answer:Since circle O' is tangent to both circles, there are two possible scenarios: 1. It is externally tangent to the smaller circle and internally tangent to the larger circle. The radius is calculated as frac{9 - 5}{2} = 2 , text{cm}. 2. It is internally tangent to both circles. The radius is calculated as frac{9 + 5}{2} = 7 , text{cm}. Therefore, the radius of circle O' could be either boxed{2 , text{cm}} or boxed{7 , text{cm}} .

answer:Page initially had 80 pairs of shoes. She decided to donate 30% of her collection. 30% of 80 is calculated as: 30/100 * 80 = 24 pairs So, she donated 24 pairs of shoes. After donating, she would have: 80 - 24 = 56 pairs left She then bought more shoes and ended up with 62 pairs. To find out how many pairs she bought, we subtract the number of pairs she had after donating from the number she has now: 62 - 56 = 6 pairs Page bought boxed{6} pairs of shoes after donating.

answer:First, we need to write the balanced chemical equation for the reaction between sulfuric acid (H2SO4) and sodium hydroxide (NaOH) to form water (H2O) and sodium sulfate (Na2SO4): [ H_2SO_4 + 2NaOH rightarrow Na_2SO_4 + 2H_2O ] From the balanced equation, we can see that 1 mole of sulfuric acid reacts with 2 moles of sodium hydroxide to produce 2 moles of water. Given that we have 3 moles of sulfuric acid, the stoichiometry of the reaction tells us that we would need 2 times that amount of sodium hydroxide to react completely, which is 6 moles of NaOH (which we have), and this would theoretically yield 2 times the amount of sulfuric acid in moles of water, which is 6 moles of water. So, the theoretical yield of water is 6 moles. The percent yield is calculated by taking the actual yield (the amount of product actually obtained from the reaction) and dividing it by the theoretical yield, then multiplying by 100 to get a percentage: [ text{Percent Yield} = left( frac{text{Actual Yield}}{text{Theoretical Yield}} right) times 100% ] Plugging in the values we have: [ text{Percent Yield} = left( frac{5.4 text{ moles}}{6 text{ moles}} right) times 100% ] [ text{Percent Yield} = left( 0.9 right) times 100% ] [ text{Percent Yield} = 90% ] Therefore, the percent yield of water in this reaction is boxed{90%} .

answer:First, factorize 360 to separate the even and odd parts: [ 360 = 2^3 times 45 ] The problem involves finding the odd factors of 360, so we focus on the factors of 45 (since odd factors are unaffected by the presence of the factor 2). Factorize 45 further: [ 45 = 3^2 times 5^1 ] The total number of factors of a number (a^b times c^d) is ((b+1)(d+1)). Here for 45: [ (2+1)(1+1) = 3 times 2 = 6 ] Thus, there are 6 odd factors of 360. boxed{6}