Mar 9, 2020 | High Park

A sunny weekend in Toronto has inspired this week's problems! Can you solve these, Mathletes?

**Lower Elementary:**

*Question: *Jack builds a sand castle with 8 different sand molds. The smallest mold holds 1^{1}/_{2} cups of sand. The next mold holds 3 cups of sand. The third mold holds 6 cups of sand, and the fourth holds 12 cups of sand. If the pattern continues, then how much sand does the biggest mold hold?

**Upper Elementary:**

*Question: *Aria, Brett, and Chelsea bounce a beach ball between them 100 times. Aria bounces the ball to Brett, Brett bounces the ball to Chelsea, Chelsea bounces the ball to Aria, and the pattern continues. Who bounces the ball the 100th time?

**Middle School:**

*Question: *Ava collects 4 more than twice as many sea shells as Ben. Ben collects 1 fewer than half as many as Carla. Carla collects 200 sea shells. How many sea shells does Ava collect?

**Algebra and Up:**

*Question: *A circular beach umbrella with an area of 16π square feet casts a circular shadow with a circumference of 16π feet. What is the circumference of an umbrella at the same height that casts a shadow whose area is 256π square feet? (For the purposes of this exercise, pretend umbrellas are flat circles instead of domes.)

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__Answers:__

**Lower Elementary:**

*Question: *Jack builds a sand castle with 8 different sand molds. The smallest mold holds 1^{1}/_{2} cups of sand. The next mold holds 3 cups of sand. The third mold holds 6 cups of sand, and the fourth holds 12 cups of sand. If the pattern continues, then how much sand does the biggest mold hold?

*Answer: *192 cups

*Solution: *Each mold holds twice as much as the last. So, the fifth holds 12 + 12 = 24 cups. The sixth holds 24 + 24 = 48 cups. The seventh holds 48 + 48 = 96 cups, and the eighth holds 96 + 96 = 192 cups.

**Upper Elementary:**

*Question: *Aria, Brett, and Chelsea bounce a beach ball between them 100 times. Aria bounces the ball to Brett, Brett bounces the ball to Chelsea, Chelsea bounces the ball to Aria, and the pattern continues. Who bounces the ball the 100th time?

*Answer: *Aria

*Solution: *There are three beach ball bouncers in the group, and Chelsea is the third. So, for each bounce that’s numbered a multiple of 3, Chelsea must bounce the ball. The closest multiple of 3 before 100 is 99, so Chelsea bounces the ball the 99th time. That means that Aria bounces the ball the 100th time because she’s next in the pattern.

**Middle School:**

*Question: *Ava collects 4 more than twice as many sea shells as Ben. Ben collects 1 fewer than half as many as Carla. Carla collects 200 sea shells. How many sea shells does Ava collect?

*Answer: *202 sea shells

*Solution: *Since Carla collects 200, Ben must collect 99 because half of 200 is 100 and one less than that is 99. Ava collects 4 more than twice as many as Ben. Twice as many as 99 is 198, and 4 more than that makes 202 sea shells.

**Algebra and Up:**

*Question: *A circular beach umbrella with an area of 16π square feet casts a circular shadow with a circumference of 16π feet. What is the circumference of an umbrella at the same height that casts a shadow whose area is 256π square feet? (For the purposes of this exercise, pretend umbrellas are flat circles instead of domes.)

*Answer: *16π feet

*Solution: *If the small umbrella has an area of 16π square feet, its radius must be 4 feet. If its shadow has a circumference of 16π feet, then its radius must be 8 feet. The radius of the shadow is double the radius of the umbrella. Since we know the area of the larger shadow is 256π square feet, its radius must be 16 feet. That means that the radius of the larger umbrella must be half of that, 8 feet, making its circumference 16π feet.