HW01 1. Before trying to solve an equation, you would like to have an idea of what the solution should be, so that the solution can be checked after solving. An equation can always be turned into a question in words, which highlights the arithmetic nature of the problem. Rephrasing an equation in words also makes it easier to estimate the value of the solution. For instance, solving the equation 4x+8=60 is the same as answering the question "What number can you multiply by 4 and then add 8 to the product to get 60?" Part A Consider the numbers 0, 10, 20, 30, and 40. Multiply each by 4 and compare the result to 60 to determine into which of the following intervals the answer to the question "What number can you multiply by 4 and then add 8 to the product to get 60?" should fall. 10 to 20 Part B The answer here, x=7, is not in the interval that you selected in the previous part. What is wrong with the work shown above? In the first step, both terms on the left should be divided by 4. Part C Recall that in Part A the equation 4x+8=60 was restated in words as, "What number multiplied by 4 and then added to 8 gives 60?" Which of the following would be an equivalent way to write this? What number can you multiply by 4 to get a number 8 less than 60? Part D You know that a number 8 less than 60 is the number 52. So, you could rewrite your answer from the previous part as, "What number can you multiply by 4 to get 52?" This is just the definition of 52 divided by 4! So, what is the number x you are looking for? 2. To use the quadratic formula to solve quadratic equations. Maria wants to plant a small tomato garden in her yard. She bought 25 tomato plants, and she has read that ideally tomatoes are planted in a square grid(Figure 1) to help them pollinate each other. Part A How many plants x should she plant in each row so that her 25 plants end up in a square (i.e., x plants in each of x rows)? 5 Part B To find x in Part A, you would need to solve the equation x2−25=0. Which of the following shows the proper values in the quadratic formula before simplifying the radical and dividing? 0±√100/2 Notice that the value of c is −25, not +25. You should always include the sign in the values of a, b, and c. Also, notice that the answer implies there are two solutions to Part A: x=5 and x=−5. Of course, you can't have −5 tomato plants, so for solving the practical problem of planting tomatoes, the only correct solution is 5 plants per row. Part C Consider the equation 2x2−3x−5=0. Plug the values for a, b, and c into the quadratic formula, but do not simplify at all. Which of the following shows the proper substitution? Part D Use the result from Part C to find the two solutions to the equation 2x2−3x−5=0. -1,2.50 3. Suppose that you need to get a heavy couch into the bed of a pickup truck.(Figure 2) You know the bed of the truck is at a height of 1.00 m and you need a ramp that makes an angle of 40∘ with the ground if you are to be able to push the couch. Use the sine function to determine how long of a board you need to use to make a ramp that just reaches the 1.00-m high truck bed at a 40∘ angle to the ground. 1.56 m You need to set up another simple ramp using the board from Part A (i.e., a board of length 1.56 m ). (Figure 3) If the ramp must be at a 25∘ angle above the ground, how far back from the bed of the truck should the board touch the ground? Assume this is a different truck than the one from Part A. 1.41m Part C Surveyors frequently use trig functions. Suppose that you measure the angle from your position to the top of a mountain to be 2.50∘. (Figure 4) If the mountain is 1.00 km higher in elevation than your position, how far away is the mountain? 22.9km Part D The 3-4-5 right triangle is a commonly used right triangle.(Figure 5) Use the inverse sine function to determine the measure of the angle opposite the side of length 3. 36.9 Part E A support wire is attached to a recently transplanted tree to be sure that it stays vertical. The wire is attached to the tree at a point 1.50 m from the ground, and the wire is 2.00 m long. (Figure 6) What is the angle ϕ between the tree and the support wire? 41.4 4. Part B Suppose that you have measured a length of 6 cm on one board and 8 cm on the other. You would adjust the two boards until the length of the string had value c to ensure that the boards made a right angle. What is c? 10cm A set of three integers that form the legs and hypotenuse of a right triangle is called a Pythagorean triple. Any Pythagorean triple multiplied by an integer is another Pythagorean triple. For instance, since this problem showed you that 6, 8, 10 is a Pythagorean triple, you also know that 12, 16, 20 is a Pythagorean triple (found by doubling 6, 8, and 10). Part C Use the Pythagorean Theorem to determine which of the following give the measures of the legs and hypotenuse of a right triangle. 8,15,17 Part D What is the length x of a side of the small inner square? X=b-a Part E Given that the side of the square has a length b−a, find the area of one of the four triangles and the area of the small inner square. (1/2)ba, (b-a)^2 Here is a simple example. The inhabitants of a small island begin exporting beautiful cloth made from a rare plant that grows only on their island. Seeing how popular the small quantity that they export has been, they steadily raise their prices. A clothing maker from New York, thinking that he can save money by "cutting out the middleman," decides to travel to the small island and buy the cloth himself. Ignorant of the local custom of offering strangers outrageous prices and then negotiating down, the clothing maker accepts (much to everyone's surpise) the initial price of 400 tepizes/m2. The price of this cloth in New York is 120 dollars/yard2. Part A If the clothing maker bought 500 m2 of this fabric, how much money did he lose? Use 1tepiz=0.625dollar and 0.9144m=1yard. 5.3 x 10^4 dollars Part B Find the value of 15.0 N in pounds. Use the conversions 1slug=14.59kg and 1ft=0.3048m. 15.0 N=3.37 lb Part A Gravity causes objects to be attracted to one another. This attraction keeps our feet firmly planted on the ground and causes the moon to orbit the earth. The force of gravitational attraction is represented by the equation F=Gm1m2/r2, where F is the magnitude of the gravitational attraction on either body, m1 and m2 are the masses of the bodies, r is the distance between them, and G is the gravitational constant. In SI units, the units of force are kg⋅m/s2, the units of mass are kg, and the units of distance are m. For this equation to have consistent units, the units of G must be which of the following? M3/kg*s^2 Part B One consequence of Einstein's theory of special relativity is that mass is a form of energy. This massenergy relationship is perhaps the most famous of all physics equations: E=mc^2, where m is mass, c is the speed of the light, and E is the energy. In SI units, the units of speed are m/s. For the preceding equation to have consistent units (the same units on both sides of the equation), the units of E must be which of the following?Kg*m^2/s^2 To solve the types of problems typified by these examples, we start with the given equation. For each symbol whose units we know, we replace the symbol with those units. For example, we replace m with kg. We now solve this equation for the units of the unknown variable. Part A To seven significant figures, the mass of a proton is 1.672623×10−27kg. Which of the following choices demonstrates correct rounding? 1.67x 10^-27 kg The number 1.672×10−27 is incorrect because when we round to four significant figures we get 1.673, not 1.672. Similarly, 1.67263×10−27 is incorrect because when we round to six significant figures we get 1.67262, not 1.67263. Part B To eight significant figures, Avogadro's constant is 6.0221367×1023mol−1. Which of the following choices demonstrates correct rounding? All options The age of the universe is around 100,000,000,000,000,000s. A top quark has a lifetime of roughly 0.000000000000000000000001s. Writing numbers out with all these zeros is not very convenient. Such quantities are usually written as powers of 10. The age of the universe can be written as 1017s and the lifetime of a top quark as 10−24s. Note that1017 means the number you would get by multiplying 10 times 10 times 10... a total of 17 times. This number, as you can see above, would be a one followed by seventeen zeros. Similarly, 10−24 is the result of multiplying 0.1 (or 1/10) times itself 24 times. As seen above, this is written as 23 zeros after the decimal point followed by a one. Part A How many top quark lifetimes have there been in the history of the universe (i.e., what is the age of the universe divided by the lifetime of a top quark)? Note that these powers of 10 follow the same rules that any exponents would follow. 1.00*10^41 Part B Compute (4.29×10^15)⋅(1.51×10^−4). 6.48*10^11 If you had been asked to multiply 4.29×1015 times 3.00×10−4, you would have found (to three significant figures) 12.9×1011. You should rewrite this in the more accepted form 1.29×1012. (Notice that you must increase the exponent to balance moving the decimal point to the left.) Part C Compute 6.28×10^13+8.30×10^11. 6.36*10^13 Part D Suppose that a certain fortunate person has a net worth of $76.0 billion ($7.60×1010). If his stock has a good year and gains $3.20 billion (3.20×109) in value, what is his new net worth? 7.92*10^10 Part E Suppose that this individual now decides to give one-eighth of a percent of his new net worth to charity. How many dollars are given to charity? 9.90*10^7 A number written in scientific notation has the form a×10k, where 1≤a<10 and k is an integer, that is, k is one of ...−3,−2,−1,0,1,2,3.... Part A Determine the values of a and k when 299,790,000 is written in scientific notation. A,k=2.9979,2 The value 2.9979×108 is the magnitude of the speed of light in meters per second, to five significant figures. This is really fast: about 675 million miles per hour! Part B Determine the values of a and k when 0.51 is written in scientific notation. A,k=5.1,-1 The value 0.51 is the magnitude of the energy associated with a stationary electron, sometimes called its "rest energy," expressed in millions of electron volts, or MeV. Part C Now consider the expression 4.0×103+4×102. Determine the values of a and k when the value of this expression is written in scientific notation. A,k=4.4,3 The value 4×103=4000 is the approximate radius of the earth in miles. Part D Finally, consider the expression (6.67×10−11)(5.97×1024)/(6.38×106)2. Determine the values of a and k when the value of this expression is written in scientific notation. A,k=9.78,0 This is the magnitude of the acceleration experienced by a freely falling body near the surface of the earth, expressed in meters per second squared (m/s2). This acceleration is usually denoted by the symbol g. You may learn later that this acceleration is related to the mass Me and radius Re of the earth as g=GMe/R2e, where G=6.67×10−11N⋅m2/kg2 is called the gravitational constant, postulated by Sir Isaac Newton and first measured by Henry Cavendish in 1798.