Linear Functions
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SAT Math › Linear Functions
A crew has paved a road length $L$ (in miles) after $d$ days, and the relationship is linear. They had completed 1.4 miles after 3 days and 5.0 miles after 11 days. To the nearest tenth, how many days will it take to reach 9.0 miles?
17.9
19
19.9
20
Explanation
The slope is $(5.0 - 1.4)/(11 - 3) = 0.45$ miles/day and $b$ satisfies $1.4 = 0.45\cdot 3 + b$, so $b = 0.05$. Solve $0.45d + 0.05 = 9.0$ to get $d = (9.0 - 0.05)/0.45 \approx 19.9$.
A store gift card balance $B$ (in dollars) decreases linearly by a constant daily subscription fee. The balance was 26 dollars on day 7 and -10 dollars on day 19. To the nearest tenth, after how many days will the balance be 0 dollars?
15
15.7
16
19
Explanation
The slope is $(-10 - 26)/(19 - 7) = -3$ dollars/day, so $B = -3d + b$ and $26 = -3(7) + b$ gives $b = 47$. Solving $0 = -3d + 47$ yields $d = 47/3 \approx 15.7$.
A taxi charges a base fee of 3.50 dollars plus 1.80 dollars per mile. Let $C(m)$ be the total cost, in dollars, for $m$ miles. Which equation models this relationship?
$C(m) = 3.50m + 1.80$
$C(m) = 1.80m - 3.50$
$C(m) = 3.50 - 1.80m$
$C(m) = 1.80m + 3.50$
Explanation
The slope is 1.80 dollars per mile and the $y$-intercept (base fee) is 3.50, so $C(m) = 1.80m + 3.50$. The other choices swap terms or use an incorrect sign.
A tank drains at a constant rate. The amount of water remaining after $t$ hours is modeled by $W(t) = 500 - 20t$. What is the $y$-intercept of this function?
-20
0
20
500
Explanation
The $y$-intercept is $W(0)$, which is 500. The -20 is the slope (rate of change), not the intercept.
A shipping service uses $C(w) = 1.8w + 4.5$ to model cost $C$ (dollars) for a package of weight $w$ (pounds). If a shipment cost 31.5 dollars, what was the package's weight?
12
15
18
20
Explanation
Solve $31.5 = 1.8w + 4.5$ to get $w = (31.5 - 4.5)/1.8 = 27/1.8 = 15$. The other choices come from ignoring the base fee or mis-subtracting.
A phone's battery percentage decreases linearly: it is 88% one hour after unplugging and 58% four hours after unplugging. What is the slope of the linear function $p(t)$ that models battery percentage $p$ as a function of time $t$ in hours?
-30
-10
-3
10
Explanation
Slope is change in percentage over change in time: $(58 - 88)/(4 - 1) = -30/3 = -10$ percent per hour. The other values use the raw change, the wrong sign, or the wrong division.
A streaming plan follows $y = 5x + 12$, where $x$ is the number of add-on playlists purchased in a month and $y$ is the total monthly cost in dollars. What is the total monthly cost if no add-on playlists are purchased?
5
12
17
60
Explanation
When $x=0$, the cost is the y-intercept, 12 dollars. The other values use the slope, add incorrectly, or multiply the slope and intercept.
A tank has 150 liters of water and drains at a constant rate of 6 liters per minute. What is the slope of a linear function $V(t)$ that gives the volume (liters) $t$ minutes after draining begins?
-150
-6
6
150
Explanation
The slope is the rate of change of volume with respect to time, which is decreasing by 6 liters per minute, so it is -6. Positive values or -150 confuse rate with an amount.
A ride-share driver's earnings $E(r)$ are linear in the number of rides $r$. She earns 142 dollars for 8 rides and 233 dollars for 15 rides. If $E(r)=mr+b$, what is $b$?
13
29
38
91
Explanation
First find $m=(233-142)/(15-8)=91/7=13$, then $b=142-13\cdot8=38$. The 13 is the slope, 91 is the total change, and 29 is an unrelated value.
A tank contains 1200 liters of water at time $t=0$ and 828 liters after 9 hours, decreasing linearly. At what time, in hours after $t=0$, will the tank be empty?
24
26.1
29
32.5
Explanation
Slope $m=(828-1200)/(9-0)=-372/9=-41.333\ldots$, so $V(t)=1200-41.333\ldots t$ and $0=1200-41.333\ldots t$ gives $t\approx29.0$ hours. Other choices result from using the wrong slope or misplacing the intercept.