The Volume V(r) (in Cubic Meters ) Of A Spherical Balloon With Radius R Meters Is Given By V(r)=(4)/(3)\pi (2024)

A) Rate of change = -3.

B) When s = 0, r has a value of 18.

C) The formula that represents r in terms of s is r = -3s + 18.

D) The formula that represents Δr in terms of s is Δr = -3.
E) i. True

ii. False

iii. True

iv. False

Given that a table representing the values of r and s we need to determine the answers asked related to their values,

A) To find the formula that represents Δr in terms of Δs, we can calculate the rate of change of r with respect to s using the given data points:

Δr = r₂ - r₁

Δs = s₂ - s₁

From the given table, we can calculate Δr and Δs for the first two data points:

Δr₁ = 9 - 15 = -6

Δs₁ = 3 - 1 = 2

Now, we can find the constant rate of change of r with respect to s:

Rate of change = Δr₁ / Δs₁

= -6/2

Rate of change = -3.

B) To find the value of r when s = 0, we need to determine the equation of the line that represents the relationship between r and s. We can use the data points given to calculate the slope (rate of change) and then find the equation using the point-slope form.

Using the first and second data points:

slope (m) = Δr/Δs = (-6)/(2) = -3

Now, we can use the point-slope form with the point (1, 15) (as it is the first data point) to find the equation:

y - y₁ = m(x - x₁)

r - 15 = -3(s - 1)

r - 15 = -3s + 3

r = -3s + 18

So, when s = 0, we can substitute s into the equation to find the value of r:

r = -3(0) + 18

r = 18

Therefore, when s = 0, r has a value of 18.

C) To define a formula that represents r in terms of s, we can use the concept of a linear equation. We can find the equation of a line passing through the given data points (1, 15) and (3, 9):

Using the point-slope form of a linear equation:

y - y₁ = m(x - x₁)

m = (r₂ - r₁) / (s₂ - s₁) = (-6) / (2) = -3

Using the point (1, 15):

r - 15 = -3(s - 1)

r - 15 = -3s + 3

r = -3s + 18

Thus, the formula that represents r in terms of s is r = -3s + 18.

D) To define a formula that represents Δr in terms of s, we can differentiate the equation for r in terms of s:

r = -3s + 18

Taking the derivative with respect to s:

d(r)/d(s) = -3

Therefore, the formula that represents Δr in terms of s is Δr = -3.

E) Let's evaluate the given statements:

i. The graph that represents r in terms of s is linear.

True. Since the equation r = -3s + 18 represents a linear relationship between r and s, the graph will be a straight line.

ii. r is proportional to s.

False. The equation r = -3s + 18 does not indicate a direct proportionality between r and s, as the coefficient of s is -3, not a constant.

iii. Δr is proportional to Δs.

True. The rate of change of r with respect to s is constant (-3), indicating that Δr is directly proportional to Δs.

iv. The graph of r in terms of s is a straight line that passes through the origin (0,0).

False. The equation r = -3s + 18 does not include the point (0,0). Therefore, the graph will not pass through the origin.

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Complete question =

The value of r is varying at a constant rate of change with respect to s.

Complete the following table.

s (value of the independent quantity) = 1, 3, 9, 12

r (value of the dependent quantity) = 15, 9, -9, -18

A) Define a formula that represent Δr in terms of Δs.

B) What is the value of r when s=0?

C) Define a formula to represent r in terms of s.

D) Define a formula to represent in terms of s.

E) Determine if the following statements are true or false:

i. The graph that represents a in terms of r is linear.

ii. r is proportional to s.

iii. Δr is proportional to Δs.

iv. Select an answer the graph of r in terms of s is a straight line that passes through the origin (0,0).