Question

kimi worked as a summer lifeguard at her local swimming pool, and she wanted to know whether hotter days attracted more swimmers. on the days she worked, she kept track of the days high temperature in degrees fahrenheit, x, and the number of swimmers, y. then, she estimated a line of best fit. the line of best fit has a y-intercept of 70 and a slope of 4.5. based on this information, evaluate each statement. | true | false | the pool got about 439 swimmers when the high temperature was 82°f. | | | about 65 fewer swimmers went to the pool when the high was 80°f than when it was 90°f. | | | if the high was 92°f, the pool got around 414 swimmers. | | | kimi can conclude that the high temperature never dipped below 70°f. | | |

Explanation:

First, the line of best fit equation is $y = 4.5x + 70$.

Step1: Check 82°F

Substitute $x=82$: $y = 4.5(82) + 70 = 369 + 70 = 439$? Wait, $4.5×82=369$, $369+70=439$. But wait, no—wait, 4.5×80=360, 4.5×2=9, so 369+70=439. But wait, is that correct? Wait no, wait the question says "about 439"—but wait, let's check the second statement first.

Step2: Check 80°F vs 90°F

Difference in x: $90-80=10$. Slope=4.5, so difference in y: $4.5×10=45$. So fewer swimmers at 80°F: 45, not 65. So statement 2 is False.

Step3: Check 92°F

$y=4.5×92 +70=414 +70=484≠414$. So statement3 is False.

Step4: Check temperature below 70°F

The y-intercept is the predicted y when x=0, not the minimum x. Kimi didn't measure temperatures below 70°F, so can't conclude. Statement4 is False.

Wait, statement1: $4.5×82 +70=369+70=439$. But wait, is that correct? Wait 82×4=328, 82×0.5=41, so 328+41=369+70=439. But wait, is the statement true? Wait the problem says "the pool got about 439 swimmers"—but the line of best fit is a prediction, not the actual number. But the question says "based on this information"—so the prediction is 439. But wait, did I miscalculate? Wait no, 4.5×82=369, plus 70 is 439. But wait, let's recheck: 82×4.5= (80×4.5)+(2×4.5)=360+9=369. Yes. So why is statement1 False? Wait no, wait maybe I made a mistake. Wait no—the problem says "the pool got about 439 swimmers"—the line of best fit predicts 439, but does that mean the actual number was about 439? No, the line is a prediction, not the actual value. So the statement claims the pool "got" (actual) about 439, but the line only predicts that. So statement1 is False.

Ah, right! The line of best fit is a prediction, not the actual number of swimmers. So all four statements are False.

Wait, let's confirm:

1. Actual number vs predicted: statement says "got about 439" (actual), but we only have predicted value. So False.
2. Difference is 45, not 65: False.
3. Predicted is 484, not 414: False.
4. Can't conclude minimum temperature: False.

So all four are False.

Final Answer:

1. False
2. False
3. False
4. False

Wait, the original answer format for multiple statements:

The line of best fit is $y = 4.5x + 70$.

Step1: Evaluate 82°F statement

Predicted $y=4.5×82+70=439$, but this is a prediction, not actual swimmers. Statement is False.

Step2: Evaluate 80°F vs 90°F

Temperature difference=10°F, so predicted swimmer difference=4.5×10=45 (not 65). Statement is False.

Step3: Evaluate 92°F

Predicted $y=4.5×92+70=484≠414$. Statement is False.

Step4: Evaluate minimum temperature

Y-intercept does not indicate minimum measured temperature. No data on temps <70°F. Statement is False.

Yes, that's correct. So the final answer is all False.

Answer:

1. The pool got about 439 swimmers when the high temperature was 82°F: False
2. About 65 fewer swimmers went to the pool when the high was 80°F than when it was 90°F: False
3. If the high was 92°F, the pool got around 414 swimmers: False
4. Kimi can conclude that the high temperature never dipped below 70°F: False