Missing Measurements of Solids from the Area

A toy company wants to replace some of the sides of their cardboard boxes with a thick transparent material to enable customers to see the packaged products inside. To order the material, the company needs to know the box’s dimensions. Given that the total area of the box is |2\ 646\ \text{cm}^2,| determine the measure of the box’s edges.

Before making a delivery, a trucker must ensure that the dimensions of the object being transported does not exceed the width of the trailer. He only has access to the measurements identified in the following image for his next delivery. Help him determine the measure of the box’s width.

Sergei wants to coat his new lidded, cylindrical trash can with an anti-rust coating. His neighbour tells him about a product that covered his |1{.}335\ \text{m}^2| garbage can with a single coat. However, Sergei's trash can is twice as tall as his neighbour's. If the diameter of his neighbour’s garbage is |0{.}5\ \text{m}|, how tall is Sergei's trash can?

Gabriela wants to buy a decorative incense burner for her living room. The burner is a regular octagonal pyramid. Gabriela doesn't know if it can fit on a shelf |15\ \text{cm}| deep.

Gabriela knows the total area of the incense burner is |789{.}12\ \text{cm}^2.| The apothem of the pyramid is |15\ \text{cm}| and the measure of one side of the octagon is |8\ \text{cm}.|

Can the incense burner fit on the shelf if the base’s apothem |(\color{#3B87CD}{a_b})| is perpendicular to the wall?

Stefan plays piano and he frequently uses an old metronome that is shaped like a square-based pyramid.

To polish the metronome, he needs to know the measure of the pyramid’s apothem. He knows that the base of the metronome measures |10\ \text{cm}| and the total area of the pyramid is |512{.}4\ \text{cm}^2.|

What is the measure of the metronome’s apothem?

Parents bought cone-shaped party hats to distribute to all the guests at their child’s birthday party. To ensure that the model found online is large enough for the children's heads, they need to know the measure of the circle’s radius at the base of the hat.

A resort wants to install a ski lift to provide a panoramic view. To complete this project, the resort needs to know the distance between the foot of the mountain and the summit.

Using topographic maps of the region, the lateral area of ​​the conical-shaped mountain is estimated to be |1{.}7\ \text{km}^2| and the measurement of the radius of the base is about |500\ \text{m}.| What should be the ski lift’s length, to the nearest metre?

During kickoffs in football games, kickers often use small cone-shaped stands to hold the ball upright.

If the tip of the ball coincides with the apex of the cone, how far is the ball from the ground when it is on the stand?