Question # 1 Solution:
As the expression [tex]13v + 8f +6b+3m[/tex] represents her cost, in dollars for the arrangements when she buys
- and m packages of marbles for the bottom of the vases
Which statement is true?
A.The term 13v represents the cost of 13 vases.
B.The coefficient 6 represents the cost of b stems of wood bark.
C.The term 8f represents the cost of f stems of flowers at $8 per stem.
D.The coefficient 3 represents the number of packages of marbles she buys.
Choosing the correct statement:
As the formula for cost of any item can be calculated by multiplying the cost of one item with the total number of items.
So, if f represents the stems of flowers, and the cost of f stems of flowers at $8 per stem. Then according to the formula for cost of any item, the statement C is true i.e. "The term 8f represents the cost of f stems of flowers at $8 per stem" is the correct statement.
Question # 2 Solution
As the inequality is given as:
12x + 9x - 3 ≤ 3(2x + 14)
and we have to find the value of x.
So,
12x + 9x - 3 ≤ 3(2x + 14)
Step 1: Simplify both sides of the inequality
21x - 3 ≤ 6x + 42
Step 2: Subtract 6x from both sides.
21x - 3 - 6x ≤ 6x + 42 - 6x
15x - 3 ≤ 42
Step 3: Add 3 to both sides.
15x - 3 + 3 ≤ 42 + 3
15x ≤ 45
Step 4: Divide both sides by 5
15x/5 ≤ 45/5
3x ≤ 9
x ≤ 9/3
So, x ≤ 9/3 is the right answer. Please note that x ≤ 9/3 can also be written as x ≤ 3.
Hence, option A i.e. x ≤ 9/3 is correct. The solution graph is also attached in figure a.
Question # 3 Solution
As we have to find the correct statements from the given graph that represents John's weight (in pounds) as a function of time t, measured in days since January 1, 2017.
So, here are the correct statements that validate the graph:
- The statement 'The independent variable is t, the number of days since January 1, 2017' is correct as t represents the independent variable because John's weight (in pounds) being the function of time t depends on the time t. Hence, time t is independent variable.
- The statement 'f(t) = 160 means that John weighed 160 lbs. on day 12, 120, and 172 after January 1, 2017' is correct because the output value i.e. f(t) of graph is showing John's weight weighed 160 lbs. i.e. f(t)=160 on the input values of day 12, 120, and 172 i.e. t = 12,120 and 172 after January 1, 2017.
- The statement 'f(0) = 140 means that John weighed 140 lbs. on January 1, 2017' is correct because the output value i.e f(0) of graph is showing John's weight weighed 140 lbs. i.e. f(0)=140 right on the day January 1, 2017 because t = 0 on January 1, 2017. And the value of t increases after January 1, 2017.
- The statement 'f(40) = 180 means that John gained 40 lbs. since January 1, 2017' is correct because on t = 40, the graph reaches to 180 i.e. the output value i.e. f(40) = 180 on t = 40. As on January 1, 2017, John was already weighed 140 i.e. f(0) = 140 on t = 0. So, f(40) = 180 means that John gained 40 lbs. since January 1, 2017.
- The statement 'f(200) - f(160) = 30 means that John gained 30 lbs. from day 160 after January 1, 2017 to day 200' is correct because the difference of f(200) and f(160) is 30 lbs. So, it specifies that on days' intervals from 160 to 200, the constitute gain of John's weight was 30 lbs.
- The statement 'The most appropriate domain for this function is all integers' is incorrect because when we talk about all integers it means it must have consisted negative integers too. But, as we cannot have negative time or days, and his weight also cannot be negative. So, to say that 'The most appropriate domain for this function is all integers' is not correct.
Keywords: inequality, equation, graph, rate of change, function
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