y= -x^2-12x+28
A painter can use the formula A=2/3bh to estimate the area of space under the parabola.
Calculate area in square ft under each arch
~ if there are 8 arches to be painted how many square ft must be painted
~ if 1 gallon paints 200 square ft , how many gallons must be needed
~ if a gallon of paint cost $27.00 and sales tax is 8% how much will it cost topaint 8 arches
y = -(x^2 + 12x - 28)
= -(x+14)(x-2)
So the parabola has x-intercepts at -14 and 2
so the vertex has an x-value of -6
(the midpoint between the intercepts)
sub in x=-6
y = -(8)(-8) = 64
so the base of the parabola is 16 and its height is 64
estimate of area = (2/3)(16)(64) = 682 2/3 square feet.
so 8 of those = 5461 1/3
number of gallons needed = 5461 1/3 ÷ 200 = 27.30666
we will need 28 gallons, can't buy .3066 of a gallon
cost = 1.08(28)(27) = $816.48
Thank you so much
To calculate the area under each arch, we need to find the x-values at the points where the parabola intersects the x-axis. These points represent the width of each arch.
Step 1: Find the x-intercepts:
To find the x-intercepts, set y = 0 and solve the quadratic equation -x^2 - 12x + 28 = 0. This can be factored as (-x + 2)(x + 14) = 0. So, either x - 2 = 0 or x + 14 = 0, giving us x = 2 and x = -14 as the x-intercepts.
Step 2: Calculate the width of each arch:
Since we have two x-intercepts, there are three arches in total. The width of each arch is the distance between consecutive x-intercepts. So, the width of the first arch is |2 - (-14)| = 16, the width of the second arch is |2 - 2| = 0, and the width of the third arch is |(-14) - 2| = 16.
Step 3: Calculate the area of each arch:
Using the formula A = (2/3) × base × height, where the base is the width of the arch and the height is the maximum value of the parabola.
For the first arch:
The base = 16 and the maximum value of the parabola can be found by substituting the x-coordinate of the vertex into the equation. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a is the coefficient of x^2 (-1) and b is the coefficient of x (-12). Plugging in the values, we get x = -(-12) / (2 * -1) = -12 / -2 = 6. Substituting x = 6 into the equation, we get y = -(6)^2 - 12(6) + 28 = -36 - 72 + 28 = -80. However, since we are interested in the absolute value, the height of the first arch is |(-80)| = 80. Therefore, the area of the first arch is A = (2/3) × 16 × 80 = 853.33 square ft (rounded to two decimal places).
For the second arch:
The base = 0 (as determined earlier) since the x-intercepts are the same. Therefore, the area of the second arch is A = 0 square ft.
For the third arch:
The base = 16 (as determined earlier) and the height is again |(-80)| = 80. Therefore, the area of the third arch is A = (2/3) × 16 × 80 = 853.33 square ft (rounded to two decimal places).
Now let's answer the specific questions:
1. If there are 8 arches to be painted, how many square feet must be painted?
Since we have three arches and the areas are 853.33 square ft and 853.33 square ft for the first and third arches respectively, and 0 square ft for the second arch, the total area to be painted is (853.33 + 853.33) × 3 = 5119.98 square ft (rounded to two decimal places). For 8 arches, the total area to be painted would be 5119.98 × (8/3) = 13653.28 square ft (rounded to two decimal places).
Therefore, if there are 8 arches to be painted, 13653.28 square ft must be painted.
2. If 1 gallon of paint covers 200 square ft, how many gallons are needed?
Divide the total area to be painted (13653.28 square ft) by the coverage of 1 gallon of paint (200 square ft per gallon).
Number of gallons needed = 13653.28 / 200 = 68.2664 gallons (rounded to four decimal places).
Therefore, approximately 68.27 gallons of paint would be needed.
3. If a gallon of paint costs $27.00 and sales tax is 8%, how much will it cost to paint 8 arches?
Multiply the number of gallons needed (68.27 gallons) by the cost per gallon ($27.00) to calculate the total cost before tax. Then, add the sales tax to get the final cost.
Cost before tax = 68.27 × $27.00 = $1842.89 (rounded to two decimal places).
Sales tax = 8% of $1842.89 = 0.08 × $1842.89 = $147.43 (rounded to two decimal places).
Total cost = Cost before tax + Sales tax = $1842.89 + $147.43 = $1990.32 (rounded to two decimal places).
Therefore, it would cost approximately $1990.32 to paint 8 arches, including sales tax.