Question

1. during a recent rain shower, zachary found that the relationship between the depth of water in his pool and the time it was raining was linear. the function ( g(t) = 2t + 30 ) represents this relationship, where ( t ) is the time, in hours, since it started raining and ( g(t) ) is the depth, in centimeters, of water in the pool. how many centimeters is the depth of water in the pool increasing for every additional hour that it rains? _______ centimeters 65. the function ( a(n) = n^2 + 5 ) (where ( n ) is a positive whole number) defines a sequence. the first term of the sequence defined by ( a(n) ) is 6, and the second term is 9. the third term of the sequence is choice 1, and the fourth term of the sequence is choice 2. choice 1: 12, 14, 15, 17; choice 2: 21, 34, 26, 28 66. the amount, in kilograms, of a certain radioactive substance remaining after ( t ) years can be modeled by the function ( a(t) = 755(0.5)^{t/21} ). what does 755 represent in this function? a. the initial amount of the substance, in kilograms b. the half - life of the substance, in years c. the percentage rate of decay d. the amount of decay of the substance, in kilograms 67. for the functions ( f(x) = |x - 2| ) and ( g(x) = x + 6 ), which value of ( x ) is a solution to the equation ( f(x) = g(x) )? a. -2 b. 2 c. 4 d. 8

Explanation:

Question 64

Step1: Identify the function type

The function \( g(t) = 2t + 30 \) is a linear function in the form \( y = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y - intercept.

Step2: Determine the rate of change

In the context of this problem, the slope \( m \) of the linear function represents the rate at which the depth of water in the pool is increasing per hour. For the function \( g(t)=2t + 30 \), the coefficient of \( t \) (the slope) is 2. This means that for each additional hour (\( t \) increases by 1), the depth \( g(t) \) increases by 2 centimeters.

Step1: Find the third term

The function for the sequence is \( a(n)=n^{2}+5 \), where \( n \) is a positive whole number. To find the third term, we substitute \( n = 3 \) into the function.
\( a(3)=3^{2}+5=9 + 5=14 \)

Step2: Find the fourth term

To find the fourth term, we substitute \( n = 4 \) into the function \( a(n)=n^{2}+5 \).
\( a(4)=4^{2}+5=16 + 5=21 \)

In an exponential decay function of the form \( A(t)=a(b)^{t/k} \), where \( A(t) \) is the amount of the substance at time \( t \), \( a \) is the initial amount of the substance, \( b \) is the base (related to the decay factor), \( t \) is the time, and \( k \) is related to the half - life. For the function \( A(t)=755(0.5)^{t/21} \), when \( t = 0 \) (the initial time, before any time has passed), \( A(0)=755(0.5)^{0/21}=755\times1 = 755 \). So 755 represents the initial amount of the substance in kilograms. Option A is correct, option B is incorrect because the half - life is related to the exponent part (21 in the exponent), option C is incorrect because the percentage rate of decay is related to the base (0.5 represents a 50% decay rate), and option D is incorrect because 755 is not a decay amount but an initial amount.

Answer:

2

Question 65