartile (Q₃) and the first quartile (Q₁).
[tex]\begin{aligned}\sf IQR &=\sf Q_3 - Q_1 \\&= \sf 86 - 18 \\&= \sf 68\end{aligned}[/tex]
Therefore, the interquartile range is 68.
A little more difficult question.
8) John went to a vending machine and bought 2 bags of chips and 2 Powerades for
$3. 50. He then went back to the same vending machine and bought another bag of
chips and 3 Powerades for $3. 75. How much does a bag of chips cost, how much
does a Powerade cost?
The cost of a Powerade is $0.75 if 2 bags of chips and 2 Powerades cost $3.50 and one bag of chips and 3 Powerades cost $3.75.
Let the cost of one Powerade be x
The cost of one bag of chips be y
Cost of 2 bags of chips and 2 Powerades = $3.50
Cost of 2 bags of chips = 2y
Cost of 2 Powerades = 2x
2x + 2y = 3.50 ----(i)
Cost of one bag of chips and 3 Powerades= $3.75
Cost of 1 bag of chips = y
Cost of 3 Powerades = 3x
x + 3y = 3.75
Multiply the above equation by 2
2x + 6y = 7.50 -----(ii)
Subtract (ii) and (i)
4y = 4
y = $1
The cost of one bag of chips is $1
x + 3 = 3.75
x = $0.75
The cost of one Powerade is $0.75
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A large box of chocolates has a width that is 2 times the height of the box and a length that is 1. 5 times the width of
the box. Each of the 48 chocolates rests in a cube with a side length of 1 inch
Let's start by using algebra to represent the relationships between the dimensions of the box:
Let h be the height of the box. Then the width of the box is 2h (since it is 2 times the height). And the length of the box is 1.5 times the width, so it is 1.5(2h) = 3h.So the dimensions of the box are: height = h, width = 2h, length = 3h.
Now let's find the volume of the box:
Volume = height x width x length Volume = h x 2h x 3h Volume = [tex]6h^3[/tex]Since we know that each chocolate rests in a cube with a side length of 1 inch, the volume of each chocolate is [tex]1^3 = 1[/tex] cubic inch. So the total volume of all 48 chocolates is 48 cubic inches.
Therefore, we can set up an equation to solve for h:
[tex]48 = (6h^3) / (1 cubic inch/chocolate)[/tex]
[tex]48 = 6h^3[/tex]
[tex]8 = h^3[/tex]
h = 2
So the height of the box is 2 inches, the width is 4 inches (since it is 2 times the height), and the length is 6 inches (since it is 1.5 times the width).
To check our work, we can calculate the volume of the box:
Volume = height x width x length
Volume = 2 x 4 x 6
Volume = 48 cubic inches
This matches the total volume of all 48 chocolates, so we can be confident in our answer.
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What is the minimum surface area of a box whose base is a square and with no top that holds 32 cm³. a 16 b 32 c 64 d 48
The minimum surface area of the box is 68 cm², which corresponds to option (d).
How to calculate the surface area of box?Let the side length of the square base be x and the height of the box be h. Then, we have:
Volume of the box = x²h = 32
h = 32/x²
The surface area of the box is given by:
S = x² + 4(xh) = x² + 4x(32/x²) = x² + 128/x
To find the minimum surface area, we can differentiate S with respect to x, set the derivative equal to zero, and solve for x:
dS/dx = 2x - 128/x² = 0
2x = 128/x²
x⁴ = 64
x = 2 cm
Note that we need to check that this critical point gives us a minimum surface area. We can do this by checking the second derivative:
d²S/dx² = 2 + 256/x³
d²S/dx² at x = 2 is positive,
indicating that this critical point gives us a minimum surface area.
Substituting x = 2 in the equation for S, we get:
S = 2² + 128/2 = 4 + 64 = 68
Therefore, the minimum surface area of the box is 68 cm², which corresponds to option (d).
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Bob and Anna are planning to meet for lunch at Sally's Restaurant, but they forgot to schedule a time. Bob and Anna are each going to randomly choose from either 1\text{ p. M. }1 p. M. 1, start text, space, p, point, m, point, end text, 2\text{ p. M. }2 p. M. 2, start text, space, p, point, m, point, end text, 3\text{ p. M. }3 p. M. 3, start text, space, p, point, m, point, end text, or 4\text{ p. M. }4 p. M. 4, start text, space, p, point, m, point, end text to show up at Sally's Restaurant. They must both choose exactly the same time in order to meet. Bob has a "buy one entree, get one entree free" coupon that he can only use if he meets up with Anna. If he successfully meets with Anna, Bob's lunch will cost him \$5$5dollar sign, 5. If they do not meet, Bob's lunch will cost him \$10$10dollar sign, 10. What is the expected cost of Bob's lunch?
The expected cost of Bob's lunch is $9.69, rounded to the nearest cent.
To figure out the probability of Bob and Anna successfully meeting up, we need to use a unitary method. The probability of Bob choosing a specific time is 1/4, and the probability of Anna choosing the same time is also 1/4. Since they both need to choose the same time, we can multiply their individual probabilities to find the probability of them meeting up:
1/4 x 1/4 = 1/16
This means that the probability of Bob and Anna successfully meeting up is 1/16 or 0.0625.
Now we can use this probability to find the expected cost of Bob's lunch. If Bob and Anna meet up, Bob will get a discount on his lunch and pay $5. If they don't meet up, he'll have to pay the full price of $10. So the expected cost of Bob's lunch is:
(Probability of meeting up x Cost if they meet) + (Probability of not meeting up x Cost if they don't meet)
(1/16 x $5) + (15/16 x $10) = $0.3125 + $9.375 = $9.69
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3. now consider equations of the form x-a = vbx+c , where a, b, and c are all positive integers and b > 1.
(a) create an equation of this form that has 7 as a solution and an extraneous solution. give the
extraneous solution.
(b) what must be true about the value of bx+c to ensure that there is a real number solution to the
equation? explain.
(a)The equation x - 7 = 2x - 14 + 1 has 7 as a solution (when v = 2) and an extraneous solution of -8.
(b) To have a real number solution, the value of bx + c should be nonzero.
(a) To create an equation of the form x - a = vb(x) + c with 7 as a solution and an extraneous solution, we can start with the equation:
x - 7 = v * (x - 7) + 1
Simplifying this equation, we have:
x - 7 = vx - 7v + 1
Rearranging the terms, we get:
x - vx = 7v - 6
Now, let's assume v = 2. Substituting this value, the equation becomes:
x - 2x = 14 - 6
Simplifying further, we have:
-x = 8
Multiplying both sides by -1, we get:
x = -8
(b) To ensure that there is a real number solution to the equation x - a = vb(x) + c, it must be true that vb(x) + c does not result in division by zero or any other mathematical operation that would lead to an undefined or imaginary number. This implies that bx + c should not be equal to zero, as dividing by zero is undefined.
Therefore, to have a real number solution, the value of bx + c should be nonzero.
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Yvette cuts a hole from a rectangular panel to make a window. She wants to determine how
much of the panel is left after she cuts the hole. She writes:
(fraction left)
(area of panel) - (area of hole)
(area of panel)
If the panel is 3 feet by 2 feet, and the hole is 1 foot by foot, what is the fraction left?
The area of the panel left after she cuts the hole is 5.215 ft²
Given that a circular hole of 1 foot by foot has been cut out of a rectangular panel of 3 feet by 2 feet,
We need to find the area of the remaining part after the cutting of the hole,
So, we will find the same by subtracting the area of the hole from the area of the panel.
So, area of the hole = π×radius² = 3.14×0.5² = 0.785 ft²
Area of the panel = length × width = 3 × 2 = 6 ft²
Area remaining part = 6-0.785 = 5.215 ft²
Hence the area of the panel left after she cuts the hole is 5.215 ft²
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