Question

1. a road is being paved where 5% is completed after day 1 and 10% after day 2.

a. based on this information, use a linear model to predict the day on which the road is 60% paved and 100% paved.
b. the roads surface is 100% paved on day 20. does this agree with the prediction you made?
c. is the percentage paved of the roads surface a linear function of the day?

1. in science class, tyler uses a graduated cylinder with water in it to measure the volume of some identical ball bearings. after dropping in 6 ball bearings so they are all under water, the water in the cylinder is at a height of 12 milliliters. after dropping in 4 additional ball bearings so they are all under water, the water in the cylinder is at a height of 13 milliliters.

a. what is the volume of 1 ball bearing?
b. how much water was in the cylinder before any ball bearings were dropped in?
c. what should be the height of the water after a total of 15 ball bearings are dropped in?
d. is the total volume measured in a linear relationship with the number of ball bearings dropped in the graduated cylinder? if so, what does the y-intercept of a line representing this relationship mean? if not, explain your reasoning.

Explanation:

Problem 1

Step1: Define linear model variables

Let $x$ = day number, $y$ = % paved. Points: $(1, 5)$, $(2, 10)$

Step2: Calculate slope $m$

$m = \frac{10-5}{2-1} = 5$

Step3: Find y-intercept $b$

Use $(1,5)$: $5 = 5(1) + b \implies b=0$. Model: $y=5x$

Step4: Predict day for 60% paved

Set $y=60$: $60=5x \implies x=12$

Step5: Predict day for 100% paved

Set $y=100$: $100=5x \implies x=20$

Step6: Compare prediction to actual

Actual day 20 matches prediction.

Step7: Evaluate linearity

Daily progress is constant 5%, so linear.

Step1: Define linear model variables

Let $x$ = number of ball bearings, $y$ = total volume (mL). Points: $(6,12)$, $(10,13)$

Step2: Calculate volume per bearing (slope)

$m = \frac{13-12}{10-6} = \frac{1}{4} = 0.25$ mL

Step3: Find initial water volume (y-intercept)

Use $(6,12)$: $12 = 0.25(6) + b \implies 12=1.5+b \implies b=10.5$ mL

Step4: Predict volume for 15 bearings

$y=0.25(15)+10.5 = 3.75+10.5=14.25$ mL

Step5: Evaluate linear relationship

Volume increases by constant 0.25 mL per bearing, so linear. Y-intercept = initial water volume.

Answer:

a. 60% paved on day 12; 100% paved on day 20
b. Yes, it agrees with the prediction.
c. Yes, it is a linear function.

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Problem 2