Question 1 Convert 9.2 × 1 0 − 5 9.2×10 −5 to standard notation. Answer 1 Step 1: Given Expression 9.2 × 1 0 − 5 9.2×10 −5 Step 2: To Convert into Standard Form Step 3: Standard Form - A standard form is always a number having decimals, constants in which there is no any multiple of exponents present. Step 4: Conversion 9.2 × 1 0 − 5 = 9.2 1 0 5 = 0.000092 9.2×10 −5 = 10 5 9.2 =0.000092 Step 5: Result Fourth Option is Correct. Question 2 Express the angle as a decimal, to the nearest ten-thousandth of a degree. 2 5 ∘ 3 9 ′ 5 2 ′ ′ 25 ∘ 39 ′ 52 ′′ Answer 1 Step 1 Use the unitary method to convert the minutes and seconds into decimals. 3 9 ′ = 39 min 39 ′ =39 min 39 60 = 0.65 60 39 =0.65 5 2 ′ ′ = 52 sec 52 ′′ =52 sec 52 60 × 60 = 0.0144 60×60 52 =0.0144 Step 2 Now, convert the angle into decimals. 2 5 ∘ 3 9 ′ 5 2 ′ ′ = 2 5 ∘ + 0.6 5 ∘ + 0.014 4 ∘ = 25.664 4 ∘ 25 ∘ 39 ′ 52 ′′ =25 ∘ +0.65 ∘ +0.0144 ∘ =25.6644 ∘ Hence, 2 5 ∘ 3 9 ′ 5 2 ′ ′ = 25.664 4 ∘ 25 ∘ 39 ′ 52 ′′ =25.6644 ∘ Question 3 What are the real numbers? Answer 1 Step 1 We need to find the definition of real numbers. First let us define what are real numbers and then provide the example for real numbers. Step 2 Any numbers that can be plotted on a number line are called real numbers. So all terminating decimals such as 0.375 or non-terminating recursive decimals such as 0.666... and non-terminating non-recursive decimals such as square root of 2 = 1.4141... =1.4141... are called real numbers. Question 4 Irrational solutions should be expressed precisely and as a decimal rounded to three places. log ⁡ 2 ( 3 𝑥 + 2 ) − log ⁡ 2 𝑥 = 3 log 2 (3x+2)−log 2 x=3 Answer 1 Step 1 Given, log ⁡ 2 ( 3 𝑥 + 2 ) − log ⁡ 2 𝑥 = 3 log 2 (3x+2)−log 2 x=3 Step 2 Now, log ⁡ 2 ( 3 𝑥 + 2 ) − log ⁡ 2 𝑥 = 3 log 2 (3x+2)−log 2 x=3 log ⁡ 2 3 𝑥 + 2 𝑥 = 3 log 2 x 3x+2 =3 3 𝑥 + 2 𝑥 = 2 3 x 3x+2 =2 3 3 𝑥 + 2 𝑥 = 8 x 3x+2 =8 3 𝑥 + 2 = 8 𝑥 3x+2=8x 5 𝑥 = 2 5x=2 𝑥 = 2 5 = 0.4 x= 5 2 =0.4 Thus, the solution is 𝑥 = 0.4 x=0.4. Question 5 Bethany needs to borrow $10,000. She can borrow the money at 5.5% simple interest for 4 years, or she can borrow at 5% with interest compounded continuously for 4 years. a) How much total interest would Bethany pay at 5.5% simple interest? b) How much total interest would Bethany pay at 5% interest compounded continuously? c) Which option results in less total interest? Answer 1 Step 1 a) Simple interest for an amount 𝑃 P, at an interest rate of 𝑟 r (in decimals) per annum and for 𝑇 T years is given by: 𝐼 = 𝑃 𝑇 𝑟 I=PTr Plugging the values: 𝐼 = ( 10000 ) ( 4 ) ( 0.055 ) I=(10000)(4)(0.055) 𝐼 = 2200 I=2200 b) Amount after continuous compounding for 𝑇 T years of principal 𝑃 P at an interest rate of 𝑟 r is given by: 𝐴 = 𝑃 𝑒 𝑟 𝑇 A=Pe rT Plugging the values: 𝐴 = ( 10000 ) 𝑒 ( 0.05 ) ( 4 ) A=(10000)e (0.05)(4) 𝐴 = ( 10000 ) 𝑒 0.2 A=(10000)e 0.2 𝐴 = ( 10000 ) ( 1.221402 ) A=(10000)(1.221402) 𝐴 = 12214.03 A=12214.03 Interest = 12214.03 − 10000 = 2214.03 Interest=12214.03−10000=2214.03 Step 3 c) Hence, simple interest results in less total interest.