Question

an artist is working on a stained - glass design for a triangular window in a cathedral. the area of the window is restricted to 240 square feet, and the artist wants the base to be 2 feet less than twice the height. determine the base and height of the triangular window.
a. ( h = 16 ) feet, ( b = 30 ) feet
b. ( h = 30 ) feet, ( b = 16 ) feet
c. ( h = 12 ) feet, ( b = 20 ) feet
d. ( h = 15 ) feet, ( b = 28 ) feet
two lawn mowers, mower a and mower b, are cutting a large field. mower a takes 3 hours less than mower b. when both mowers are used simultaneously, they can finish the job in 2 hours. find the time it takes for each mower to complete the job alone.

Explanation:

First Sub - Question (Triangular Window)

Step 1: Recall the area formula for a triangle

The area formula for a triangle is $A=\frac{1}{2}bh$, where $A$ is the area, $b$ is the base and $h$ is the height. We know that $A = 240$ square feet and $b=2h - 2$.

Step 2: Substitute the values into the area formula

Substitute $A = 240$ and $b = 2h-2$ into $A=\frac{1}{2}bh$:

$$ 240=\frac{1}{2}(2h - 2)h $$

Multiply both sides by 2 to get rid of the fraction:

$$ 480=(2h - 2)h $$

Expand the right - hand side:

$$ 480 = 2h^{2}-2h $$

Divide both sides by 2:

$$ 240=h^{2}-h $$

Rearrange the equation to form a quadratic equation:

$$ h^{2}-h - 240=0 $$

Step 3: Solve the quadratic equation

We can solve the quadratic equation $h^{2}-h - 240 = 0$ by factoring. We need two numbers that multiply to $- 240$ and add up to $-1$. The numbers are $-16$ and $15$.

$$ h^{2}-16h + 15h-240=0 $$

$$ h(h - 16)+15(h - 16)=0 $$

$$ (h - 16)(h + 15)=0 $$

So, $h=16$ or $h=-15$. Since height cannot be negative, $h = 16$ feet.

Step 4: Find the base

Now that we know $h = 16$ feet, we can find the base using $b = 2h-2$.

$$ b=2\times16 - 2=32 - 2=30 $$

So the height $h = 16$ feet and the base $b = 30$ feet.

Second Sub - Question (Lawn Mowers)

Step 1: Define variables

Let the time taken by Mower B to complete the job alone be $x$ hours. Then the time taken by Mower A to complete the job alone is $(x - 3)$ hours.
The rate of work of Mower B is $\frac{1}{x}$ (jobs per hour) and the rate of work of Mower A is $\frac{1}{x - 3}$ (jobs per hour). When they work together, their combined rate is $\frac{1}{2}$ jobs per hour (since they complete the job in 2 hours when working together).

Step 2: Write the rate equation

The combined rate of work of Mower A and Mower B is the sum of their individual rates. So,

$$ \frac{1}{x}+\frac{1}{x - 3}=\frac{1}{2} $$

Step 3: Solve the equation

First, find a common denominator, which is $2x(x - 3)$. Multiply each term by $2x(x - 3)$:

$$ 2(x - 3)+2x=x(x - 3) $$

Expand the left - hand side:

$$ 2x-6 + 2x=x^{2}-3x $$

Combine like terms on the left - hand side:

$$ 4x-6=x^{2}-3x $$

Rearrange the equation to form a quadratic equation:

$$ x^{2}-7x + 6=0 $$

Step 2: Solve the quadratic equation

Factor the quadratic equation:

$$ x^{2}-6x - x + 6=0 $$

$$ x(x - 6)-1(x - 6)=0 $$

$$ (x - 6)(x - 1)=0 $$

So, $x = 6$ or $x = 1$. But if $x = 1$, then the time taken by Mower A would be $x-3=1 - 3=-2$ hours, which is not possible. So $x = 6$ hours.

Step 3: Find the time for each mower

The time taken by Mower B is $x = 6$ hours.
The time taken by Mower A is $x - 3=6 - 3 = 3$ hours.

Answer:

• For the triangular window: A. $h = 16$ feet, $b = 30$ feet
• For the lawn mowers: Mower A takes 3 hours and Mower B takes 6 hours.