Digital SAT Math: Hard Grid-In Practice (Set 2)

The key to this problem is recognizing the perfect square trinomials and utilizing the formal definition of the principal square root: $\sqrt{u^2} = |u|$.

Step 1: Simplify the radicals.

The expression inside the first radical is a perfect square: $x^2 - 6x + 9 = (x - 3)^2$.

The expression inside the second radical is also a perfect square: $x^2 + 4x + 4 = (x + 2)^2$.

Substitute these into the original equation:

$$ \sqrt{(x - 3)^2} + \sqrt{(x + 2)^2} = 7 $$

$$ |x - 3| + |x + 2| = 7 $$

Step 2: Solve the absolute value cases.

Check the intervals defined by the critical points $x = 3$ and $x = -2$.

• Case 1 ($x \ge 3$): Both expressions are non-negative.
  $(x - 3) + (x + 2) = 7 \Rightarrow 2x - 1 = 7 \Rightarrow 2x = 8 \Rightarrow x = 4$. (This is valid since $4 \ge 3$).
• Case 2 ($-2 \le x < 3$): The first is negative, the second is positive.
  $-(x - 3) + (x + 2) = 7 \Rightarrow -x + 3 + x + 2 = 7 \Rightarrow 5 = 7$. (No solution here).
• Case 3 ($x < -2$): Both expressions are negative.
  $-(x - 3) - (x + 2) = 7 \Rightarrow -x + 3 - x - 2 = 7 \Rightarrow -2x + 1 = 7 \Rightarrow -2x = 6 \Rightarrow x = -3$. (This is valid since $-3 < -2$).

The solutions are $-3$ and $4$. The question asks for the positive solution.

Answer: 4

First, find the period of the function by applying the rule to itself.

$$ f(x+6) = f((x+3)+3) = -f(x+3) = -(-f(x)) = f(x) $$

The function strictly repeats its value every 6 units. Therefore, it has a period of 6.

To find $f(20)$, subtract multiples of 6 until you reach the known input of 2:

$$ f(20) = f(20 - 6) = f(14) $$

$$ f(14) = f(14 - 6) = f(8) $$

$$ f(8) = f(8 - 6) = f(2) $$

Since $f(20)$ is equivalent to $f(2)$, and we are given that $f(2) = 4$:

Answer: 4

This is a recursive functional relationship. Every time the $x$-value increases by 2, the output is multiplied by 3, and then 1 is subtracted. We must step forward from our known value.

• Step 1 (Find $f(3)$): Let $x = 1$.
  $f(3) = 3f(1) - 1 \Rightarrow f(3) = 3(2) - 1 = 5$
• Step 2 (Find $f(5)$): Let $x = 3$.
  $f(5) = 3f(3) - 1 \Rightarrow f(5) = 3(5) - 1 = 14$
• Step 3 (Find $f(7)$): Let $x = 5$.
  $f(7) = 3f(5) - 1 \Rightarrow f(7) = 3(14) - 1 = 41$

Answer: 41

The rule $f(x) = f(x+5)$ means the function has a period of 5. To evaluate $f(37)$, we must "walk backward" by intervals of 5 until we find an equivalent $x$-value that falls within the defined base domain of $-2 \le x < 3$.

Divide 37 by 5, which gives 7 with a remainder of 2. Therefore, subtracting 35 ($7 \times 5$) from 37 yields 2.

$$ f(37) = f(37 - 35) = f(2) $$

The value $x = 2$ falls within the restricted domain $[-2, 3)$. Now, evaluate the base polynomial function at $x = 2$:

$$ f(2) = (2)^3 - 2(2) $$

$$ f(2) = 8 - 4 = 4 $$

Answer: 4

To find the centers, complete the square for both circle equations.

Circle 1: $x^2 - 10x + y^2 + 24y = 0$

$$ (x^2 - 10x + 25) + (y^2 + 24y + 144) = 25 + 144 $$

$$ (x - 5)^2 + (y + 12)^2 = 169 $$

Center $C_1$ is $(5, -12)$.

Circle 2: $x^2 + 6x + y^2 - 6y = -2$

$$ (x^2 + 6x + 9) + (y^2 - 6y + 9) = -2 + 9 + 9 $$

$$ (x + 3)^2 + (y - 3)^2 = 16 $$

Center $C_2$ is $(-3, 3)$.

Calculate Distance (d):

Use the distance formula between $(5, -12)$ and $(-3, 3)$:

$$ d = \sqrt{(-3 - 5)^2 + (3 - (-12))^2} = \sqrt{(-8)^2 + (15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17 $$

Answer: 17

When a plane cuts a pyramid parallel to the base, it creates a smaller, similar pyramid at the top. The cross-section is the base of this smaller pyramid.

Step 1: Determine the height of the smaller pyramid.

The total height is 15. The cut is made 9 units from the bottom base. Therefore, the height of the smaller pyramid from the apex is $15 - 9 = 6$.

Step 2: Find the ratio of similarity.

The ratio of the heights of the small pyramid to the large pyramid is $6 / 15$, which simplifies to $2 / 5$.

Step 3: Calculate the area.

Because the pyramids are similar 3D figures, the ratio of their base areas is the square of the ratio of their linear dimensions (heights).

$$ \text{Area Ratio} = \left(\frac{2}{5}\right)^2 = \frac{4}{25} $$

$$ \text{Cross-Section Area} = 100 \times \frac{4}{25} = 4 \times 4 = 16 $$

Answer: 16

According to the Intersecting Chords Theorem, when two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord.

$$ AP \times PB = CP \times PD $$

Let the length of segment $CP$ be $x$. Because the entire length of chord $CD$ is 11, the remaining segment $PD$ must be $11 - x$.

Substitute the known values into the theorem's equation:

$$ 4 \times 6 = x(11 - x) $$

$$ 24 = 11x - x^2 $$

Rearrange this into a standard quadratic equation:

$$ x^2 - 11x + 24 = 0 $$

Factor the quadratic equation:

$$ (x - 3)(x - 8) = 0 $$

The two segments of chord $CD$ have lengths of 3 and 8. The question asks for the length of the longer segment.

Answer: 8

To find the length of a common external tangent, you can construct a right triangle by drawing a line from the center of the smaller circle parallel to the tangent line.

Let the centers of the circles be $A$ and $B$, with radii $r_1 = 3$ and $r_2 = 8$. The distance between centers is $AB = 13$. Let the tangent points be $T_1$ and $T_2$. The length we want is $T_1T_2$, which we will call $L$.

Draw a segment from center $A$ perpendicular to the radius of the larger circle at $B$. This forms a right triangle with:

• Hypotenuse: The distance between the centers ($13$).
• Leg 1: The difference between the two radii ($8 - 3 = 5$).
• Leg 2: The length of the common tangent ($L$).

Use the Pythagorean theorem:

$$ L^2 + 5^2 = 13^2 \Rightarrow L^2 + 25 = 169 \Rightarrow L^2 = 144 \Rightarrow L = 12 $$

Answer: 12

The length of an arc $s$ is a fraction of the circle's total circumference, proportional to the central angle $\theta$ in degrees:

$$ s = 2\pi r \times \left(\frac{\theta}{360}\right) $$

Substitute the given arc length ($10\pi$) and radius ($12$):

$$ 10\pi = 2\pi(12) \times \left(\frac{\theta}{360}\right) $$

$$ 10\pi = 24\pi \times \left(\frac{\theta}{360}\right) $$

Divide both sides by $\pi$ to simplify:

$$ 10 = 24 \times \left(\frac{\theta}{360}\right) $$

$$ \frac{10}{24} = \frac{\theta}{360} $$

Simplify the fraction on the left to $5 / 12$, then solve for $\theta$:

$$ \frac{5}{12} = \frac{\theta}{360} \Rightarrow \theta = 360 \times \frac{5}{12} = 30 \times 5 = 150 $$

Answer: 150

This is a combinations probability problem. First, find the total number of ways to draw 3 balls from the entire box.

The total number of balls is $5 + 3 + 2 = 10$.

Total possible outcomes = $\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$.

Now, calculate the number of successful outcomes (drawing exactly 1 red, 1 blue, and 1 green). Since the draws are independent combinations, multiply the individual possibilities:

Ways to choose 1 Red = $\binom{5}{1} = 5$

Ways to choose 1 Blue = $\binom{3}{1} = 3$

Ways to choose 1 Green = $\binom{2}{1} = 2$

Total successful outcomes = $5 \times 3 \times 2 = 30$.

The probability is the number of successful outcomes over the total outcomes:

$$ P = \frac{30}{120} = \frac{1}{4} $$

Answer: 1/4 (or .25)

When dealing with averages, it is almost always best to convert them back into sums.

Step 1: Find the sum of the original 10 numbers.

$$ \text{Sum}_{10} = 10 \times 45 = 450 $$

Step 2: Find the sum of the remaining 8 numbers.

$$ \text{Sum}_8 = 8 \times 48 = 384 $$

Step 3: Determine the sum of the removed numbers.

The difference between the two sums must be exactly equal to the values that were removed ($x$ and $y$):

$$ x + y = 450 - 384 = 66 $$

Step 4: Find the average of $x$ and $y$.

The average of these two numbers is their sum divided by 2:

$$ \text{Average} = \frac{66}{2} = 33 $$

Answer: 33

This situation can be modeled by the exponential growth formula:

$$ P(t) = P_0 \times 2^{t/k} $$

Where $P_0$ is the initial population, $k$ is the doubling time, and $t$ is the total time elapsed.

Substitute the known values into the model:

$$ 32000 = 500 \times 2^{t/4} $$

Isolate the exponential term by dividing both sides by 500:

$$ \frac{32000}{500} = 2^{t/4} $$

$$ 64 = 2^{t/4} $$

Recognize that 64 is a power of 2: $64 = 2^6$. Substitute this into the equation:

$$ 2^6 = 2^{t/4} $$

Because the bases are identical, you can set the exponents equal to each other:

$$ 6 = \frac{t}{4} \Rightarrow t = 24 $$

Answer: 24