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 and earlier this year, you can press the button below.

Alex is looking to buy a lego set. They want the lego set which will give him the most pieces for his money. One such set has 283 pieces and is $25 while another set has 4835 and is $470. Which lego set should they buy?

To find this, we will need to find the cost per piece of each set. In order to find this, we will need to divide the price of the lego set with the amount of pieces in the set.

For the first set, we will calculate 25/283 = .0883

For the second set, we will calculate 470/4835 = .0972

This means that the pieces in the second set are more expensive, meaning the first set is the set that Alex should buy.

What is the ones unit of the following expression: (3^53 + 8^3) * (7^3 + 9^83)

Since the question is only asking for the ones unit, we can ignore most of the expression. For the large exponents, all we have to do if find the pattern of the ones unit. For example, for 3, the pattern is 3, 9, 7, 1, and then it repeats. For 8, it is 8, 4, 2, 6, and then it repeats. The pattern for 7 is 7, 9, 3, 1 and the pattern for 9 is 9, 1. From here we can find the ones units for all of the exponents. So you can divide the numbers by 4, and ones unit using the remainder.

(3 + 2) * (3 + 9) = (5) * (12) = 60

This means the ones unit is 0.

A ship is setting sail in the Atlantic Ocean. The ship travels at 5 knots per hour 37 degrees northeast, measuring the angle from the north axis for 7 hours. The ship then changes direction to traveling due east while maintaining the same speed and sails for another 5 hours. How far did the ship sail?

To find the distance sailed by the ship, we have to multiply the speed of the ship by the time the ship sailed. The ship sailed 5 nautical miles per hour (or knots) for 7 hours, meaning it sailed 35 miles. It then sailed for the same speed for another 5 hours which means it sailed 25 miles. In total, the ship sailed 60 nautical miles.

Looking at the previous question below, how far did the ship sail from its origin point?

A ship is setting sail in the Atlantic Ocean. The ship travels at 5 knots per hour 37 degrees northeast, measuring the angle from the north axis for 7 hours. The ship then changes direction to traveling due east while maintaining the same speed and sails for another 5 hours. How far did the ship sail?

In order to solve this problem, we can use triangles. Assume that the boat leaves at a port which is the first vertex of the triangle. The second vertex is where the ship starts going due east and the third vertex is where the boat stops sailing. This means that you know 2 sides of the triangle and one of the angles. The two sides are the same length, 35, and the angle at the port is 53, since the angle is measured from the north axis. Using the law of cosines, we can find the length of the third side, or the distance from the port.

c^2 = a^2 + b^2 - 2*a*b*cos(c)

35^2 = 35^2 + b^2 - 2*35*b*cos(53)

b = 42.127

Oscar wants to go to a farm and meet animals. If Oscar counts 42 heads at the farm, 118 legs, and 36 wings, how many chickens, cows, and humans are there?

Here we can use a system of equations to find the amount of animals. Let’s say that x is chickens, y is cows, and z is humans.

Since there’s 42 heads, x + y + z = 42

Since there’s 118 legs, and cows have 4 legs while humans and chickens have 2 legs, 2x + 4y + 2z = 118

Since there’s 36 wings, and only chickens have wings, 2x = 36. This means x = 18.

We can plug in y into the other 2 equations to get:

y + z = 24

4y + 2z = 82

We can then substitute the z variable using the first equation.

z = 24 - y

4y + 2(24-y) = 82

4y - 2y = 82 - 48

y = 17

18 + 17 + z = 42

z = 7.

A customer wants to create a box that can hold 3450 cubic inches of grain. What are the dimensions of the box that satisfy this volume while having the least surface area?

To do this, we can use the volume formula to find one of the dimensions of the box.

V = w^2h

3450 = w^2h

w = (3450/(h))^.5

Now, we can plug this into the surface area formula.

SA = w^2 + 4wl

SA = (3450/h)^2 + 4(3450/(w^2))^.5w

Using a graphing calculator, the h value with the lowest surface area is the 132.645 inches. This means the width is 5.100 inches.

George wants to go on vacation to Alaska and has to choose between 3 airlines, 4 hotels, and 2 rental cars. These are all the same price and offer the same amenities. How many different combinations can George make?

For airlines, George has 3 airlines. From each airline, he has 4 options as a hotel and for each hotel, he has 2 options for rental cars. This means that he has 3*4*2 total combinations. 3*4*2 is 24, therefore, he has 24 different combinations.

George wants to get a pet snake. His parents ask him to tally up how much the snake would cost over the first year. The snake is $150, the enclosure for the snake is $299.99, the interior decoration and plants are $36.86. The food for the snake are $23.45 every week and the water cost is $2 a month. How much does the snake cost for the first year?

So the upfront costs are the snake, enclosure, decoration, and plants.

$299.99 + $150 + $36.86 = $486.85

The food costs $23.45 a week, and there’s 52 weeks in a year which means that the food cost is $23.45*52 = $1,219.40

The water cost is $2/month which means that the water cost is $24 a year.

If we add all of these together, we can find that the snake costs $1730.25.

Charles is in a Formula 1 racecar which accelerates at 11.1m/s^2. Logan is American and doesn’t understand imperial units. How fast is Charles going in miles/hour^2? (1609m = 1mi)

To convert this from m/s^2 to miles/hour^2, we need to convert the seconds squared to hours squared and convert meters to miles.

60sec * 60min = 1 hour = 3600sec. Since it’s squared, we need to square the 3600, which equals 12960000 seconds. Now we have to convert the meters to miles:

11.1m/1609m = .000684 miles. Now we just multiply the two terms to get an acceleration of 8,860mi/hour^2.

There are two similar triangles, one of which has a side length of 3.7. The corresponding side of the other triangle is 17.3. What is the area of the larger triangle if the area of the smaller triangle is 14.9?

So we have to find the ratio of the sides of the triangle.

17.3/3.7 = 4.7

This means the sides of the smaller triangle are 4.7x larger than the smaller triangle. This means the area of the larger triangle is 4.7^2 times larger.

4.7^2 * 14.9 = 329.1

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