Ski Resort: Lift Lines and Snowfall Estimates

Use rational estimates to locate, compare, and interpret irrational values.

Bound the Recorded Values

Work without a calculator. Use nearby perfect squares or square the decimal bounds. All square roots mean the nonnegative square root.

A snowfall model gives a depth of √58 inches. Between which consecutive whole numbers does this depth lie?

< √58 <

A lift-line model gives a wait of √118 minutes. Which pair of bounds is correct?

A. 10.8 < √118 < 10.9B. 10.9 < √118 < 11.0C. 11.8 < √118 < 11.9D. 10.7 < √118 < 10.8

A snow-depth calculation gives √7 inches. Use 2.64² = 6.9696 and 2.65² = 7.0225 to estimate √7 to the nearest hundredth.

Estimated depth: inches

For a lift wait of √92 minutes, which calculation establishes that the wait is between 9.5 and 9.6 minutes?

A. 9.5² = 90.25 and 9.6² = 92.16.B. 9.4² = 88.36 and 9.5² = 90.25.C. 9² = 81 and 10² = 100.D. 9.6² = 92.16 and 9.7² = 94.09.

Locate and Compare

Use rational benchmarks to locate the resort's modeled values. Each marked interval on a number line represents 0.1 unit.

A snowfall estimate is √43 inches. Between which adjacent marks should it be placed?

A. Between 6.3 and 6.4.B. Between 6.4 and 6.5.C. Between 6.6 and 6.7.D. Between 6.5 and 6.6.

The signed difference between a modeled snowfall total and the seasonal average is −√19 inches. Write the two adjacent marks that surround this value, smaller first.

< −√19 <

Four modeled lift waits, in minutes, are π, √15, 3.9, and √16.2. Which list orders them from least to greatest?

A. π < √15 < √16.2 < 3.9B. √15 < π < 3.9 < √16.2C. π < √15 < 3.9 < √16.2D. π < 3.9 < √15 < √16.2

A modeled snowfall depth is √26 inches. Select every rational depth that is greater than √26 inches.

Use Estimates to Make Decisions

Apply rational estimates to the resort reports. Round only as directed.

The morning snowfall model gives √82 inches. The resort posts depths to the nearest tenth. Use 9.05² = 81.9025 and 9.06² = 82.0836.

What depth should the resort post? inches

10.

Noor can choose the Pine lift, with a modeled wait of √150 minutes, or the Summit lift, with a wait of 12.3 minutes. Which comparison is correct?

A. The Pine wait is shorter because √150 < 12.0.B. The Pine wait is shorter because √150 < 12.3.C. The Summit wait is shorter because √150 > 12.3.D. The waits are equal because √150 = 12.3.

11.

A resort planning spreadsheet includes the expression π². Use 3.14 as a rational approximation for π, then square that approximation.

Estimate π² to the nearest hundredth:

12.

A snowfall model predicts √37 inches during each of two equal storm periods. Estimate the combined depth, 2√37, to the nearest tenth. You may use 6.08² = 36.9664 and 6.09² = 37.0881.

Combined depth: inches

13.

Eli's modeled lift wait is √170 minutes. He wants a time allowance that is at least this long. His timer accepts only multiples of 0.1 minute. Use 13² = 169 and 13.1² = 171.61.

What is the smallest time allowance he can enter? minutes

14.

A snowfall model gives a depth of √(49 + 36) inches. Sam writes, “√(49 + 36) = √49 + √36 = 13 inches.” Explain why Sam's method is incorrect. Use consecutive whole-number bounds to show why 13 inches is not a reasonable estimate.