Question

A manufacturer of rectangular tarpaulins uses the following combined function to calculate the total area of different sizes of tarpaulins: h(x) = x^2 + 10x + 21. What two functions could represent length and width for given values of x based on the combined function?

a) f(x) = x + 7 and g(x) = x + 3
b) f(x) = x - 7 and g(x) = x - 1
c) f(x) = x^2 + 7 and g(x) = x^3 + 3
d) f(x) = x - 7 and g(x) = x - 3

Answer 1

The functions that represent length and width whose product equals the combined function h(x) = x^2 + 10x + 21 are f(x) = x + 7 and g(x) = x + 3.

Step-by-step explanation:

The student has asked to find two functions that could represent length and width such that their product is the combined function given by h(x) = x^2 + 10x + 21. To find such functions, we need to factor the quadratic equation, looking for two binomials that multiply to give us h(x). By factoring, we can rewrite h(x) as (x + 7)(x + 3), which means the two functions representing length and width can be:

• f(x) = x + 7
• g(x) = x + 3

This corresponds to option a). These functions satisfy the conditions as their product is indeed the combined function h(x).