Question

hirotos texting plan costs $20 per month, plus $0.05 per text message that is sent or received. emilias plan costs $10 per month and $0.25 per text. using the graph below, which statement is true?
○ hirotos plan costs more than emilias plan when more than 50 texts are sent
○ both plans cost the same when 22 texts are sent
○ emilias plan costs more than hirotos plan when more than 22 texts are sent
○ both plans cost the same when 50 texts are sent

Explanation:

Step1: Define cost functions

Let $x$ = number of texts.
Hiroto's cost: $C_H = 20 + 0.05x$
Emilia's cost: $C_E = 10 + 0.25x$

Step2: Find equal cost point

Set $C_H = C_E$:
$$20 + 0.05x = 10 + 0.25x$$
$$20 - 10 = 0.25x - 0.05x$$
$$10 = 0.20x$$
$$x = \frac{10}{0.20} = 50$$
At $x=50$, $C_H = C_E = 20 + 0.05(50) = 22.5$, matching the graph point $(50, 22.5)$.

Step3: Analyze each option

• Option1: For $x>50$, $C_H < C_E$ (false, Hiroto's is cheaper)
• Option2: At $x=22$, $C_H=20+1.1=21.1$, $C_E=10+5.5=15.5$ (not equal, false)
• Option3: For $x>22$, $C_E < C_H$ until $x=50$ (false)
• Option4: Confirmed in Step2, true.

Answer:

Both plans cost the same when 50 texts are sent.