Final answer:
To find the value of n for the right-angled triangle with an area of 30 cm², where the width is n + 5 cm and the height is 7 cm less than the width, solve the quadratic equation derived from the area formula. After simplification, n is found to be 7 cm.
Step-by-step explanation:
The student is asking to solve for the variable n in a right-angled triangle where the width is n + 5 cm and the height is 7 cm less than the width. With a given area of 30 cm², we can use the formula for the area of a triangle:
Area = 1/2 × base × height
Given that the base width) is n + 5 cm and the height is (n + 5) - 7 cm, we can plug in these expressions and the given area into the formula:
30 = 1/2 × (n + 5) × ((n + 5) - 7)
By simplifying and solving the quadratic equation, we can find the value of n. Here is how:
30 = 1/2 × (n + 5) × (n - 2)
60 = (n + 5)(n - 2)
60 = n² + 3n - 10
n² + 3n - 70 = 0
The quadratic equation can be solved through factoring or using the quadratic formula. Factoring the expression, we get:
(n + 10)(n - 7) = 0
Therefore, n can be either -10 or 7. Since we are dealing with lengths, which must be positive, the value of n is 7 cm.