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

There are 2 racecar drivers. In order for a racecar to be as fast as possible, the engineers need to make the car just above the minimum weight, which is 742 kilograms. There are 2 drivers, one of which is 63kg and the other being 52kg. Considering this, how much more weight could be spent on the aero and car components with the lighter driver?

This answer is actually quite simple. Since the minimum car weight is the same, the difference in the aero weight would just be the difference between the weight of the two drivers.

63kg - 52kg = 11kg

The lighter driver could spend 11 more kilograms compared to the heavier driver.

There are 2 similar triangle. One such triangle is triangle ABC, while the other is triangle XYZ. A corresponds to X and B corresponds to Y. If angle A is 48 degrees, side AB has a length of 6, and side XY has a length of 15, and angle Y is 83 degrees, what is the value of the angle C?

To find the value of angle C, we can use the values from angle A and Y. Since these triangles are similar, the angles must be the same for the corresponding angles. This means that:

A = X = 48 degrees

B = Y = 83 degrees

C = Z = ?

The angles in a triangle will always add up to 180, which means that to find the last angle, we can subtract (48 + 83) from 180.

180 - (48 + 83) = 49 degrees.

A spaceship is travelling through outer space at a speed of 40m/s until he finds an unexpected planet. While descending through the planet's atmosphere, he follows a path represented by the graph f(x) = -.000025x^3 - .2x + 100 where x is the time after entry in seconds and y is the spaceship's altitude in kilometers. What altitude is the rocket at after 20 seconds?

In order to do this question, we can just plug in the value, 20, into the original function.

f(20) = -.000025(20)^3 - .2(20) + 100

f(20) = .000025(8000) - 4 + 100 = .2 - 4 +100 = 95.8km

A train is moving at a rate that can be expressed using the function v(x) = 7x^2-3x+4. What is the function that represents the velocity?

To find the function that represents the acceleration, you take the derivative of the velocity function. This is because the acceleration of an object is the derivative of its velocity. To find the derivative of v(x), you can use power rule.

v(x) = 7x^2 - 3x + 4

v'(x) = 14x - 3

This is the function that represents the acceleration.

Pete is a cyclist who likes riding his bike in rose curves. He starts in the center of the field and reaches a maximum of 9km from the center of the field and meets this maximum 7 times during his ride. What is the equation of his path?

To find the answer to this problem, we can use the elements given in the question to find the characteristics of the equation. Since the maximum distance the n. Since Pete reaches a maximum of 9 km away from the center, the coefficient in front of the equation is 9. In addition, since Pete meets this maximum 7 times during his ride, the equation is (+ or -)cos(7x). This means that the equation of Pete’s path can either be:

8cos(7x)

-8cos(7x)

Jerry wants to see if he can roll 5 even numbers in a row on a 9 sided die. What are the chances of Jerry rolling 5 even numbers in a row? (Assuming the die is fair)

In order to do this, we will first need to find out the chance of rolling an even number once. Since there are 4 even numbers on the die: 2, 4, 6, and 8, the chances of rolling an even number is 4/9. In addition, since each instance is independent, to find the odds of Jerry rolling 5 in a row, we can just put (4/9) to the 5th power. The chance that Jerry rolls an even number 5 times in a row ends up being about 1.73%.

A company says that 62% of products fail quality control while the remaining 38% pass. However, their quality control system needs improvement considering the fact that 34% of their products have wrongly passed quality control and 3% of those that failed the quality control should have passed. What are the chances of a product falsely failing quality control?

To find these chances, we can use a tree diagram. To find the chances of a product wrongly failing quality control, we can multiply the 62% of products that fail quality control by the 3% of products that are falsely failed. This means the chances of the product falsely failing the quality is .0207.

It takes Joey 4 hours to mow a lawn with an area of 32 m^2 and it takes Jimmy 2 hours to mow a lawn with an area of 20 m^2. If they combine forces, how long would it take for them to mow a lawn with an area of 42 m^2?

To find the time it takes for them to mow the lawn, we can find the rate at which each individual mows the lawn. Joey mows lawns at a rate of 32/4 m^2/hr and Jimmy mows lawns at a rate of 20/2 m^2/hr. If they combine forces, they will mow the lawn at 72/4 m^2/hr, or 13m^2/hr. Now to find the time it takes for them to mow the lawn, you can evaluate (42m^2) / (13m^2/hr). This ends up being 3.23 hours.

A particle's velocity can be expressed by the function V(t) = 2t^3 - 7t^2 - 8t. When does the particle have a negative acceleration?

While this question dives into physics a little bit, this can be easily solved using math. All you need to know is that the acceleration is the derivative of velocity. This means to find when the particle has a negative acceleration, you have to find the derivative of the velocity function, find the critical values, and test whether the intervals are above or below the x-axis.

V’(t) = 6t^2-14t-8

V’(t) = 2(3t^2-7t-4)

t = 2.808, -.475 (calculator)

Now we can make a number line and plug in values from each interval into the derivative equation. Once we do this, we find that the particle has a negative acceleration from the interval (-.475,2.808).

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