Q. 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 $$ 112 \pi \mathrm{~cm}{ }^{3} $$. 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.

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### Problem 1
The amount of gasoline $$ G $$ used by a car varies jointly as the distance $$ D $$ travelled and the square root of the speed $$ S $$. This can be expressed as:
$$
G = k \cdot D \cdot \sqrt{S}
$$
where $$ k $$ is a constant.

Given that $$ G = 25 $$ liters, $$ D = 100 $$ km, and $$ S = 100 $$ km/hr, we can find $$ k $$:
$$
25 = k \cdot 100 \cdot \sqrt{100}
$$
$$
25 = k \cdot 100 \cdot 10
$$
$$
25 = 1000k
$$
$$
k = \frac{25}{1000} = 0.025
$$

Now, we want 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 $$ h $$:
$$
A = k \cdot b \cdot h
$$

Given that $$ A = 36 $$ cm², $$ b = 8 $$ cm, and $$ h = 9 $$ cm, we can find $$ k $$:
$$
36 = k \cdot 8 \cdot 9
$$
$$
36 = 72k
$$
$$
k = \frac{36}{72} = \frac{1}{2}
$$

Now, we want to find the area when $$ b = 10 $$ cm and $$ h = 7 $$ cm:
$$
A = \frac{1}{2} \cdot 10 \cdot 7
$$
$$
A = \frac{1}{2} \cdot 70 = 35 \text{ cm}^2
$$

### Problem 3
The volume $$ V $$ of a right circular cylinder varies jointly as the height $$ h $$ and the square of the radius $$ r $$:
$$
V = k \cdot r^2 \cdot h
$$

Given that $$ V = 112\pi $$ cm³, $$ r = 4 $$ cm, and $$ h = 7 $$ cm, we can find $$ k $$:
$$
112\pi = k \cdot 4^2 \cdot 7
$$
$$
112\pi = k \cdot 16 \cdot 7
$$
$$
112\pi = 112k
$$
$$
k = \pi
$$

Now, we want to find the volume for $$ r = 8 $$ cm and $$ h = 14 $$ cm:
$$
V = \pi \cdot 8^2 \cdot 14
$$
$$
V = \pi \cdot 64 \cdot 14
$$
$$
V = 896\pi \text{ cm}^3
$$

### Problem 4
The mass $$ M $$ of a rectangular sheet of wood varies jointly as the length $$ L $$ and the width $$ W $$:
$$
M = k \cdot L \cdot W
$$

Given that $$ M = 200 $$ grams, $$ L = 20 $$ cm, and $$ W = 10 $$ cm, we can find $$ k $$:
$$
200 = k \cdot 20 \cdot 10
$$
$$
200 = 200k
$$
$$
k = 1
$$

Now, we want 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 $$ L $$, $$ W $$, and $$ T $$ to find the weight. If you provide those values, I can help you calculate the weight.
1. The car will use 38.4 liters of gasoline on a 192 km trip at 64 km/hr. 2. The area of the triangle with a base of 10 cm and a height of 7 cm is 35 cm². 3. The volume of the cylinder with a radius of 8 cm and a height of 14 cm is 896π cm³. 4. The mass of the rectangular sheet of wood with a length of 15 cm and a width of 10 cm is 150 grams.

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