ACT Math Practice

The square root of 63 is 7.937. Both 6 and 7 are less than that, therefore 8 is the smallest integer greater than 7.937.

If the motorcycle is losing 5 ounces every minute, it is losing 300 ounces per hour since there are 60 minutes in an hour. (5 × 60 = 300). Since the motorcycle can hold 600 fluid ounces total it will take 2 hours for the motorcycle to run out of oil (600 ÷ 300 = 2). If the motorcycle is traveling at 45 mph, in two hours it will have traveled 90 miles total. (45 𝑚𝑖𝑙𝑒𝑠 × 2 ℎ𝑜𝑢𝑟𝑠 = 90 𝑚𝑖𝑙𝑒𝑠 𝑡𝑜𝑡𝑎𝑙)

Since Josie started with 65 wreaths, and had 57 left after day one, she sold 8 wreaths on day one (65 − 57 = 8). After day two, she sold 8 wreaths on day two (57 − 49 = 8). Since Josie had 41 wreaths left after day three, she sold 8 wreaths on day three (49 − 41 = 8). Continuing this pattern, of selling 8 wreaths per day, at the end of day four Josie would have 41-8= 33 wreaths. After day five, 33-8= 25 wreaths. Finally, at the end of day six, Josie would have 25-8= 17 wreaths. Thus, Josie would have 17 wreaths left after day six.

At the start of the run 𝑡 = 0 since the skier has not started skiing. If we substitute 0 in to find the skier’s elevation we have $$E=\frac{36,000}{t+200}=\frac{36,000}{0+200}=\frac{36,000}{200}=180$$ feet.

To solve for 𝑥 we need to distribute to eliminate parentheses. After distributing, we have 3𝑥 − 24 + 𝑥 = 24. Next, we can combine the like terms on the left side of the equation. The 3𝑥 and 𝑥 can be combined to have 4𝑥 − 24 = 24. We use inverse operations to eliminate the − 24, so we add 24 to both sides which gives us 4𝑥 = 48. We use inverse operations again and divide both sides by 4 to get x by itself. This gives us 𝑥 = 12.

If an 18 mph wind can exert a force of 45 kg on a 5 sq.ft surface, then we know our values are 𝑆 = 5 and 𝑣 = 18. We can create an equation with those values and set it equal to our force of 45. Our equation is $$kSv^2$$ , and 2 = $$k\left(5\right)\left(18\right)^2=45$$ we can now solve this for our unknown constant, 𝑘. Solving this, 𝑘(5)(324) = 45, gives us the value $$k=\frac{1}{36}$$ . Now that we have the value of , we can find the maximum 1 36 𝑘 force that a 36 mph wind can apply on a 2 sq.ft. surface by substituting these values into the expression $$kSv^2$$ . $$\left(\frac{1}{36}\right)\left(2\right)\left(36\right)^2=\left(\frac{1}{36}\right)\left(2\right)\left(1,296\right)=\left(\frac{1}{36}\right)\left(2,592\right)=72$$

We must use the formula for finding the midpoint of a line segment, $$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$ . We substitute the points $$\left(x_1,\ y_1\right),\ \left(x_2,\ y_2\right)$$ into the formula to calculate the midpoint. In this question, $$x_1=6,\ y_1=2,\ x_2=2,$$ and $$y_2=-2$$ . When we substitute our points into our formula, we have $$\left(\frac{6+2}{2},\frac{2+\left(-2\right)}{2}\right)$$ . This simplifies to (4,0).

Using the properties of parallelograms, we know that ∠𝐹 is equal to ∠𝐷, so ∠𝐷 = 50°. We must then know that the sum of the angles of a triangle always equals 180°. This means ∠𝐶 + ∠𝐷 + ∠𝐸 = 180°. We will then substitute the measure of our angles into this equation to get 54° + 50° + ∠𝐸 = 180°. At this point, we will solve our equation to find ∠𝐸 = 76°.

There are two options to solve this problem. The first option is to use the Pythagorean Theorem, which is $$a^2+b^2=c^2$$ . Using the measures of the legs of our triangle, we would substitute these values in for 𝑎 and 𝑏 to solve for our hypotenuse, 𝑐. Our equation, after substituting our values in, becomes $$\left(16\right)^2+\left(12\right)^2=c^2$$ . We can simplify to $$256+144=c^2$$ which becomes $$400=c^2$$ . Taking the square root of both sides to get 𝑐 by itself, gives us 𝑐 = 20. Our second option is to use Pythagorean Triples, and recognize that this is an example of a 3-4-5 triangle. The GCF of the legs of the triangle is 4. So, 5 × 4 = 20 gives us the measure of the hypotenuse of our triangle. Both methods give us a hypotenuse of 20. To find the midpoint of our hypotenuse, we would find half of the measure, so 20 ÷ 2 = 10.

The minimum or maximum value of a quadratic equation is found at the vertex. Using the general form of a parabola, $$ax^2+bx+c$$ , we can find the vertex using the formula $$x=\frac{-b}{2a}$$ . For our equation, $$p\left(d\right)=1,800d-5d^2$$ , we can substitute 𝑎 =− 5, 𝑏 = 1, 800, and 𝑐 = 0 into our equation. Substituting our values into $$x=\frac{-b}{2a}$$ gives us the value of d for which the monthly profit will be the highest is $$x=\frac{-b}{2a}=\frac{-1800}{2\left(-5\right)}=\frac{-1800}{-10}=180$$