Math  /  Algebra

QuestionQuestion At the beginning of the year, Valeria had $20\$ 20 in savings and saved an additional $15\$ 15 each week thereafter. Jackson started the year with $70\$ 70 and saved $5\$ 5 every week. Let VV represent the amount of money Valeria has saved tt weeks after the beginning of the year and let JJ represent the amount of money Jackson has saved tt weeks after the beginning of the year. Graph each function and determine the who has more money saved after 2 weeks.

Studdy Solution

STEP 1

What is this asking? We need to figure out how much money Valeria and Jackson have saved after a certain number of weeks and see who has more money after 2 weeks! Watch out! Don't mix up Valeria's and Jackson's savings rates or their starting amounts!

STEP 2

1. Define the functions
2. Calculate savings after 2 weeks
3. Compare the savings

STEP 3

Let's **define** Valeria's savings, VV, as a function of the number of weeks, tt.
She starts with $20\$20 and saves $15\$15 each week.
So, her savings can be represented as: V(t)=20+15tV(t) = 20 + 15 \cdot t

STEP 4

Now, let's **define** Jackson's savings, JJ, as a function of time, tt.
He begins with $70\$70 and adds $5\$5 each week.
His savings can be represented as: J(t)=70+5tJ(t) = 70 + 5 \cdot t

STEP 5

Let's **calculate** Valeria's savings after t=2t = 2 weeks.
We'll substitute t=2t = 2 into Valeria's savings function: V(2)=20+152V(2) = 20 + 15 \cdot 2 V(2)=20+30V(2) = 20 + 30V(2)=50V(2) = 50So, Valeria has $50\$50 after **2 weeks**.

STEP 6

Now, let's **calculate** Jackson's savings after t=2t = 2 weeks.
Substituting t=2t = 2 into Jackson's savings function: J(2)=70+52J(2) = 70 + 5 \cdot 2 J(2)=70+10J(2) = 70 + 10J(2)=80J(2) = 80So, Jackson has $80\$80 after **2 weeks**.

STEP 7

After 2 weeks, Valeria has $50\$50 and Jackson has $80\$80.
Since 80>5080 > 50, Jackson has more money saved.

STEP 8

After 2 weeks, Jackson has more money saved than Valeria.
Jackson has $80\$80, while Valeria has $50\$50.

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