Garden Club: Market Volume Plans

Use volumes to plan produce portions, packages, and displays.

Model the Produce

Use geometric models for the produce. Keep π exact unless told otherwise. When π follows a blank, enter only its coefficient.

The school garden club models a straight cucumber as a cylinder with radius 2.5 cm and length 12 cm. Which expression gives its volume in cubic centimeters?

A. π × (1.25)² × 12B. π × 2.5 × 12C. (1/3) × π × (2.5)² × 12D. π × (2.5)² × 12

For a market sign, Priya models a round melon as a sphere with diameter 15 cm. Find its volume.

Volume = π cm³

The club sells small peppers in cone-shaped paper sleeves. One sleeve has an inside opening radius of 4.5 cm and a perpendicular height of 16 cm. Find its full capacity. Ignore the paper's thickness.

Capacity = π cm³

A cylindrical display jar for dried beans has an inside diameter of 7 cm and an inside height of 8 cm. Find its capacity.

Capacity = π cm³

Andre models a cucumber as a cylinder with radius 2 cm and length 18 cm. He cuts it across its length into six equal cylindrical pieces for a market demonstration. Ignore any material lost in cutting.

What is the volume of each piece? π cm³

Plan Portions and Packages

Use inside dimensions for capacities. Treat each stated geometric model as exact.

The club sells melon halves. Model one whole melon as a sphere with radius 6 cm. It is cut through its center into two equal hemispheres. A hemisphere is half a sphere.

What is the volume of one half? π cm³

A new cone-shaped pepper sleeve must have a capacity of 100π cm³ and a perpendicular height of 12 cm. What inside radius will give that capacity?

Radius = cm

The club replaces a cylindrical bean jar of inside radius 3 cm and inside height 16 cm with a wider cylinder of inside radius 6 cm. What inside height gives the new jar exactly the same capacity?

A. 8 cmB. 4 cmC. 16 cmD. 32 cm

A bag contains 10 radishes, each modeled as a sphere with radius 3 cm. The bag's total interior volume is 500π cm³. Use the sum of the radish volumes to find the percent of the bag's interior occupied by radish material. Do not count air gaps as radish material.

Radish material occupies % of the bag's interior.

Check the Market Plans

Combine or compare volumes as needed. For the final question, explain your reasoning using calculations.

10.

A spherical tomato with radius 4 cm fits inside a cube-shaped gift box with inside edge length 8 cm. Ignore the box material. How much space inside the box is not occupied by the tomato? Use 3.14 for π and round to the nearest tenth of a cubic centimeter.

A. 445.0 cm³B. 311.0 cm³C. 244.1 cm³D. 267.9 cm³

11.

A wide produce sleeve is made by cutting the pointed end off a cone. The original cone has inside radius 6 cm and perpendicular height 18 cm. The removed tip is a smaller cone with radius 2 cm and perpendicular height 6 cm. Ignore the paper's thickness. Find the capacity of the remaining sleeve.

Remaining capacity = π cm³

12.

The club needs a cylindrical display container with capacity 275π cm³. Its inside radius will be 5 cm. Find the required inside height.

Height = cm

13.

The club compares three geometric models for a market display. Which list puts their volumes in order from least to greatest?

Market Display Models
Model	Solid	Radius (cm)	Height (cm)
Pepper sleeve	Cone	5	10
Round tomato	Sphere	5	—
Bean jar	Cylinder	5	10

A. Pepper sleeve, round tomato, bean jarB. Bean jar, round tomato, pepper sleeveC. Pepper sleeve, bean jar, round tomatoD. Round tomato, pepper sleeve, bean jar

14.

A cone-shaped pepper sleeve stands with its point down. Its inside radius at the opening is 6 cm, and its perpendicular height is 18 cm. Peppers reach a level 9 cm above the tip. The space below that level is a smaller cone with radius 3 cm. Elena says, “The peppers reach half the height, so they fill half the sleeve's capacity.” Model the peppers as filling all the space below that level, with no air gaps. Is Elena correct? Calculate the full capacity and the occupied volume, then state what fraction of the capacity is occupied.