Answer:
Step-by-step explanation:
The best-supported statement by the information in the table is:
C. The fee at Bank K will always be greater than the fee at Bank F.
This is because for any given average daily balance, Bank K has a higher fee than Bank F. For example, for an average daily balance of less than $500, Bank F charges $8 while Bank K charges $12. Similarly, for an average daily balance between $500 and $750, Bank F charges no fee while Bank K charges $12. Therefore, the fee at Bank K will always be greater than the fee at Bank F.
Engineers built an arch bridge across a river. The arch bridge
makes a parabola shape that has the equation
y=-0. 1(x – 5)2 + 12, where x and y are measured in
meters. If the bridge makes contact with both banks at a height
of 4 meters, how long is the distance between the two banks
of the river where the bridge is? Round your answer to the
nearest whole number.
To find the distance between the two banks of the river where the arch bridge is, we first need to determine the points where the bridge makes contact with the banks at a height of 4 meters.
We are given the parabola shape of the arch with the equation y = -0.1(x - 5)^2 + 12.
1. Set the height y equal to 4 meters:
4 = -0.1(x - 5)^2 + 12
2. Subtract 12 from both sides:
-8 = -0.1(x - 5)^2
3. Divide both sides by -0.1:
80 = (x - 5)^2
4. Take the square root of both sides:
sqrt(80) = x - 5
5. Now, we find the two x-values where the bridge contacts the banks:
x1 = sqrt(80) + 5
x2 = -sqrt(80) + 5
6. Calculate the distance between the two banks by subtracting x2 from x1:
Distance = x1 - x2 = (sqrt(80) + 5) - (-sqrt(80) + 5)
7. Simplify the expression:
Distance = 2 * sqrt(80)
8. Round your answer to the nearest whole number:
Distance ≈ 18 meters
The distance between the two banks of the river where the arch bridge is, is approximately 18 meters.
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the produce manager at the local pig & whistle grocery store must determine how many pounds of
bananas to order weekly. based upon past experience, the demand for bananas is expected to be 100,
150, 200, or 250 pounds with the following probabilities: 100lbs 0.20; 150lbs 0.25, 200lbs 0.35, 250lbs 0.20.
the bananas cost the store $.45 per pound and are sold for $.085 per pound. any unsold bananas at the
end of each week are sold to a local zoo for $.30 per pound. use your knowledge of decision analysis to
model and solve this problem in order to recommend how many pounds of bananas the manager should
order each week
As per the probability, the expected demand for bananas per week is 182.5 pounds.
To model this problem, we can use decision analysis, which involves identifying the possible outcomes, assigning probabilities to each outcome, and calculating the expected value of each decision.
In this case, the possible outcomes are the demand for bananas, which can be 100, 150, 200, or 250 pounds per week. The probabilities of each demand level are given as 0.20, 0.25, 0.35, and 0.20, respectively.
Let X denote the demand for bananas in pounds. Then, the expected demand for bananas, denoted as E(X), can be calculated as follows:
E(X) = 100(0.20) + 150(0.25) + 200(0.35) + 250(0.20) = 182.5
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Hanif is 14 years old. he plans to do up to 70% training intensity. while jogging, hanif took his resting pulse rate for two days in a row. so hanif found that his resting heart rate was 76 beats per minute. what is hanif's training pulse rate?
Hanif's training pulse rate at 70% intensity is approximately 142 beats per minute.
To find Hanif's training pulse rate at 70% intensity, we first need to calculate his maximum heart rate (MHR) using the formula:
MHR = 220 - age
Substituting Hanif's age, we get:
MHR = 220 - 14 = 206
Next, we need to calculate Hanif's target heart rate (THR) range at 70% intensity. This range is between 70% and 85% of his MHR. To calculate the lower end of the range, we multiply his MHR by 0.7:
THR lower = 0.7 × MHR = 0.7 × 206 = 144.2 (rounded to one decimal place)
To calculate the upper end of the range, we multiply his MHR by 0.85:
THR upper = 0.85 × MHR = 0.85 × 206 = 175.1 (rounded to one decimal place)
So Hanif's target heart rate range at 70% intensity is between 144.2 and 175.1 beats per minute.
To find his training pulse rate, we add his resting pulse rate (76 beats per minute) to the percentage of his target heart rate range which corresponds to 70% intensity. This is given by:
Training pulse rate = resting pulse rate + (0.7 × (THR upper - resting pulse rate))
Substituting the values we calculated, we get:
Training pulse rate = 76 + (0.7 × (175.1 - 76)) ≈ 142
Therefore, Hanif's training pulse rate at 70% intensity is approximately 142 beats per minute.
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Given the following information about two triangles, triangle CAT and triangle DOG:
Which postulate can be used to prove triangle CAT and triangle DOG are congruent?
SSS Postulate
SAS Postulate
SSA Postulate
ASA Postulate
AAS Postulate
Choose all that apply
To determine which postulate can be used to prove that triangle CAT and triangle DOG are congruent, we need information about the side lengths and angles of each triangle. Unfortunately, the given information about triangles CAT and DOG is not provided in your question.
However, I can briefly explain each of the mentioned postulates to help you understand how they can be applied to prove congruence:
1. SSS (Side-Side-Side) Postulate: If all three sides of one triangle are equal in length to the corresponding sides of another triangle, the triangles are congruent.
2. SAS (Side-Angle-Side) Postulate: If two sides and the included angle of one triangle are equal to the corresponding sides and included angle of another triangle, the triangles are congruent.
3. SSA (Side-Side-Angle) Postulate: This is not a valid postulate for proving triangle congruence.
4. ASA (Angle-Side-Angle) Postulate: If two angles and the included side of one triangle are equal to the corresponding angles and included side of another triangle, the triangles are congruent.
5. AAS (Angle-Angle-Side) Postulate: If two angles and a non-included side of one triangle are equal to the corresponding angles and non-included side of another triangle, the triangles are congruent.
Once you have the necessary information about triangles CAT and DOG, you can apply the appropriate postulate to prove their congruence.
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Boris needs to read 2 novels each month.let n be the number of novels boris needs to read in m months.write an equation relating n to m. then use this equation to find the number of novels boris needs to read in 17 months.equation:number of novels in 17 months: i novels
Solving the equation, Boris needs to read 34 novels in 17 months.
Given that Boris needs to read 2 novels each month.
To relate the number of novels Boris needs to read (n) to the number of months he has to read them (m), we can use the equation:
n = 2m
This equation states that the number of novels (n) is equal to two times the number of months (m) since Boris needs to read 2 novels each month.
Now, to find the number of novels Boris needs to read in 17 months, we can substitute m = 17 into the equation:
n = 2m
n = 2(17)
n = 34
Therefore, Boris needs to read 34 novels in 17 months to meet his goal of reading 2 novels each month.
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(4y + z)^2 what is the a value and what is the b value
Answer:
a = 16
b = 8z
Step-by-step explanation:
Expanding the given expression, we get:
(4y + z)^2 = (4y + z) × (4y + z)
= 16y^2 + 8yz + z^2
Comparing this with the general form of a quadratic expression, ax^2 + bx + c, we can see that:
a = 16
b = 8z
Therefore, the value of a is 16 and the value of b is 8z.
In a scale model of a boat 1 inch represents 5 feet
The height of the real boat is 3 inches and length of the boat is 45 feet
What is Unit of Measurement?
A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity.
In a scale model of a boat 1 inch represents 5 feet
1 inch = 5 feet
The height of the real boat is 15 feet
We have to find in inches
1/5=x/15
x=3 inches
So height of the real boat is 3 inches
The length of the boat is 9 inches
We have to find in feet
1/5 = 9/x
x=45 feet
Hence, the height of the real boat is 3 inches and length of the boat is 45 feet
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help pls rlly fast i will give good points
Answer: less than
Step-by-step explanation:
The unit in a volume are always ________.
Answer:
The unit in a volume are always cubic meter
Step-by-step explanation:
Chain of Thought Reasoning: Volume is a three dimensional space, which is measured in three dimensions (length, width, and depth). The SI unit for length is meters, so the SI unit for volume is cubic meters (m^3). I hope this helps you
the human body contains about bacteria.
the human body contains 1 × 1012 about genes. the number of bacteria contained in the human body is how 4 × 104 many times as great as the number of genes contained in the human body?
explain how you arrived at your answer.
The number of bacteria contained in the human body is 40 times as great as the number of genes contained in the human body.
To find out how many times greater the number of bacteria in the human body is than the number of genes, we need to divide the number of bacteria by the number of genes:
4 × 10^13 (number of bacteria) ÷ 1 × 10^12 (number of genes)
= 40
Therefore, the number of bacteria contained in the human body is 40 times as great as the number of genes contained in the human body.
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1. If Cos A = 1/4, what is Sin B?
2. Simplify Sin A + Cos B ÷ 2
3. If Tan B = 1/5, what is Tan A? Which angle is bigger, angle A or angle B?
Please answer all 3
a. Sin A = opposite/hypotenuse = √15/4
b. (Sin A + Cos B) ÷ 2 = (Sin A + Cos B)/2
c. Tan A = 1/5
The both angle could be of same size
How do we calculate?Since Cos A = adjacent/hypotenuse,
Applying the Pythagorean theorem,
we can find the opposite side:
opposite^2 + 1^2 = 4^2
opposite^2 = 16 - 1
opposite = √15
Now we can find the value of Sin A:
Sin A = opposite/hypotenuse = √15/4
b.
Sin A + Cos B = (Sin A) + (Cos B)
(Sin A + Cos B) ÷ 2 = (Sin A + Cos B)/2
c. Since Tan B = opposite/adjacent,
we use Pythagorean theorem to find the hypotenuse:
hypotenuse^2 = 5^2 + 1^2
hypotenuse = √26
Tan A = opposite/adjacent = 1/5
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Of 125 students attending a college orientation session, 18 are criminal justice majors. If 4 students at the orientation are selected at random, determine the probability that each of the 4 is a criminal justice major. Assume that selection is to be done without replacement Set up the problem as if it were to be solved, but do not solve. P(4 criminal justice majors selected) N
The probability that each of the 4 is a criminal justice major is equal to 0.0003 (rounded to four decimal places).
The probability of selecting 4 criminal justice majors from a group of 125 students, without replacement,
Using the hypergeometric probability distribution.
Start by calculating the total number of ways to choose 4 students from the group of 125.
C(125,4) = 125! / (4! (125-4)!)
= 125 x 124 x 123 x 122 / (4 x 3 x 2 x 1)
= 9,691,375
Next, calculate the number of ways to choose 4 criminal justice majors from the group of 18.
C(18,4) = 18! / (4! (18-4)!)
= 18 x 17 x 16 x 15 / (4 x 3 x 2 x 1)
= 3060
Finally,
Probability of selecting 4 criminal justice majors
= number of ways to choose 4 criminal justice majors / total number of ways to choose 4 students:
P(4 criminal justice majors selected) = C(18,4) / C(125,4)
⇒P(4 criminal justice majors selected) = 3060 / 9,691,375
= 0.0003157
Therefore, probability that each of the 4 students selected at random from the group of 125 students are criminal justice majors, without replacement is 0.0003 (rounded to four decimal places).
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A circle is placed in a square with a side length of 8 m, as shown below. Find the area of the shaded region.
Use the value 3.14 for pi, and do not round your answer. Be sure to include the correct unit in your answer.
The area of the shaded region is the expression 16(4 - π) square metres.
How to evaluate for the shaded regionThe shaded region is the remaining area in the square which is outside the circle, so it is derived by subtracting the area of the circle from the area of the square as follows:
area of the square = 8 m × 8 m
area of the square = 64 m²
area of the circle = π × 4 m × 4 m
area of the circle = 16π m²
area of the shaded region = 64 m² - 16π m²
area of the shaded region = 16(4 - π) m²
Therefore, the area of the shaded region is the expression 16(4 - π) square metres.
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If arc wvx=(13x+9) and angle wxz=(5x+36) find angle wxy
Angle WXY is vertical to angle WXZ, they are equal, therefore, angle WXY is also 141 degrees.
How to find the angle WXY using information about the arc WVX and angle WXZ in a circle?To find the angle WXY, we need to use the properties of angles formed by intersecting chords in a circle. The angles formed by intersecting chords are related to the arcs intercepted by those chords.
Given that the arc WVX is equal to (13x + 9) and the angle WXZ is equal to (5x + 36), we can set up an equation:
Angle WXZ = [tex]\frac{1}{2}[/tex] * Arc WVX
(5x + 36) = [tex]\frac{1}{2}[/tex] * (13x + 9)
To solve for x, we'll multiply both sides of the equation by 2 to eliminate the fraction:
2(5x + 36) = 13x + 9
10x + 72 = 13x + 9
Subtracting 10x and 9 from both sides, we get:
72 - 9 = 13x - 10x
63 = 3x
Dividing both sides by 3, we find:
x = 21
Now that we have the value of x, we can substitute it back into the equation for the angle WXZ to find its value:
Angle WXZ = 5x + 36 = 5(21) + 36 = 105 + 36 = 141 degrees
Since angle WXY is vertical to angle WXZ, they are equal. Therefore, the angle WXY is also 141 degrees.
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What is the measure of an angle that goes through 2/8 of a circle?
The measure of an angle that goes through 2/8 of a circle is 90°
A circle is a 2-dimensional shape that is round in shape it is equidistant from the center.
A circle has a total angle of 360°
That is the whole complete angle of the circle = 360°
The 2/8 th of the complete angle of the circle = 360 * 2/8
= 360 * 1/4
= 360/4
=90°
Thus, the 90° of the circle is given as 2/8th of an angle of the circle or we can say that the quarter angle of a circle comes out to 90°.
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Determine the values of a, b, and c in the following matrix equation.
4 a
[34]
30
52
[3
2 b5
1 8
LC
a.
b.
C.
54=
16.
The values of a, b, and c in the matrix equation are a = 3, b = 1 and c = 8
Determining the values in the matrix equation.To determine the values of a, b, and c in the matrix equation:
4 a + 3 0 = 7 3
3 4 5 b c 5
By addition operation, we have
a + 0 = 3
3 + 5 = c
4 + b = 5
When the equations evaluated, we have
a = 3
c = 8
b = 1
The above are the values of a b and c
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Complete question
Determine the values of a, b, and c in the following matrix equation.
4 a + 3 0 = 7 3
3 4 5 b c 5
Angle BCD is a right triangle. The length of the hypotenuse is 18 centimeters. The length of one of the legs is 13 centimeters. What is the length of the other leg? Enter your answer, as a decimal rounded to the nearest tenth, in the box.
Answer:12.4
Step-by-step explanation:
18^2-13^2=155
Square root of 155 to the nearest tenth is 12.4
If (x+1/x)² = 3, find x³+1/x³.
The value of the algebraic expression from the given parameters is:
x³ + 1/x³ = 0
How to solve Algebraic Expressions?The given problem is simply based on the expansion.
In expansion, what we do is that we expand the mathematical terms by first of all removing all the brackets that are in that mathematical expression.
In expanding a mathematical expression, what we have to do is that we have to make use some of the identities that can be gotten by multiplying one binomial with the another one and then this type of identities are called as Standard Identities.
For example:
(x + a)(x + b) = x² + (a + b)x + ab
Thus:
(x + 1/x)² = 3
x + 1/x = √3
(x+1/x)³ = x³ + (1/x)³ + 3(x)(1/x) (x + 1/x)
√3³ = x³ + 1/x³ + 3(√3)
x³ + 1/x³ = 3√3 - 3√3
x³ + 1/x³ = 0
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Solve the following problems:
given: circle k(o), diameter us, mru=50°, mut=30°
find: m
The measure of angle M is 20°.
To solve the problem, we need to find the measure of angle M, given the information about Circle K with center O, diameter US, angle MRU = 50°, and angle MUT = 30°.
Step 1: Determine the relationship between angles MRU and MUT.
Since MRU and MUT are both inscribed angles in Circle K, they share the same intercepted arc, which is arc MU.
Step 2: Calculate the measure of arc MU.
The measure of an intercepted arc is twice the measure of the inscribed angle. Since angle MRU = 50°, the measure of arc MU will be 2 * 50° = 100°.
Step 3: Find the measure of angle M.
We know that angle MUT = 30°, and the measure of an intercepted arc is twice the measure of the inscribed angle. Therefore, the measure of arc MT = 2 * 30° = 60°. Now, since arc MU = 100°, we can determine the measure of arc MS (arc MS = arc MU - arc MT) which is 100° - 60° = 40°.
Step 4: Calculate the measure of angle M.
Finally, the measure of angle M can be found using the intercepted arc MS. Since the measure of an intercepted arc is twice the measure of the inscribed angle, angle M = 1/2 * arc MS = 1/2 * 40° = 20°.
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Pls help with these two equations. Pls
Answer:
11. x = 16
12. x = 26
Step-by-step explanation:
11. ∠1 + ∠2 = 90°
∠1 = 42°
∠2 = 90° - 42° = 48°
3x = 48
x = 16
12. ∠C + ∠D = 180°
∠C = 128°
∠D = 180° - 128° = 52°
2x = 52
x = 26
2.
A painting company will paint this wall of a building. The owner gives them the following dimensions:
Window A is 6-ft x 5
6 ft x 5 ft.
Window Bis 3 ft x 4 ft.
Window Cis 9ft?
Door D is 4 ft x 8 ft.
33 ft
What is the area of the painted part of
the wall?
577 square feet is the area of the painted part of the wall.
To calculate the area of the painted part of the wall, you'll first need to find the total area of the wall and then subtract the areas of the windows and door. Let's assume the wall has a height of 33 ft and a width of 20 ft (since the other dimensions aren't provided).
1. Calculate the total area of the wall:
Area of wall = Height x Width = 33 ft x 20 ft = 660 sq ft
2. Calculate the areas of the windows and door:
Window A = 6 ft x 5 ft = 30 sq ft
Window B = 3 ft x 4 ft = 12 sq ft
Window C = 9 sq ft (already provided)
Door D = 4 ft x 8 ft = 32 sq ft
3. Subtract the areas of the windows and door from the total wall area:
Painted area = Wall area - (Window A + Window B + Window C + Door D) = 660 sq ft - (30 sq ft + 12 sq ft + 9 sq ft + 32 sq ft) = 660 sq ft - 83 sq ft = 577 sq ft
The area of the painted part of the wall is 577 square feet.
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A musician charges C (x) = 64x + 20,000 where x is the total - number of attendees at the concert. The venue charges $80 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?
The venue breaks even when 1,250 people buy tickets, and the total value of tickets sold at that point is $100,000.
To find the break-even point for the venue, we need to set the musician's charges (C(x) = 64x + 20,000) equal to the venue's earnings from ticket sales ($80 per ticket). Hence,
1. Set the musician's charges equal to the venue's earnings:
64x + 20,000 = 80x
2. Subtract 64x from both sides:
20,000 = 16x
3. Divide both sides by 16:
x = 1,250
At the break-even point, 1,250 people need to buy tickets. To find the value of the total tickets sold at this point:
1. Multiply the number of attendees (x) by the ticket price:
Total ticket sales = x * ticket price
2. Substitute the values:
Total ticket sales = 1,250 * $80
3. Calculate the total ticket sales:
Total ticket sales = $100,000
So, the breaks even point is 1,250 people buying tickets, and corresponding total value of tickets sold is $100,000.
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Look at the image below.
What is the area of the triangle?
Answer:
60
Step-by-step explanation:
Area of a triangle: 1/2(bh)
Base = 12
Height = 10
1/2(12*10)
1/2(120) = 60
The roof of a building is in the shape of a hyperbola, y^2-x^2=38, where x and y are in meters. Determine the height of the outside. The distance between the center of the hyperbola and the walls is 3m.
a) -29. 1
b) 47. 3
c) 35. 2
d) 6. 9
The height of the outside is given as 17.44 meters
How to solveThe equation of hyperbola is :
[tex]y^2 - x^2 = 38[/tex]
=>[tex]y^2/38 - x^2/38 = 1[/tex]
(of the form [tex]y^2/a^2 - x^2/b^2 = 1[/tex] and transverse axis is y-axis.)
Here, [tex]a^2 = b^2 = 38[/tex]
[tex]c^2 = a^2 + b^2 = 38+38 = 76[/tex]
( a is the distance of vertices from the center and c is the distance of foci from the center.)
Distance between walls = 2 a = [tex]2*\sqrt(38) = 12.33[/tex] meters at the center
and = [tex]2c = 2*\sqrt(76) = 17.44[/tex] meters at the end when the line joining
end points of the wall on one side is through the foci point.
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The value of lim a^x-x^a/x^x-a^a is
lim (1 - a^(1-a)) / (ln(a)) as x -> a This is the value of the given limit.
To find the value of the given limit, which can be represented as lim (a^x - x^a) / (x^x - a^a) as x approaches 'a', you can apply L'Hôpital's Rule, which states that if the limit of the ratio of the derivatives of two functions exists, then the limit of the original functions is equal to that limit.
First, differentiate the numerator and denominator with respect to x:
Numerator: d(a^x - x^a) / dx = a^x * ln(a) - a * x^(a-1)
Denominator: d(x^x - a^a) / dx = x^x * ln(x)
Now, we can find the limit of the ratio of the derivatives as x approaches 'a':
lim (a^x * ln(a) - a * x^(a-1)) / (x^x * ln(x)) as x -> a
After substituting 'a' for 'x' in the limit:
lim (a^a * ln(a) - a * a^(a-1)) / (a^a * ln(a)) as x -> a
Now, cancel out the common term a^a * ln(a):
lim (1 - a^(1-a)) / (ln(a)) as x -> a
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If Wendy is 63 inches tall and her hand is 6. 5 inches long, what is the residual if the formula to predict h, height in inches, from x, hand length in inches?
If the formula predicted Wendy's height to be 65 inches based on her hand length of 6.5 inches, the residual would be -2 inches
A residual is the difference between the predicted value of a variable (in this case, height) and the actual value of that variable. Residuals are often used in statistical analysis to assess the accuracy of a prediction or model.
In this case, if we were given the formula for predicting height from hand length, we could use it to predict Wendy's height and compare that to her actual height of 63 inches. The residual would be the difference between the predicted height and her actual height. If the prediction overestimated her height, the residual would be negative. If it underestimated her height, the residual would be positive.
For example, if the formula predicted Wendy's height to be 65 inches based on her hand length of 6.5 inches, the residual would be -2 inches (predicted height minus actual height). If the formula predicted her height to be 61 inches, the residual would be +2 inches.
Overall, residuals are a useful tool for assessing the accuracy of predictions or models, but the specific calculation of a residual depends on the formula being used to make the prediction.
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Write a function to describe the following scenario.
Jonathan is selling his old trading cards.
Each customer that buys gets the first
box they purchase for $10, and each
additional box for only $5.
y = [?]x + [?]
Answer:
y = 5x + 5
Step-by-step explanation:
If x is the number of boxes sold and y is the cost
The first box costs $10
Each additional box costs $5.
If the total number of boxes sold is x, then after selling the first box for $10, there will be x - 1 boxes left to be sold
The cost of x -1 boxes at $5 per box = 5(x - 1) = 5x - 5
Therefore for a total of x boxes sold the total cost, y in dollars is
y = 10 (for the first box) + 5x - 5 (for the remaining x - 1 boxes)
= 10 + 5x - 5
= 5 + 5x
which in standard form is written as
y = 5x + 5
We can verify our equation using specific numbers for x
For x = 1
y = 5 + 5(1) = 10 ; since only one box has been sold, the cost is fixed at $10
For x = 2 y = 5 + 5(2) = 5 + 10 = $15
This works out to since first box is sold at $10 and the second box at $5
Leave it to you to work out for other numbers
George says his bicycle has a mass of 15 grams. If he takes the front wheel off what could be the mass?
If George's bicycle has a mass of 15 grams, then it is highly unlikely that he has stated the correct mass, as 15 grams is an extremely low mass for a bicycle.
If George`s bicycle weighs 15 grams, what would be the resulting weight of it if he removes the front wheel?Determine if the givens mass of 15 grams is reasonable for a bicycle.A typical bicycle weighs anywhere from 7 to 15 kilograms. It is highly unlikely that a bicycle would weigh only 15 grams, as this is much lighter than the lightest bicycle ever made.
Therefore, it is reasonable to assume that George made a mistake and meant to say 15 kilograms instead of grams.
Calculate the mass of the bicycle without the front wheel.Assuming the mass of the bicycle is 15 kilograms, removing the front wheel will decrease the mass slightly, but not by a significant amount.
The front wheel typically accounts for around 1-2 kilograms of the total mass of the bicycle, so removing it would leave a mass of approximately 14 kilograms.
However, assuming he made a mistake and meant to say 15 kilograms, then the mass of the bicycle without the front wheel would be approximately 14 kilograms.
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A number greater than 9 is called cute if when we add the product of the digits to
the sum of the digits, the result is the original number. For example 29 is cute since
2 + 9 + 2 × 9 = 29, but 513 isn’t cute since 5 + 1 + 3 + 5 × 1 × 3 6= 513. How many
cute numbers are there?
There are 6 cute numbers in total which are 14, 19, 49, 55, 79, 85.To find the cute numbers, we need to check all numbers greater than 9 and see if they satisfy the cute condition.
Let's start by analyzing the digits of a number. Suppose the number has two digits, x and y. The cute condition requires:
x + y + xy = 10x + y
Rearranging this equation, we get:
xy - 9x = y - x
xy - x - y = -9x
(x - 1)(y - 1) = 9x - 1
For a number to be cute, the right-hand side of the equation must be divisible by the left-hand side. Since 9x - 1 is odd, the left-hand side must also be odd, which means one of the factors (x - 1) or (y - 1) must be odd and the other even.
We can now check all possible pairs of (x,y) that satisfy this condition. We find that the cute numbers are:
14, 19, 49, 55, 79, 85. Therfore, there are total 6 cute numbers.
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how do you know when to use the Rule of Sum or Fundamental Counting Principle for probability problems?
If the events are exclusive, use the Rule of Sum. If the events are independent, use the Fundamental Counting Principle.
The Rule of Sum and the Fundamental Counting Principle are two common methods used in probability to calculate the total number of possible outcomes. Knowing which method to use depends on the nature of the problem and the type of events involved.
The Rule of Sum is used when we have two or more exclusive events. This means that only one of the events can happen at a time. For example, when rolling a die, the events of rolling a 2 or a 4 are exclusive because you cannot roll both at the same time.
The rule of sum states that the total number of possible outcomes is the sum of the number of outcomes for each event.
On the other hand, the Fundamental Counting Principle is used when we have a sequence of events that are independent of each other. This means that the outcome of one event does not affect the outcome of the next event.
For example, when flipping a coin, the outcome of the first flip does not affect the outcome of the second flip. The fundamental counting principle states that the total number of possible outcomes is the product of the number of outcomes for each event.
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