Question

A dispenser holds 4650 cm^3 of hand sanitizer and is now full. The radius of the dispenser is 6.5 what is the difference between the height of the soap in the full dispenser and the height when 3239 cm^3 of soap remains in the dispenser? Use 3.14 to approximate pi. Full sanitizer Volume = pir^2h partially full sanitizer Volume = pir^2h difference ____?

Answer 1

the difference in height between the full dispenser and the dispenser with 3239 cm³ remaining is approximately 10.63 cm.

Explanation:

The volume of a cylinder is given by the formula
`\(V = \pi r^2 h\)`, where
`\(r\)` is the radius and
`\(h\)` is the height.

For the full dispenser:

`\[V_{\text{full}} = \pi * (6.5)^2 * h_{\text{full}}\]`
For the partially full dispenser:

`\[V_{\text{partial}} = \pi * (6.5)^2 * h_{\text{partial}}\]`
The difference in height is given by subtracting the heights:

`\[ \text{Difference} = h_{\text{full}} - h_{\text{partial}} \]`

Now, we know that
`\(V_{\text{full}} - V_{\text{partial}} = 4650 - 3239\)` (as the volume decreases when soap is used).


`\[ \pi * (6.5)^2 * h_{\text{full}} - \pi * (6.5)^2 * h_{\text{partial}} = 4650 - 3239 \]`

Factoring out common terms:

`\[ \pi * (6.5)^2 * (h_{\text{full}} - h_{\text{partial}}) = 1411 \]`

Now, we can solve for the difference in height:

`\[ h_{\text{full}} - h_{\text{partial}} = (1411)/(\pi * (6.5)^2) \]`

Calculate this expression to find the difference in height.

To compute the difference in height using the provided information:

`\[ h_{\text{full}} - h_{\text{partial}} = (1411)/(\pi * (6.5)^2) \]`

Substitute the values:


`\[ h_{\text{full}} - h_{\text{partial}} = (1411)/(3.14 * (6.5)^2) \]`

Calculate the expression:


`\[ h_{\text{full}} - h_{\text{partial}} \approx (1411)/(3.14 * 42.25) \]\[ h_{\text{full}} - h_{\text{partial}} \approx (1411)/(132.745) \]\[ h_{\text{full}} - h_{\text{partial}} \approx 10.63 \]`

So, the difference in height between the full dispenser and the dispenser with 3239 cm³ remaining is approximately 10.63 cm.