Q. Activity 19 : Think Deeper! Solve the following: 1. The amount of gasoline used by a car varies jointly as the distance travelled and the square root of the speed. Suppose a car used 25 liters on a 100 kilometer trip at \( 100 \mathrm{~km} / \mathrm{hr} \). About how many liters will it use on a 192 kilometer trip at \( 64 \mathrm{~km} / \mathrm{hr} \) ? 2. The area of triangle varies jointly as the base and the height. A triangle with a base of 8 cm and a height of 9 cm has an area of 36 square centimeters. Find the area when the base is 10 cm and the height is 7 cm . 3. The volume of a right circular cylinder varies jointly as the height and the square of the radius. The volume of a right circular cylinder, with radius 4 centimeters and height 7 centimeters , is 112 cm . Find the volume of another cylinder with radius 8 centimeters and height 14 centimeters. 4. The mass of a rectangular sheet of wood varies jointly as the length and the width. When the length is 20 cm and the width is 10 cm , the mass is 200 grams. Find the mass when the length is 15 cm and the width is 10 cm . 5. The weight of a rectangular block of metal varies jointly as its length, width and thickness.

Let's solve each of the problems step by step. ### Problem 1: The amount of gasoline used \( G \) varies jointly as the distance \( D \) and the square root of the speed \( S \). This can be expressed as: \[ G = k \cdot D \cdot \sqrt{S} \] where \( k \) is the constant of variation. From the information given: - \( G = 25 \) liters - \( D = 100 \) km - \( S = 100 \) km/hr Substituting these values into the equation: \[ 25 = k \cdot 100 \cdot \sqrt{100} \] \[ 25 = k \cdot 100 \cdot 10 \] \[ 25 = 1000k \] \[ k = \frac{25}{1000} = 0.025 \] Now, we need to find the gasoline used for a trip of \( D = 192 \) km at \( S = 64 \) km/hr: \[ G = 0.025 \cdot 192 \cdot \sqrt{64} \] \[ G = 0.025 \cdot 192 \cdot 8 \] \[ G = 0.025 \cdot 1536 \] \[ G = 38.4 \text{ liters} \] ### Problem 2: The area \( A \) of a triangle varies jointly as the base \( b \) and the height \( h \): \[ A = k \cdot b \cdot h \] From the information given: - \( A = 36 \) cm² - \( b = 8 \) cm - \( h = 9 \) cm Substituting these values: \[ 36 = k \cdot 8 \cdot 9 \] \[ 36 = 72k \] \[ k = \frac{36}{72} = 0.5 \] Now, we need to find the area when \( b = 10 \) cm and \( h = 7 \) cm: \[ A = 0.5 \cdot 10 \cdot 7 \] \[ A = 0.5 \cdot 70 = 35 \text{ cm²} \] ### Problem 3: The volume \( V \) of a right circular cylinder varies jointly as the height \( h \) and the square of the radius \( r \): \[ V = k \cdot h \cdot r^2 \] From the information given: - \( V = 112 \) cm³ - \( r = 4 \) cm - \( h = 7 \) cm Substituting these values: \[ 112 = k \cdot 7 \cdot 4^2 \] \[ 112 = k \cdot 7 \cdot 16 \] \[ 112 = 112k \] \[ k = 1 \] Now, we need to find the volume for \( r = 8 \) cm and \( h = 14 \) cm: \[ V = 1 \cdot 14 \cdot 8^2 \] \[ V = 14 \cdot 64 = 896 \text{ cm³} \] ### Problem 4: The mass \( M \) of a rectangular sheet of wood varies jointly as the length \( L \) and the width \( W \): \[ M = k \cdot L \cdot W \] From the information given: - \( M = 200 \) grams - \( L = 20 \) cm - \( W = 10 \) cm Substituting these values: \[ 200 = k \cdot 20 \cdot 10 \] \[ 200 = 200k \] \[ k = 1 \] Now, we need to find the mass when \( L = 15 \) cm and \( W = 10 \) cm: \[ M = 1 \cdot 15 \cdot 10 \] \[ M = 150 \text{ grams} \] ### Problem 5: The weight \( W \) of a rectangular block of metal varies jointly as its length \( L \), width \( W \), and thickness \( T \): \[ W = k \cdot L \cdot W \cdot T \] To solve this problem, we would need specific values for the dimensions and the weight of the block to find the constant \( k \) and then calculate the weight for another set of dimensions. If you provide those values, I can help you with that calculation!