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Sagot :
Of course! Let's break down and solve each part of the question step by step.
### Part 1: Calculating the Volume of the Larger Tank
1. Understanding the Ratio of the Sides:
- The ratio of the lengths of the corresponding sides of two similar rectangular water tanks is given as 3:5.
2. Finding the Ratio of the Volumes:
- The ratio of the volumes of two similar geometric shapes is the cube of the ratio of their corresponding side lengths.
- Hence, the volume ratio [tex]\( V_{\text{ratio}} \)[/tex] can be calculated as:
[tex]\[ V_\text{ratio} = \left( \frac{3}{5} \right)^3 \][/tex]
3. Volume of the Smaller Tank:
- The volume of the smaller tank is given as [tex]\( 8.1 \, \text{m}^3 \)[/tex].
4. Calculating the Volume of the Larger Tank:
- Let [tex]\( V_{\text{smaller}} = 8.1 \, \text{m}^3 \)[/tex].
- The volume of the larger tank [tex]\( V_{\text{larger}} \)[/tex] is found by inverting the volume ratio and multiplying by the smaller tank's volume:
[tex]\[ V_{\text{larger}} = \frac{V_{\text{smaller}}}{( \frac{3}{5} )^3} \][/tex]
- Solving this gives you the volume of the larger tank:
[tex]\[ V_{\text{larger}} = 37.5 \, \text{m}^3 \][/tex]
### Part 2: Finding the Coordinates of Point Q
1. Understanding the Scale Factor:
- An enlargement scale factor specifies how much a shape is magnified from its original size.
- The scale factor given is 3.
2. Coordinates of Point P:
- Point P is initially at coordinates [tex]\( (0, 2) \)[/tex].
3. Effect of Enlargement on Point P:
- The point P is scaled by a factor of 3, producing a new point [tex]\( P1 \)[/tex] at coordinates [tex]\( (4, 6) \)[/tex].
4. Finding Coordinates of Q:
- To find the coordinates of Q under the same enlargement factor, multiply the coordinates of P by the scale factor:
- [tex]\( Q_x = 0 \times 3 = 0 \)[/tex]
- [tex]\( Q_y = 2 \times 3 = 6 \)[/tex]
- Thus, the coordinates of point Q are:
[tex]\[ Q = (0, 6) \][/tex]
### Summary of Results
1. The volume of the larger tank is [tex]\( 37.5 \, \text{m}^3 \)[/tex].
2. The coordinates of point Q are [tex]\( (0, 6) \)[/tex].
### Part 1: Calculating the Volume of the Larger Tank
1. Understanding the Ratio of the Sides:
- The ratio of the lengths of the corresponding sides of two similar rectangular water tanks is given as 3:5.
2. Finding the Ratio of the Volumes:
- The ratio of the volumes of two similar geometric shapes is the cube of the ratio of their corresponding side lengths.
- Hence, the volume ratio [tex]\( V_{\text{ratio}} \)[/tex] can be calculated as:
[tex]\[ V_\text{ratio} = \left( \frac{3}{5} \right)^3 \][/tex]
3. Volume of the Smaller Tank:
- The volume of the smaller tank is given as [tex]\( 8.1 \, \text{m}^3 \)[/tex].
4. Calculating the Volume of the Larger Tank:
- Let [tex]\( V_{\text{smaller}} = 8.1 \, \text{m}^3 \)[/tex].
- The volume of the larger tank [tex]\( V_{\text{larger}} \)[/tex] is found by inverting the volume ratio and multiplying by the smaller tank's volume:
[tex]\[ V_{\text{larger}} = \frac{V_{\text{smaller}}}{( \frac{3}{5} )^3} \][/tex]
- Solving this gives you the volume of the larger tank:
[tex]\[ V_{\text{larger}} = 37.5 \, \text{m}^3 \][/tex]
### Part 2: Finding the Coordinates of Point Q
1. Understanding the Scale Factor:
- An enlargement scale factor specifies how much a shape is magnified from its original size.
- The scale factor given is 3.
2. Coordinates of Point P:
- Point P is initially at coordinates [tex]\( (0, 2) \)[/tex].
3. Effect of Enlargement on Point P:
- The point P is scaled by a factor of 3, producing a new point [tex]\( P1 \)[/tex] at coordinates [tex]\( (4, 6) \)[/tex].
4. Finding Coordinates of Q:
- To find the coordinates of Q under the same enlargement factor, multiply the coordinates of P by the scale factor:
- [tex]\( Q_x = 0 \times 3 = 0 \)[/tex]
- [tex]\( Q_y = 2 \times 3 = 6 \)[/tex]
- Thus, the coordinates of point Q are:
[tex]\[ Q = (0, 6) \][/tex]
### Summary of Results
1. The volume of the larger tank is [tex]\( 37.5 \, \text{m}^3 \)[/tex].
2. The coordinates of point Q are [tex]\( (0, 6) \)[/tex].
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