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Sagot :
To solve this problem, let's break it down step-by-step:
### Step 1: Understand the Relationship
Given that the mass ([tex]\( m \)[/tex]) of a cylinder varies jointly as the square of the radius ([tex]\( r \)[/tex]) and inversely as the square root of the height ([tex]\( h \)[/tex]), we have the mathematical relationship:
[tex]\[ m \propto \frac{r^2}{\sqrt{h}} \][/tex]
This relationship can be written as:
[tex]\[ m = k \frac{r^2}{\sqrt{h}} \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
### Step 2: Define Initial Values
Let's assume the initial radius and height of the cylinder are:
[tex]\[ r_{\text{initial}} = r \][/tex]
[tex]\[ h_{\text{initial}} = h \][/tex]
Using these initial values, the initial mass [tex]\( m_{\text{initial}} \)[/tex] can be defined as:
[tex]\[ m_{\text{initial}} = k \frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}} \][/tex]
### Step 3: Apply the Changes
The radius is reduced by 25%, and the height is increased by 21%. Hence, the new radius and height will be:
[tex]\[ r_{\text{new}} = r_{\text{initial}} \times (1 - 0.25) = 0.75 \times r_{\text{initial}} \][/tex]
[tex]\[ h_{\text{new}} = h_{\text{initial}} \times (1 + 0.21) = 1.21 \times h_{\text{initial}} \][/tex]
### Step 4: Calculate the New Mass
Using the new values of radius and height, the new mass [tex]\( m_{\text{new}} \)[/tex] is:
[tex]\[ m_{\text{new}} = k \frac{r_{\text{new}}^2}{\sqrt{h_{\text{new}}}} \][/tex]
Substituting the new values:
[tex]\[ m_{\text{new}} = k \frac{(0.75r_{\text{initial}})^2}{\sqrt{1.21h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{\sqrt{1.21} \sqrt{h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}} \][/tex]
### Step 5: Compare the Initial and New Mass
Let's express [tex]\( m_{\text{initial}} \)[/tex] and [tex]\( m_{\text{new}} \)[/tex] in terms of each other:
[tex]\[ m_{\text{initial}} = k \frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}} \][/tex]
By simplifying the ratio of [tex]\( m_{\text{new}} \)[/tex] to [tex]\( m_{\text{initial}} \)[/tex]:
[tex]\[ \frac{m_{\text{new}}}{m_{\text{initial}}} = \frac{\frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}}}{\frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}}} = \frac{0.5625}{1.1} \][/tex]
[tex]\[ \frac{m_{\text{new}}}{m_{\text{initial}}} = 0.5113636 \][/tex]
### Step 6: Calculate the Percentage Change in Mass
The percentage change in mass is given by:
[tex]\[ \text{Percentage change} = \left( \frac{m_{\text{new}} - m_{\text{initial}}}{m_{\text{initial}}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( \frac{0.5113636 m_{\text{initial}} - m_{\text{initial}}}{m_{\text{initial}}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( 0.5113636 - 1 \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = -48.8636\% \][/tex]
### Final Answer
The percentage change in the mass of the cylinder is -48.86%. This means the mass of the cylinder is reduced by approximately 48.86% due to the changes in radius and height.
### Step 1: Understand the Relationship
Given that the mass ([tex]\( m \)[/tex]) of a cylinder varies jointly as the square of the radius ([tex]\( r \)[/tex]) and inversely as the square root of the height ([tex]\( h \)[/tex]), we have the mathematical relationship:
[tex]\[ m \propto \frac{r^2}{\sqrt{h}} \][/tex]
This relationship can be written as:
[tex]\[ m = k \frac{r^2}{\sqrt{h}} \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
### Step 2: Define Initial Values
Let's assume the initial radius and height of the cylinder are:
[tex]\[ r_{\text{initial}} = r \][/tex]
[tex]\[ h_{\text{initial}} = h \][/tex]
Using these initial values, the initial mass [tex]\( m_{\text{initial}} \)[/tex] can be defined as:
[tex]\[ m_{\text{initial}} = k \frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}} \][/tex]
### Step 3: Apply the Changes
The radius is reduced by 25%, and the height is increased by 21%. Hence, the new radius and height will be:
[tex]\[ r_{\text{new}} = r_{\text{initial}} \times (1 - 0.25) = 0.75 \times r_{\text{initial}} \][/tex]
[tex]\[ h_{\text{new}} = h_{\text{initial}} \times (1 + 0.21) = 1.21 \times h_{\text{initial}} \][/tex]
### Step 4: Calculate the New Mass
Using the new values of radius and height, the new mass [tex]\( m_{\text{new}} \)[/tex] is:
[tex]\[ m_{\text{new}} = k \frac{r_{\text{new}}^2}{\sqrt{h_{\text{new}}}} \][/tex]
Substituting the new values:
[tex]\[ m_{\text{new}} = k \frac{(0.75r_{\text{initial}})^2}{\sqrt{1.21h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{\sqrt{1.21} \sqrt{h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}} \][/tex]
### Step 5: Compare the Initial and New Mass
Let's express [tex]\( m_{\text{initial}} \)[/tex] and [tex]\( m_{\text{new}} \)[/tex] in terms of each other:
[tex]\[ m_{\text{initial}} = k \frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}} \][/tex]
[tex]\[ m_{\text{new}} = k \frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}} \][/tex]
By simplifying the ratio of [tex]\( m_{\text{new}} \)[/tex] to [tex]\( m_{\text{initial}} \)[/tex]:
[tex]\[ \frac{m_{\text{new}}}{m_{\text{initial}}} = \frac{\frac{0.5625 r_{\text{initial}}^2}{1.1 \sqrt{h_{\text{initial}}}}}{\frac{r_{\text{initial}}^2}{\sqrt{h_{\text{initial}}}}} = \frac{0.5625}{1.1} \][/tex]
[tex]\[ \frac{m_{\text{new}}}{m_{\text{initial}}} = 0.5113636 \][/tex]
### Step 6: Calculate the Percentage Change in Mass
The percentage change in mass is given by:
[tex]\[ \text{Percentage change} = \left( \frac{m_{\text{new}} - m_{\text{initial}}}{m_{\text{initial}}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( \frac{0.5113636 m_{\text{initial}} - m_{\text{initial}}}{m_{\text{initial}}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( 0.5113636 - 1 \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = -48.8636\% \][/tex]
### Final Answer
The percentage change in the mass of the cylinder is -48.86%. This means the mass of the cylinder is reduced by approximately 48.86% due to the changes in radius and height.
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