2023 1-100 math challenge answers

The result of the calculation was $93.41 per week.

To find the average amount of money spent on food by American college students per week, a researcher conducted a survey of 300 college students across the U.S. The researcher collected data on the amount of money spent by each student on food per week and then calculated the average using the formula: Average = (sum of all the scores)/(number of scores). The researcher then added up the money spent by all 300 students and divided it by 300 to get the average amount of money spent on food per week by American college students. The result of the calculation was $93.41 per week.

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The number of tricycles in the preschool play yard is given as follows:

10.

The variables for the system of equations are given as follows:

There are 30 seats, hence:

x + y = 30

x = 30 - y.

There are 70 wheels, hence:

2x + 3y = 70.

Replacing x = 30 - y on the second equation, the number of tricycles is obtained as follows:

2(30 - y) + 3y = 70.

y = 10.

A preschool playground has both bicycles and tricycles. There is a total of 30 seats and 70 wheels. What is the number of tricycles?

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The given expression will not have real factors. The correct option is A.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the quadratic expression is  -2x² - 11x - 20. The expression will be solved as below:-

E = -2x² - 11x - 20

Apply the quadratic factorization formula,

x = [ -b ± √( b² - 4ac ) ] / 2a

x = [ 11 ± √ ( 121 - ( 4 x - 2 x - 20 ) ] / ( 2 x - 2 )

x = [ 11 ± 39i } / -4

The roots of the quadratic equation are not real.

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Answer:

no real factors

Step-by-step explanation:

i took the test

The segments MN, BN, and MD are the sides of a triangle.

Let AM intersect BC at point P and let AN intersect CD at point Q. Spin A about PQ to obtain E where AE = AB. We want to prove that EMN is the required triangle. For that, it is adequate to show that EM = MD and EN = BN.

Let h be the distance of A from the BCD. Then the distance of A from the line PQ is minimum h, so point E exists provided that the side length maximum is 2h. It is √(3/2) h.

We have by construction AD = AE and EP = AP.  Also, since P lies on the line BC, then PA = PD. Hence PA = PA, PD = PE, AD = AE, so PAD and PAE are congruent triangles. Hence E can also be acquired by spinning D about AP. Since M lies on AP, it gives EM = DM.

Similarly, with B and D interchanged and M and N interchanged, we get EN = BN.

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Answer: 9

Step-by-step explanation:

21 - 12 = 9

a) One possible basis for U is {(1/6,1,0,0,0), (0,0,1,-2,3)}.

b) A basis of C5 is {(1/6,1,0,0,0), (0,0,1,-2,3), (1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.

c) The basis of W is {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.

a) To find a basis of U, we can use the two given equations to solve for two of the variables in terms of the others. From the first equation, we have z1 = z2/6. Substituting this into the second equation, we get z3 + 2z4 + 3z5 = 0. Therefore, one possible basis for U is {(1/6,1,0,0,0), (0,0,1,-2,3)}.

b) To extend this basis to a basis of C5, we need to add three more vectors that are not in the subspace U. These vectors can be any linearly independent vectors that are not in U. One possible set of additional vectors is {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}. Therefore, a basis of C5 is {(1/6,1,0,0,0), (0,0,1,-2,3), (1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.

c) To find a subspace W of C5 such that C5 = U direct sum W, we can take the complement of U in C5. The complement of U is the set of all vectors in C5 that are not in U. Since we have already found a basis for C5, we can simply take the subset of basis vectors that are not in U to form a basis for W. In this case, the basis of W is {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.

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It is possible for Angela to have spent 80 minutes playing soccer if she plays for a total of exactly 125 minutes and plays soccer for 45 minutes longer than she plays golf because 80+45 adds up to 125.

A linear equation is an equation that has the variable of the highest power of 1. The standard form of a linear equation is of the form Ax + B = 0.

Angela plays soccer and golf for a total of 125 minutes every day. She plays soccer for 45 minutes longer than she plays golf.

Part A:  A pair of linear equations to show the relationship between the number of minutes Angela plays soccer (x) and the number of minutes she plays golf (y) every day.

x = y + 45

Part B: A pair of linear equations to show how much Angela spends playing golf every day.

125 - 45 = 80,

80/2 = 40

Part C: Yes, it is possible for Angela to have spent 80 minutes playing soccer if she plays for a total of exactly 125 minutes and plays soccer for 45 minutes longer than she plays golf because 80+45 adds up to 125.

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The size of the crowd is 7392 people.

The ratio is a numerical relationship between two values that demonstrates how frequently one value contains or is contained within another.

20 people fit comfortably in a 10 feet by 10 feet area.

So,

the ratio of the number of people per unit area = 20/(10 × 10) = 20/100 = 1/5

The area, A, occupied by the crowd = 7 feet × 1 mile × 1

1 mile = 5280 feet

A = 7 feet × 5280 feet × 1 = 36960 feet².

Now, number of people = 36960 x 1/5

= 7392 people.

Therefore, the required value is 7392 people.

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The expression 18-x represents the length of the candle that has been burned. As x increases (representing more inches being burned), 18-x decreases (representing less inches remaining).

b) x represents the inches burned from the candle, y represents the inches of candle remaining, 18 represents the initial height of the candle before it is lit, and 18-x represents the length of the candle that has been burned.

c) y = 18 - x (The height of the candle remaining is 18 inches minus the inches that have been burned, represented by x)

d) It is not possible to write the formula in part (c) without using a constant quantity. Constant quantities, such as the initial height of the candle (18 inches), help to define the relationship between the two varying quantities (x and y) and provide a starting point for the calculation.

Without the constant quantity, it would not be clear what the relationship is between x and y.

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The values of each type of energy the object has:

Kinetic energy is 67.5J and Gravitational Potential Energy is 435 J.

KE: Mass= 15 kg Velocity = 3 m/s Velocity2= 9 m²/s²

KE = ½ (15) (9) = 67.5 J

GPE: Mass= 15 kg Gravity (g)=9.8 m/s² Height = 2.5 m

GPE= (15) (9.8) (2.5) = 367.5 J

KE+GPE= 67.5 + 367.5 = 435 J

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The exact circumference for the circles are;

5. C = 94.26mi

6. C = 879.76in

7. C = 91.118in

8. C = 62.84m

9. C = 12.568m

10. C = 29.535cm

The circumference of a circle is its boundary or the length of the complete arc of a circle. The circumference of a circle is the perimeter of the circle. It is the total length of the boundary of the circle. The circumference of a circle is the product of the constant π and the diameter of the circle.

Circumference = πd

or

Circumference = 2πr

where π = 3.142 or 22/7

r = radius

d = diameter

5. C = 2πr

C = 2 x 3.142 x 15

C = 94.26mi

6. C = 2πr

C = 2 x 3.142 x 140

C = 879.76in

7. C = πd

C = 3.142 x 29

C = 91.118in

8. C = 2πr

C = 2 x 3.142 x 10

C = 62.84m

9. C = 2πr

C = 2 x 3.142 x 2.1

C = 12.568m

10. C = πd

C = 3.142 x 9.4

C = 29.535cm

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The length of time it would have taken for both garden hoses to fill the pool would be 6.67 hours.

To solve this problem, we can use the formula:

1/rate = time

In the case of Joe's hose, the rate is 1/12 of the pool per hour. For his neighbor's hose, the rate is 1/15 of the pool per hour.

So, the combined rate is:

= (1/12) + (1/15)

= (5/60) + (4/60)

= 9/60 of the pool per hour

So, the time it will take to fill the pool using both hoses is:

= 1 / (9/60)

= 60/9 hours

= 6.7 hours

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The full question is:

Joe's garden hose fills the pool in 12 hours. His neighbor has thinner hose that fills the pool in 15 hours. How long will it take to fill the pool using both hoses?​

The minimum number of aunts that Shamar must have based on the information provided is 10.

It is known that uncles will take a photograph in groups of 10, while aunts will take a photograph in groups of 5. Based on this, the possible number for each are:

Uncles: 10, 20, 30, 40, 50

Aunts: 5, 10, 15, 20, 25

In this case, the minimum number is the LCM or Least Common Multiple and based on the previous list, this number is 10.

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Answer:

We get that x > -5

Step-by-step explanation:

1. Split up the inequality into two parts

We think of it as a > b > c and know that it is the same as a > b and b > c.

Apply this to the compound inequality

3x + 2 > 2x-3 and 2x - 3 > x - 12

2. Start solving 3x + 2 > 2x-3

Subtract 2x from both sides

x + 2 >-3

Subtract 2 from both sides

x > -5

3. Start solving 2x - 3 > x - 12

Subtract x from both sides

x - 3 > -12

Add 3 to both sides

x > -9

4. Finish

Now we have to think how we started with 3x + 2 > 2x-3 and 2x - 3 > x - 12

Since it is an and we need to identify the interval in which it is true on both inequalities which is when the x overlap. In this case is from x > -5.

All of the following are possible steps of scientific investigation except for

C. assumption of conclusions without prior experimentation or observation

All of the following are possible steps of scientific investigation except for.

A. the development of one or more working hypotheses or models to explain facts

B. development of observations and experiments to test the hypotheses

C. assumption of conclusions without prior experimentation or observation

D. the collection of scientific facts through observation and measurement

All of the following are possible steps of scientific investigation except for

C. assumption of conclusions without prior experimentation or observation

At the core of biology and other sciences lies a problem-solving approach called the scientific method. The scientific method has five basic steps, plus one feedback step:

Make an observation.

Ask a question.

Form a hypothesis, or testable explanation.

Make a prediction based on the hypothesis.

Test the prediction.

Iterate: use the results to make new hypotheses or predictions.

The scientific method is used in all sciences—including chemistry, physics, geology, and psychology. The scientists in these fields ask different questions and perform different tests. However, they use the same core approach to find answers that are logical and supported by evidence.

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A 3x3 matrix A, with nonzero entries, and a vector b in R3 such that b is not in the set spanned by the  of A= | 1 1 1 | | 1 | . |2 2 2 | b = | 2 |

In order for b to not be in the set spanned by the columns of A, we need to ensure that the system of equations Ax=b has no solution. This means that the system of equations needs to be inconsistent. We can achieve this by making sure that the determinant of A is not equal to zero, i.e. the matrix must be invertible.

The matrix given in option C satisfies this requirement. A = | 1 1 1 | | 2 2 2 | | 3 3 3 |

The corresponding vector b is | 1 | | 2 | | 3 |

Thus, the correct answer is C.

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The integral[tex]$\int_{-\infty}^{\infty} f(x)^2 dx$[/tex] expresses the volume of the solid, where f(x) is the side length of the square cross sections taken perpendicular to the x-axis in the region r.

The given integral [tex]$\int_{-\infty}^{\infty} f(x)^2 dx$[/tex] expresses the volume of the solid whose base is the region r. The integral is calculated by taking the cross sections of the solid perpendicular to the x-axis and consider them as squares. The side length of each square is denoted by f(x). The integral is calculated over the region r, which is the base of the solid. This integral is the sum of the areas of all the squares, which is equivalent to the volume of the solid. The integral is calculated by taking the lower limit of integration as[tex]$-\infty$[/tex] and the upper limit of integration as[tex]$\infty$[/tex]. Hence, the integral [tex]$\int_{-\infty}^{\infty} f(x)^2 dx$[/tex] expresses the volume of the solid with the base region r and square cross sections.

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All of the following contribute to the success of incumbent members of Congress n election campaigns EXCEPT Incumbents tend to understand national issues better than do their challengers.

The term "incumbent" has several meanings depending on the context, but it generally refers to someone who holds a particular position or office. The most common usage of this term is in the context of politics, where an incumbent is the current holder of a political office, such as a president, governor, or member of Congress. An incumbent candidate is the candidate who is currently holding a political office and is running for re-election.

In other fields, the term incumbent is used to describe someone who holds a particular job, position or role and is responsible for certain tasks or duties, for example an incumbent employee, an incumbent contractor, an incumbent provider, etc.

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Complete question not found

Answer: Let's call the width of the rectangular playing field "w" and the length "l". From the problem, we know that:

l = 3w + 4 (the length is 4 yards longer than triple the width)

We also know that the perimeter of a rectangle is the sum of the lengths of all four sides, so:

P = 2l + 2w

We are given that the perimeter is 416 yards, so we can substitute that into the perimeter equation:

416 = 2(3w + 4) + 2w

Simplifying and solving for w:

16 = 6w + 8

8 = 6w

w = 8/6 = 4/3

We can now substitute that back into the equation for the length:

l = 3w + 4

l = 3(4/3) + 4

l = 4 + 4

l = 8

So the dimensions of the rectangular playing field are 4/3 yards wide and 8 yards long.

Step-by-step explanation:

To be certain that the angles created form a square inside a circle, it is important to first understand the properties of a square and a circle.

A square is a four-sided polygon with all sides of equal length and all angles of 90 degrees. This means that if we draw a square inside a circle, the four corners of the square should be tangent to the circle.

A circle is a shape with all points on the circumference equidistant from the center. This means that if we draw a square inside a circle, the distance from the center of the circle to the midpoint of each side of the square should be equal to the radius of the circle.

By verifying that the above two conditions are met, we can be certain that the angles created form a square inside the circle. Additionally, we can also check that all the four angle of the square is 90 degrees and the opposite sides are parallel.

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The positive interval and the negative interval for the function given is (-∞, 3) and (3, ∞).

A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.

Given function is f(x) = [tex]-2\sqrt{x+1} +4[/tex]

Positive interval of the function is the values of x when function > 0.

To get the function > 0, the value [tex]2\sqrt{x+1}[/tex] < 4.

[tex]\sqrt{x+1}[/tex] < 2

x + 1 < 2²

x + 1 < 4

x < 3

Positive interval is (-∞, 3).

Negative interval of the function is the values of x when function < 0.

To get the function value < 0

[tex]-2\sqrt{x+1} +4[/tex] < 0

[tex]2\sqrt{x+1}[/tex] > 4

[tex]\sqrt{x+1}[/tex] > 2

x + 1 > 2²

x + 1 > 4

x > 3

Negative interval is (3, ∞).

Hence the positive interval of the function is (-∞, 3) and the negative interval is (3, ∞).

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Answer:

the total average monthly expenditure in 2020 was £2400.

Step-by-step explanation:

c) To find the total average monthly expenditure in 2020, we need to add up all the individual expenditures in each category. From the given data, we have:

Average monthly expenditure in 2020 = 600 + 500 + 400 + 300 + 200 + 100 = £2400

So, the total average monthly expenditure in 2020 was £2400.

The probability that the pirate hits the Captain's ship, but the Captain misses is 1/6.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

We have,

P(Captain Ben hit the ship) = 2/3

P(Captain Ben misses to hit the ship) = (1 - 2/3) = 1/3

P(Pirate hit the ship) = 1/3

P(Pirate misses to hit the ship) = (1 - 1/3) = 2/3

Now,

The probability that the pirate hits the Captain's ship, but the Captain misses.

= P(Pirate hit the ship) x P(Captain Ben misses to hit the ship)

= 1/3 x 1/3

= 1/6

Thus,

The probability is 1/6.

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Answer:

The total budget for all categories is $49,200 and the total actual is $50,175, for a percent change of 1.8%. Rounded to the nearest tenth, the percent change is 1.9%.

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The length of each equal piece is 17/6 feet, the mixed fraction is 2 5/6.

There is a fraction, containing numerator(upper value) and denominator(lower value).

When the numerator is less than the denominator, the fraction is called proper fraction, otherwise it is called improper fraction.

A proper fraction is also called fraction which is less than 1.

And an improper fraction is ≥ 1

A mixed fraction contains a sum of whole number and a proper fraction.

Given;

The board is 10 feet long

First he has to cut off 1 1/2 feet from one end of the board

Now, after the first cut remaining board will be;

=10-1 1/2=17/2

After cutting it in 3 equal pieces

=17/2/3

=17/6

=2 5/6

Therefore, the answer in mixed fraction will be 2 5/6

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The percentage is 40%, decimal point of blue candies is 0.4, and the fraction of blue candies is 2/5. The correct option is H.  40%, 0.4, 2/5.

Given,

Julia has 10 candies in green, 10 candies in red, 10 candies in pink and 20 candies in blue. There are 50 candies in Julia's hand.

We have to find percentage, decimal and fraction of blue candies = ?

Percentage = 20/50 x 100 = 40%

Decimal = 20/50 = 2/5 = 0.4

Fraction = 20/50 = 2/5.

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The distance the race car traveled during this time period using the velocities at the beginning of the time intervals is 23.886 mi

Use areas of rectangles to approximate the area under the curve. The width of each rectangle is 10 sec = 1/36 hrs and the lengths are the velocities.

LH = 1/36 ( 183.9 + 167 + 107.6 + 98.8 + 125.5 + 177.1)

LH=1/36(862.9)

LH≈ 23.886 mi

To find the right-hand approximation:

RH = 1/36 (167 + 107.6 + 98.8 +125.5 +177.1 + 176.6)

RH=1/36(852)

RH≈ 23.683 mi

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Answer: $14.04

Step-by-step explanation: -21.46+35.50 = 14.04

It will take 4 months to pay off the debt for Farah.

Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.

Given that Farrah borrowed $155 from her brother. She has already paid back $15. She plans to pay back $35 each month until the debt is paid off.

The expression can be written as:-

35x + 15 = 155

35x = 155 - 15

35x = 140

x = 140 / 35

x = 4 months

Hence, the number of months will be 4.

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