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Its the start of the New Year! This is where we'll keep with the math problems in 2023. If you want to access the problems from 2022, you can press the button below.
You're getting ready for New Years! You invite 6 friends over and you want to make 3 cookies per every person, including yourself. The recipe calls for 2 cups of flour for 9 cookies. How many cups of flour do you need?
To find the amount of cookies you will need to make, you should multiply the amount of people by the number of cookies you need to make.
(5 friends + 1 yourself) * 3 cookies per person = 18 cookies
Since the recipe accounts for 9 cookies, you would need 2 batches of cookies. This means you need 2 batches * 2 cups of flour per batch = 4 cups of flour.
If a hockey team scores 13% of the time, on average, how many times would they score if they had 307 shots on goal in a season (round to the nearest goal)?
If a team scores 13% of the time, they would score 13/100 shots taken. From here we can set up a proportion:
13/100 = x/307
100x = (13)(307)
x = 3991/100 = 39.91
Since we round to the nearest goal, the answer is 40 goals.
A basketball team averages 14 points per player during a game. There are 6 players on the team who have scored. A player scores a quarter of the points during a game. How many points did they score?
If there are 14 points per player during a game, and there are 6 players who have scored, then there were 84 points scored by that team. Since the player scored a quarter of the points, that means they scored 84/4 points, or, 21 points.
A figure skater does a jump. They are in the air for .63 seconds and complete 4.5 revolutions. About how many rpm is the figure skater spinning? (ignore air resistance and friction)
To find the amount of time it takes to complete one rotation, you would divide the revolutions by the amount of air time:
4.5 revolutions / .63 seconds = 7.143 rev/sec
To convert this to revolutions per minute, one would have to multiply this number by 60.
7.143 rev/sec * 60 sec/min = 428.571 rpm
Someone is standing on the ground 450 feet away from the base of a cell tower. If the person looks towards the top of the cell tower looks up at an angle of 32 degrees, how tall is the tower?
The person makes a right triangle with the cell tower. We know one angle and the adjacent side. From this, we can use trigonometry to determine the height of the tower.
sin(32 degrees) = x/450
x = 238.464
If 74% of a compound dissolves when coming into contact with water, how much of a 251 gram sample would dissolve in water?
In order to solve this problem, you would need to find the percent of 251 grams that will dissolve. Since 74% of the compound dissolves, to find the percent of the compound that dissolves, all you have to do is multiply the mass by the percent that dissolves.
.74 percent * 251 grams = 185.74 grams that dissolves
Someone looks at the top of a skyscraper from 2 different points where the distance between the two points is 60 feet. If the angles of elevation are 30 and 60 degrees, how tall is the skyscraper?
There are a few ways to solve this problem, but I will be using the law of sines.
The law of sines are a set of ratios that can be used to find different sides or angles in any triangle.
sin30/x = sin30/60ft
sin60 = x/60
x = 103.9 feet
If a boat docked at a pier and travels 12 knots (nautical miles per hour) west for 4 hours and then 10 knots north for 6.4 hours, how far from the start would the boat be?
The shape of the boat’s path would be a right triangle. To find the distances that the boat travelled west and north, you can multiply the speed of the boat by the amount of time it has travelled.
12 knots * 4 hours = 48 nautical miles west
10 knots * 6.4 hours = 64 nautical miles north
These numbers create a 3-4-5 right triangle, meaning the distance from the boat to the port is 80 nautical miles.
A power washer charges a flat rate of $50 in addition to $25/hour of work. Write an equation for the price of the service, P, as a function of hours of work, w.
Since the power washer charges $25/hour, you can assume the price to be 25*hours of work without the flat rate. Since the flat rate of $50 doesn’t change with the amount of time the job takes, you can just add the 50 to the equation.
P = 25w + 50
Someone hires a contractor to fix 3 broken windows. If the contractor charges $40 per window with a $65 down payment. How much would the job be?
Since there’s 3 windows, that would cost 3*$40 which is $120. In addition to the $120, you would need the $65 down payment. The total comes to $185.
Suppose there is an orchestra that's hoping to buy licenses for 3 songs to play at a concert. They have a selection of music whose prices are $26, $73, $92, $72, $62, and $82. Assuming price is directly correlated to how good the music is, what is the best program the orchestra can come up with if their budget is $160?
In order for the program to be the best possible, you would buy the most expensive pieces that can fit into the budget. The $92 and $82 would leave little room for the other 2 pieces so we’d start with the $73 piece. After that, we would most likely choose the $62 since the $72 would leave little room for a third. $73 + $62 = $135. Now this leaves us $1 short of the $26 piece. So we would have to swap the $73 piece with the $72 piece.
The program you would come up with is the $72 piece, the $62 piece, and the $26 piece.
If you put $100 in a bank account that has 8% interest compounded quarterly, how much money would be in the account after 5 years?
For this question, you can use the compound interest formula.
A = P(1 + r/n)^nt
Where r is the rate, n is the number of times interest is applied per time period, t is time, and p is the initial amount.
In our case r is .08, or 8 percent, t is 5 years, n is 4 since it’s compounded quarterly, and p is 100 since that is the initial deposit.
A = 100(1 + .08/4)^(4*5) = 148.59
If someone wants to buy a house, but cant afford to pay for it outright, they will have to get a mortgage. If your house is $200k, with a down payment of $5k, and you pay it over 15 years, how much would your monthly payment be?
For this problem, you can use the mortgage formula.
A = P(r(1 + r)^n)/((1 + r)^n - 1)
This means that the monthly payment would be:
A = $195000(.03(1 + .03)^(12))/((1 + .03)^(12) - 1) = $16334 a year
If a hyperbola has a center at the origin, a major axis length of 8 and a focus at (0,6), what is the equation of the hyperbola?
First, we need to use the hyperbola equation to determine what we need to find.
y/a^2 - x/b^2 = 1
Since the focus is on the y-axis, we can determine that the y axis is major. Using the major axis length, 8, we can find the a value. The major axis length is equivalent to 2a, so the a value is 4.
y/4^2 - x/b^2 = 1
To find the b^2 value, we will use the focus. In order to find this value, we can use the formula:
c^2 = a^2 + b^2
6^2 = 4^2 + b^2
36 = 16 + b^2
b^2 = 20
Now that we know both the a^2 and b^2 values, we can find the equation of the hyperbola.
y/16 - x/20 = 1
Find the units digit of the following expression: (36^25 + 62^87)^7
Since the question is only asking about the units digit, we can ignore the other numbers.
Effectively, this would be (6^25 + 2^87)^7
Now this looks challenging, but all powers have a pattern for the ones digit. The ones digit for a power for a number ending in 6, will always be 6. The pattern for 2 is 2, 4, 8, 6 and then it repeats. So we know the ones digit for 6^25 will be 6. For 2, we will have to find out 87/4 to find where in the pattern the ones digit will be. 87/4 = 21 R3, meaning it will be the third digit in the pattern: 8. We can add these two ones digits to create 14, and then ignoring the tens digit, we can simplify it to 4. The pattern for 4 is 4, 6, and then it repeats. This means that the ones digit will be 4.
If there is a worker who spends 8 hours a day, 5 days a week, working for a rate of $34/hour, how much would they make per year?
If they worked 8 hours a day, 5 days a week, then they work 40 hours a week.
40 hours a week x $34 dollars an hour = $1360 a week.
There are 52 weeks in a year, which means they make $1360 per week x 52 weeks. This is equal to $70,720.
There is a 9 sided regular polygon with side lengths of 8, what is the measurement of the internal angles of the polygon?
The side lengths of this polygon are irrelevant, it is extra information. In order to find the internal angles, you can use the formula:
S = ((n-2)*180)/n
For a 9 sided polygon, this would be
S = ((9 - 2)*180)/9 = 140 degrees.
Find the limit of f(x) = (x^2 - 5x + 6)/(x-3) as x -> 3.
To find the limit, we will first try to substitute. In the case, this wouldn’t work as the denominator of the function would be 0, leading to an undefined solution. Since substitution doesn’t work, we will try to factor the numerator to find a hole in the graph.
Here we can see that there is a hole in the graph at x = 3. Now that we know there is a hole, we can find the limit to the graph by cancelling out the common factors and substituting.
(3-2) = 1
Is the function, f(x) = (x^2+2x-15)/(x^2-2x-3) continuous at x=3. If it is discontinuous, is the discontinuity removable or nonremovable?
If we factor both the numerator and denominator, we will find a hole at x=3
((x + 5)(x - 3))/((x - 3)(x +1))
This makes the graph discontinuous at x = 3. Since the discontinuity is a hole, it is removable. Therefore, the graph has a removable discontinuity at x = 3