Mathematics Flashcards

A team of five medical personnel from a hospital will travel to a far-off island. There are 12 nurses and eight doctors available at the facility. How many five-person teams are possible if each team must have at least one doctor and two nurses?

Answer: 13816

With 8 doctors and 12 nurses, valid 5-person teams must satisfy at least 1 doctor and at least 2 nurses, giving three cases: (1 doctor, 4 nurses), (2 doctors, 3 nurses), and (3 doctors, 2 nurses). Computing each with combinations gives 8·495 + 28·220 + 56·66 = 3960 + 6160 + 3696 = 13816. The other answers result from missing a case or miscounting the combinations.

Depending on the individual's income, a specific community is divided into three classes: lower, middle, and higher. You want to research the community's residents' standard of living. Which sampling strategy should be employed to produce a result representative of the entire population?

Answer: Stratified sampling is the best method to use to ensure that subjects from different classes will be well-represented.

Explanation: Stratified sampling is the best sampling technique to ensure that the study results are representative of the entire population with regards to income classes. In stratified sampling, the population is divided into homogeneous subgroups or strata based on a specific characteristic, in this case, the income classes (lower class, middle class, and upper class).

Calculate the total of 13 + 9 + 5 +... + (-99) +. (-103).

Answer: -1350

This is an arithmetic series with first term 13, common difference −4, and last term −103, which yields 30 terms (since 13 + (n−1)(−4) = −103 gives n = 30). The sum is (30/2)(13 + (−103)) = 15 × (−90) = −1350. The other values come from using the wrong term count or sign.

Three hundred fifty feet separate you from the cliff's base when you are looking at it. What is the ridge's height if the elevation angle from your location to its peak is 60 degrees?

Answer: 606 feet

The setup forms a right triangle where the 350-foot distance is the adjacent side and the height is the opposite side, so height = 350 × tan(60°) = 350 × 1.732 ≈ 606 feet. The smaller answers result from using the wrong trig ratio (such as cos or dividing instead of multiplying).

Which of the following uses of a line's slope is practical?

Answer: Interpolating values from data that follows a linear trend

Explanation: Interpolating values from data that follows a linear trend is a practical application of the slope of a line. The slope of a line represents the rate of change between two variables. In the context of data that follows a linear trend, the slope can provide valuable information about how one variable changes with respect to another.

What practical applications of mathematics did the Babylonians have in antiquity?

Answer: They used algebra to write formulas to represent a general situation to arrive at a more specific answer

Explanation: While the Babylonians did not have algebra in the modern sense, they did develop mathematical techniques and methods that allowed them to solve day-to-day problems. One way mathematics helped the Babylonians was through the use of numerical calculations and systematic methods for solving practical problems.

Pigs to goats are a 3:5 ratio, while goats to cows are a 4:9 ratio. Given that there are eight pigs, how many cows are there?

Answer: 30

Explanation: To solve the problem, we need to find the number of cows based on the given ratios. Given that the ratio of pigs to goats is 3:5, we can set up a proportion: Pigs/Goats = 3/5 If there are 8 pigs, we can substitute the value: 8/Goats = 3/5 Cross-multiplying, we get: 5 * 8 = 3 * Goats 40 = 3 * Goats Now, we need to find the number of goats based on the second ratio. The ratio of goats to cows is 4:9, so we set up another proportion: Goats/Cows = 4/9 Substituting the value of goats from the previous equation (Goats = 40/3): (40/3)/Cows = 4/9 To solve for the number of cows, we can cross-multiply: 9 * (40/3) = 4 * Cows 120 = 4 * Cows Cows = 120/4 Cows = 30 Therefore, if there are 8 pigs, the number of cows would be 30.