Launch Day: See More in Each Expression

Use equivalent expressions to interpret rocket designs and launch plans.

Rocket Design

Read each model carefully. Use the rewritten form to describe the quantities, not just the calculations.

A rocket has the two mass components shown below, where x is positive. Its total mass is 4x + x = 5x grams.

Rocket Mass
Component	Mass (grams)
Rocket without payload	4x
Payload	x

What percent of the total mass is the payload? %

Another rocket has a mass of m grams without its payload. The payload has a mass of 0.25m grams. The total is m + 0.25m = 1.25m. Rewriting m as 0.80(1.25m) reveals which relationship?

A. The rocket without its payload is 80% of the total mass.B. The payload is 25% of the total mass.C. The rocket without its payload is 125% of the total mass.D. The payload is 80% of the total mass.

Four rocket parts originally have a mass of n grams each, where n > 5. Three parts are each made 2 grams lighter, and the fourth is made 5 grams lighter. The new combined mass is 3(n − 2) + (n − 5) = 4n − 11 grams.

Compared with the original combined mass of 4n grams, how many grams were removed altogether? grams

A rocket's total mass is m grams. Its body has mass 0.70m, its engine has mass 0.20m, and its recovery system has mass 0.10m. The rewrite 0.70m + 0.20m + 0.10m = m describes the complete rocket. Select every statement supported by these expressions.

A camera adds 9 grams to a rocket with an original mass of m grams. Jules uses m + 9, while Ren uses 1.09m = m + 0.09m. Which statement correctly compares their models for positive values of m?

A. Jules adds 9 grams; Ren adds 9% of the original mass.B. Both models add 9 grams for every original mass.C. Jules adds 9% of the original mass; Ren adds 9 grams.D. Both models add 109% of the original mass.

Flight Records

Interpret the relationships shown by each rewrite. Give fractions in simplest form.

A flight model gives an ascent time of n seconds and a descent time of 2n seconds, where n is positive. The total flight time is n + 2n = 3n seconds.

What fraction of the total flight time is spent descending?

Two rocket tests have predicted peak heights of x + 15 meters and 2x + 15 meters, where x is positive. Their difference is (2x + 15) − (x + 15) = x. What does this rewrite reveal?

A. The second prediction is exactly twice the first prediction.B. The second prediction is exactly 15 meters above the first.C. The second prediction is exactly x meters above the first.D. The second prediction is exactly x + 15 meters above the first.

A ground crew measures a distance as 100x + 40 centimeters. Since 100 centimeters equals 1 meter, they rewrite it as 100(x + 0.40) centimeters. Which expression gives the same distance in meters?

A. x + 40B. 0.01x + 0.40C. 100x + 0.40D. x + 0.40

For positive n, a rocket's predicted peak height is 6n meters, but its measured peak height is 5n meters. The shortfall is 6n − 5n = n meters. Rewriting n as (1/6)(6n) compares the shortfall with the prediction.

What fraction of the predicted height is the shortfall?

10.

A launch clock begins at 0 seconds. A rocket leaves the pad at n + 4 seconds and lands at n + 19 seconds, where n is a nonnegative launch delay. Ari says that increasing n by 10 must make the flight last 10 seconds longer. Rewrite (n + 19) − (n + 4) and explain whether Ari is correct.

Ground-Crew Decisions

Choose or interpret the expression form that makes each planning relationship clear.

11.

The crew starts with 30 batteries. Each launch uses 2 batteries. After n launches, 30 − 2n = 2(15 − n) batteries remain, where n is a whole number from 0 to 15. What does 15 − n represent?

A. The number of launches already completed.B. The number of batteries already used.C. The number of additional launches the batteries can support.D. The number of individual batteries remaining.

12.

Two storage layouts hold rockets upright. The first has 5 racks with n + 4 slots each. The second has 4 racks with n + 5 slots each. Subtracting their capacities gives 5(n + 4) − 4(n + 5) = 5n + 20 − 4n − 20.

The first layout has how many more slots when n = 12? slots

13.

A rectangular landing zone is n meters wide and n + 8 meters long. Its perimeter is 2(n + 8) + 2n = 4(n + 4) meters. Which statement does the factored form make easiest to see?

A. It has the same area as a square with side length 4n + 4 meters.B. It has the same perimeter as a square with side length n + 4 meters.C. It has the same area as a square with side length n + 4 meters.D. It has the same perimeter as a square with side length n + 8 meters.