The area of a triangle varies jointly as its base and altitude. By what percent will the area change if the base is increased by 15% and the altitude decreased by 25%?

A1 = 0.5bh,

A2 = 1.15*0.5b*1.25h = 0.71875bh.

A2 / A1 = 0.71875bh / 0.5bh = 1.4375 =
143.75%.

% Change = 143.75 - 100 = 43.75.

To find the percent change in the area of a triangle when the base is increased by 15% and the altitude is decreased by 25%, we first need to understand the relationship between the base, altitude, and area of a triangle.

The formula to find the area of a triangle is:

Area = (1/2) * base * altitude

In this case, the area varies jointly as the base and altitude, which means that it is directly proportional to the base and altitude. Mathematically, this can be expressed as:

Area = k * base * altitude

Where k is the constant of proportionality.

Now, let's calculate the initial area of the triangle. Let's assume that the initial base is represented by b and the initial altitude is represented by a. So, the initial area (A1) can be calculated as:

A1 = k * b * a

Now, let's calculate the final area of the triangle when the base is increased by 15% and the altitude is decreased by 25%. The new base (b2) can be calculated by multiplying the initial base (b) by 1 + 15% (or 0.15), and the new altitude (a2) can be calculated by multiplying the initial altitude (a) by 1 - 25% (or 0.75):

b2 = b * (1 + 0.15)
a2 = a * (1 - 0.25)

The new area (A2) can be calculated using the same formula as before:

A2 = k * b2 * a2

Now, we can calculate the percent change in the area by finding the ratio of the difference between A2 and A1 to A1, and then multiplying by 100:

Percent change = (A2 - A1) / A1 * 100

Substituting the calculated values, the formula becomes:

Percent change = (k * b2 * a2 - k * b * a) / (k * b * a) * 100

Since k is the constant of proportionality and cancels out, we can simplify the formula to:

Percent change = (b2 * a2 - b * a) / (b * a) * 100

Now, we can substitute the calculated values for b2, a2, b, and a to find the percent change in the area.