Question

(09.01) A biologist created the following graph to show the relationship between the temperature of water (x), in degrees Celsius, and the number of insect larvae (y) in the water: graph of y equals minus 2 times the square of x plus 20 times x plus 400 What do the x-intercepts of the graph represent?

Answer 1

Answer:

x-intercept:

A line that crosses the graph at x-axis.

i.e substitute y = 0 and solve for x.

As per the statement:

A biologist created the following graph to show the relationship between the temperature of water (x), in degrees Celsius, and the number of insect larvae (y) in the water

Given the graph:

$y = -2x^2+20x+400$

Substitute y = 0  we have;

$-2x^2+20x+400 = 0$

⇒$-2(x^2-10x-200) = 0$

⇒$x^2-10x-200 =0$

⇒$x^2-20x+10x-200 =0$

⇒$x(x-20)+10(x-20)=0$

⇒$(x-20)(x+10)=0$

By zero product property we have;

x-20 =0 and x+10 = 0

⇒x = 20 and x = -10

Therefore,  the x-intercept represents the water has no larvae at −10 degrees Celsius and 20 degrees Celsius.

Answer 2

Answer:

The temperature of the water in degrees Celsius, after which there is no insect.

Step-by-step explanation:

Given function,

$y = -2x^2 + 20x + 400$

Where,

x = the temperature of water,

y =  the number of insect larvae (y) in the water when  the temperature of the water is x,

If x = 0 minutes,

$y = -2(0)^2 + 20(0) + 400 = 0 + 0 + 400 = 400$

Thus, when the temperature of the water is 0°Celsius there is no insect.

But, 400 is the x-intercept of the function $y = -2x^2 + 20x + 400$,

Hence, x-intercept represents temperature of the water  (x), in degrees Celsius, after which there is no insect.