Describing Repeated Roots on a Graph

You are designing a rectangular poster to contain 7575 in2 of printing with a 33-in margin at the top and bottom and a 11-in m

erma4kov [3.2K]

Answer:

The dimensions of the rectangular poster is 15 in by 5 in.

Step-by-step explanation:

Given that, the area of the rectangular poster is 75 in².

Let the length of the rectangular poster be x and the width of the rectangular poster be y.

The area of the poster = xy in².

$\therefore xy=75$

$\Rightarrow y=\frac{75}{x}$ ....(1)

1 in margin at each sides and 3 in margin at top and bottom.

Then the length of printing space is= (x-2.3) in

=(x-6) in

The width of printing space is = (y-2.1) in

=(y-2) in

The area of the printing space is A =(x-6)(y-2) in²

∴ A =(x-6)(y-2)

Putting the value of y

$\Rightarrow A =(x-6)(\frac{75}{x}-2)$

$\Rightarrow A = 87-\frac{450}{x}-2x$

Differentiating with respect to x

$A '= \frac{450}{x^2}-2$

Again differentiating with respect to x

$A''=-\frac{900}{x^3}$

To find the minimum area of printing space, we set A' = 0

$\therefore \frac{450}{x^2}-2=0$

$\Rightarrow 450 =2x^2$

$\Rightarrow x^2=225$

$\Rightarrow x=\pm 15$

Now putting x=±15 in A''

$A''|_{x=15}=-\frac{900}{15^3}$

$A''|_{x=-15}=-\frac{900}{(-15)^3}=\frac{900}{(15)^3}>0$

Since at x=15 , A"<0 Therefore at x=15 , the area will be minimize.

From (1) we get

$y=\frac{75}{x}$

Putting the value of x

$y=\frac{75}{15}$

=5 in

The dimensions of the rectangular poster is 15 in by 5 in.

4 0

3 years ago

Help me please I’ll give brainlest if it right

beks73 [17]

Answer:

it' should be 0.15 have a great day

8 0

3 years ago

Read 2 more answers

The varsity basketball team started selling T-shirts online in 1994. The number of T-shirts sold online, s, is modeled by the gr

-Dominant- [34]

(PART-A): s-intercept in this case represents the y-intercept of the graph, whereas the t-intercept in this case represents the x-intercept of the graph.

Always remember that y-intercept represents the value of y where the curve (or line) crosses the y-axis, and x-intercept represents the value of x where the curve (or line) crosses the x-axis.

In the graph attached with the question, at point (0,6) the curve crosses the y-axis. Therefore, the y-intercept (s-intercept) in this case is 6. It means that in year 2000 (t=0), 6 hundred shirts were sold.

However, there is no point on the graph where the curve crosses the x-axis, meaning there is no x-intercept (t-intercept). It means that there is not a single year when the number of shirts sold is 0.

(PART B): As you can see in the graph (attached with the question) that the f(t) is increasing (exponentially) with the increase in t. Therefore, we can safely say that as t increases without bound, the f(t) increases (also). In this context, it means that the sale of shirts increases as the years go by. Hence, the correct blank is "increases."

(PART C): The average rate of change is actually the slope. To find the slope, we can use the following formula:

slope = $\frac{y_2 - y_1}{x_2 - x_1}$ --- (X)

Given points: (5, 12.5) and (7, 22). Plug the values in equation (X),

slope = $\frac{22-12.5}{7-5} = 4.75$

Hence, the average rate of change for the function between t=5 and t=7 is 4.75 (answer).

(PART D): This part is bit tricky. Therefore, read the explanation carefully!

You can see in the graph that number of Shirts sold is in hundreds. It means that 1 unit (of y-axis in a graph) represents 100 shirts. Therefore, as in the question it is mentioned that there are 1000 T-shirts sold, it will become 10 units (since 10*100 = 1000). So, the function g(t) will become the following:

$g(t) = f(t) + 10$ --- (Y)

Why did I add f(t)? Because in the question, the word "and" is underlined. It means that g(t) represents not only 1000 T-shirts (10*100 = 1000) sold at the basketball games each year, but it also has the number of T-shirts sold ONLINE, which is f(t).

Now insert f(t) in (Y) and solve:

$g(t) = f(t) + 10\\g(t) = (1.5)^t + 5+ 10\\g(t) = (1.5)^t + 15$