Tutor profile: Laura C.
Subject: Physics (Newtonian Mechanics)
An airplane takeoff the runway with a speed of 90 m/s and requires 1200 m to reach that speed. Find the acceleration of the airplane on the runway and the time required to reach this speed.
1) We identify the variables that we have from the problem. Vi = 0 m/s Vf = 90 m/s d = 1200 m And then, we identify the unknown variables. a = ? t = ? 2) Before we start solving the problem, we need to make sure that all the units are in the same unit system. In this case, we will be using the International System of Units (SI) and all the units are consistent with this system. Otherwise, we will need to do some conversions of units firsts. 3) To find the acceleration we need to use the formula: Vf² = Vi² + 2ad Let's replace the known variables in the formula: (90 m/s)² = (0 m/s)² + 2 x (a) x (1200 m) Then we solve the equation to find a: Solve the powers: 8100 m²/s² = 0 m²/s² + 2 x (a) x (1200 m) Remove the 0: 8100 m²/s² = 2 x (a) x (1200 m) Solve the multiplication: 8100 m²/s² = (2400 m) x a Divide both sides by 2400 m: 8100 m²/s² ÷ 2400 m = (2400 m ÷ 2400 m) x a 8100 m²/s² ÷ 2400 m = a Solve the division: 3.375 m/s² = a The acceleration of the airplane on the runway is 3.375 m/s² 4) To find the time spend, we need to use the formula: Vf = Vi + at Let's replace the known variables in the formula: 90 m/s = 0 m/s + (3.375 m/s²) x t Remove the 0: 90 m/s = 0 m/s + (3.375 m/s²) x t Divide both sides by 3.375 m/s²: 90 m/s ÷ 3.375 m/s² = (3.375 m/s² ÷ 3.375 m/s²) x t 90 m/s ÷ 3.375 m/s² = t Solve the division: 26.67 s = t The time required to reach the speed is 26.67 s.
Subject: Basic Math
Carlos went to the supermarket to buy the ingredients to make milkshakes for his birthday party. He bought 2 bottles of milk, 1 box of ice cream, and 1.5 kg of strawberries. Find how much did Carlos pay in total for the ingredients. Take into account the following prices. 1 bottle of milk: $3 USD 1 box of ice cream: $7.5 USD 1 kg of strawberries: $5 USD
To solve this math problem, first, we need to find how much did Carlos pay for each item, and then add these values to find the total. 1) Let's find out how much did Carlos pay for each item. Milk: He bought 2 bottles and each bottle has a value of $3 USD. We need to multiply 2 by 3 to find the value for the 2 bottles. 2 x 3 = 6 ---> Carlos paid $6 USD for the milk. Ice cream: He bought 1 box and each box has a value of $7.5 USD. So he paid $7.5 USD for the ice cream. Strawberries: He bought 1.5 kilograms and each kilogram has a value of $5 USD. We need to multiply 1.5 by 5 to find how much did he pay for the strawberries. To multiply a decimal number by a whole number. First, we need to multiply the two numbers ignoring the decimal point. 15 x 5 = 75. Then, we count the number of decimal places we have in the two factors of the initial multiplication, the answer (product) should have the same number of decimal places. 1.5 has one decimal place and 5 has no decimal places, so in total, we have one decimal place. So the answer to the multiplication should have one decimal place. 1.5 x 5 = 7.5 Carlos paid $7.5 USD for the strawberries. 2) Now we need to add the value that Carlos paid for each ingredient to find the total. Milk: $6 USD Ice cream: $7.5 USD Strawberries: $7.5 USD To add whole numbers with decimal numbers we need to line up the place values, units with units, tens with tens, tenths with tenths, and so on. Let's start adding the value of the milk and the ice cream. 6 + 7.5 If we line up the place values, we will have 6 6 . 0 + 7 . 5 Which is the same as + 7 . 5 --------------- ---------------- 1 3 . 5 Now we can add the value of the strawberries. 1 3 1 3 . 5 + 7 . 5 Which is the same as + 7 . 5 ------------------- ------------------- 2 1 . 0 We have already added the 3 ingredients Carlos bought. In total, Carlos paid $21.0 USD for the ingredients.
Find the equation of the line that passes through the points (-4,-2) and (0,5).
1) Let's remember the general equation of a line: y = mx + c where m is the slope, and c is the value where the line intercepts with the y-axis. 2) To find the slope (m) we need to find the quotient of the change in the y-axis by the change in the x-axis. m = (y2 - y1) / (x2 - x1) Let's find the slope m. We have two points (-4,-2) and (0,5). Remember that the nomenclature of the point is (x,y). Then, x1 = -4 y1 = -2 x2 = 0 y2 = 5 We can replace these values in the slope formula. m = (y2 - y1) / (x2 - x1) ---> m = (5 - (-2)) / (0 - (-4)) To solve this equation we need to solve first the internal parenthesis. The rules for integers tell us that minus times minus is equal to plus. ( - x - = + ) So -(-2) is equal to +2. And -(-4) is equal to +4. If we apply this to our slope equation m = (5 + 2) / (0 + 4) Then we can solve the parenthesis in the dividend (5 + 2) = 7. So m = 7 / (0 + 4) Then we can solve the parenthesis in the divisor (0 + 4) = 4. So m = 7 / 4 And we already found the slope. m = 7/4 = 1.75 3) To find the y-intercept (c) we use the slope (m) we found and one of the points to solve the line equation and find c. It doesn't matter which of the 2 points we use, we answer will be the same. The slope we found is m = 1.75, and let's use the point (0,5). Then we need to replace these values in the general formula of the line. y = mx + c ---> 5 = 1.75(0) + c First, we solve the parenthesis. 1.75(0) = 1.75 x 0 = 0 Then 5 = 0 + c, which is the same as 5 = c So c is equal to 5 4) Finally, we replace the values we found of m and c in the general equation of a line: y = mx + c ---> y = 1.75x + 5 And this is the solution.