Question

a manager of a logging company compares the total weight, in tons, of two trucks based on the amount of time, x, in hours, after logging begins for the day. the functions representing the total weight of each truck are shown in the table. truck 1: ( p(x) = 8x + 15 ) truck 2: ( r(x) = 6(x + 2) ) which function, ( t(x) ), can be used to find how much more truck 1 weighs than truck 2 based on the number of hours, x?
options:
( t(x) = 2x + 3 )
( t(x) = 2x + 13 )
( t(x) = 14x + 17 )
( t(x) = 14x + 27 )

Explanation:

Step1: Expand \( R(x) \)

First, expand \( R(x) = 6(x + 2) \). Using the distributive property \( a(b + c)=ab + ac \), we get \( R(x)=6x + 12 \).

Step2: Find \( T(x)=P(x)-R(x) \)

We know \( P(x)=8x + 15 \) and \( R(x)=6x + 12 \). To find how much more Truck 1 weighs than Truck 2, we subtract \( R(x) \) from \( P(x) \). So \( T(x)=(8x + 15)-(6x + 12) \).

Step3: Simplify the expression

Simplify \( (8x + 15)-(6x + 12) \). Distribute the negative sign: \( 8x + 15-6x - 12 \). Combine like terms: \( (8x - 6x)+(15 - 12)=2x + 3 \). Wait, no, wait, maybe I made a mistake. Wait, let's re - check. Wait, the problem is to find how much more Truck 1 weighs than Truck 2, so \( T(x)=P(x)-R(x) \). Wait, \( P(x)=8x + 15 \), \( R(x)=6(x + 2)=6x+12 \). Then \( T(x)=(8x + 15)-(6x + 12)=8x + 15-6x - 12 = 2x+3 \)? But wait, maybe I misread the functions. Wait, maybe \( P(x) \) is \( 8x + 15 \)? Wait, no, looking at the image again, maybe \( P(x)=8x + 15 \) and \( R(x)=6(x + 2) \). Wait, but let's check the options. Wait, maybe I made a mistake in the function of Truck 1. Wait, maybe \( P(x)=8x + 15 \) is wrong. Wait, looking at the image, the first truck's function is \( P(x)=8x + 15 \)? Wait, no, maybe it's \( P(x)=8x + 15 \) and the second truck's function is \( R(x)=6(x + 2) \). Wait, but when we calculate \( T(x)=P(x)-R(x) \), we have \( 8x + 15-6(x + 2)=8x + 15-6x - 12 = 2x+3 \). But wait, the options have \( T(x)=2x + 3 \) as the first option. Wait, but let's check again. Wait, maybe the first truck's function is \( P(x)=8x + 15 \) and the second is \( R(x)=6(x + 2) \). So the difference is \( (8x + 15)-6(x + 2)=8x + 15-6x - 12 = 2x + 3 \). So the function \( T(x)=2x + 3 \).

Wait, but let's check the options. The first option is \( T(x)=2x + 3 \), second is \( T(x)=2x + 13 \), third is \( T(x)=14x + 17 \), fourth is \( T(x)=14x + 27 \). So according to our calculation, \( T(x)=2x + 3 \).

Answer:

\( T(x)=2x + 3 \) (the first option among the given options)